Consequences and examples

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).

The coefficient \(q+1\) in the simplest spherical Hecke product counts intermediate lattices. In the sheaf calculation it comes from the two cohomology groups of a projective line. Passing between these descriptions requires retaining both the perverse shift and the Frobenius action on the top cohomology group. This lesson calculates the point counts and proper-fibre traces explicitly, proves the corresponding IC identifications from actual supported orientations, proves the scalar Frobenius operator on every Satake weight, the general Satake character and algebra formulas, normalized IC trace convolution, split Tate Frobenius on all IC stalks and costalks, the full orbit-indicator Satake formula, and the graded IC q-analogue. The Hall–Littlewood transition is proved for the whole dual root datum and lattice; the Whittaker and factorization statements are then formulated with their hypotheses.

1. Frobenius, shifts and half twists

Let \(G\) be split and connected reductive over \(\mathbf F_q\), with a split maximal torus \(T\subset B\). Write \(N\) for the unipotent radical of \(B\), \(L=X_*(T)\), and

\[ d_\lambda=\langle2\rho,\lambda\rangle, \qquad h_\nu=\langle2\rho,\nu\rangle. \]

Here \(\lambda\) is dominant, whereas \(\nu\) can be any coweight. Fix \(\ell\ne\operatorname{char}\mathbf F_q\) and a finite extension \(E/\mathbf Q_\ell\) containing a square root \(a\) of \(q\). Choose an embedding of the algebraic numbers involved into \(\mathbf C\) for which \(a=\sqrt q>0\). All Frobenius operators below are geometric Frobenius. Thus \(E(1)\) has eigenvalue \(q^{-1}\).

Use the rank-one Weil object with eigenvalue \(a^{-1}\) to define \(E(1/2)\). The notation \((d/2)\) means its \(d\)-th tensor power, including negative powers. This choice concerns the Weil structure; it does not change the underlying geometric sheaf. Define

\[ \mathrm{IC}^{\mathrm{raw}}_\lambda =j_{\lambda!*}E[d_\lambda], \qquad \mathcal I_\lambda =\mathrm{IC}^{\mathrm{raw}}_\lambda(d_\lambda/2). \tag{1.1} \]

The first object has the constant Weil structure on its open orbit. The second is the usual weight-zero normalization when the IC purity theorem applies. The phrase “weight zero” means that its degree-\(i\) stalk cohomology has weights at most \(i\), and the corresponding dual inequalities hold; it does not mean that Frobenius is the identity in every degree.

For a Weil complex \(K\) and \(x\in Y(\mathbf F_q)\), put

\[ t_K(x)=\sum_i(-1)^i \operatorname{Tr}(F_x;H^i(K_{\bar x})). \tag{1.2} \]

We use bounded complexes with finite-dimensional stalks. The shift convention is \(H^i(K[r])=H^{i+r}(K)\).

Lemma 1.1. Trace functions are additive on distinguished triangles and multiply on tensor products. Moreover

\[ t_{K[r](s/2)}=(-1)^r a^{-s}t_K. \tag{1.3} \]

Proof. For an invariant subspace \(U\subset V\), choose a basis of \(U\) and extend it to one of \(V\). The Frobenius matrix is block triangular, so its trace is the sum of the traces on \(U\) and \(V/U\). Apply this to kernels and images in the finite long exact cohomology sequence of a triangle. Adjacent image terms have opposite signs and cancel. This proves additivity. Over a field, the cohomology of the tensor product of two bounded complexes is the direct sum of their cohomology tensor products: split each complex into its cohomology and two-term contractible summands. The trace on \(V\otimes W\) is \(\operatorname{Tr}(F_V)\operatorname{Tr}(F_W)\), as a product basis shows. Multiplying the alternating sums proves the tensor assertion. A shift changes the sign by \((-1)^r\), and the Weil twist multiplies every eigenvalue by \(a^{-s}\). \(\square\)

These calculations agree with Perverse sheaves, D-modules and the function-sheaf dictionary, §§1–2. Its general pushforward formula has an additional trace-theorem premise. In the explicit convolution below we compute every fibre's cohomology, so no all-dimensional trace formula is needed for that product.

On \(\mathrm{Gr}^{\lambda}(\mathbf F_q)\), formula (1.1) gives

\[ t_{\mathcal I_\lambda}=(-1)^{d_\lambda}a^{-d_\lambda}. \tag{1.4} \]

It vanishes outside \(\mathrm{Gr}^{\leq\lambda}\). Consequently, whenever these IC objects are defined with finite-dimensional stalks, their trace functions are a basis of the finitely supported spherical functions. Indeed their expansion in orbit indicators is triangular, with nonzero diagonal entries (1.4). Every interval below a dominant coweight is finite, by Orbits and Schubert varieties, §3; inversion of a finite triangular matrix on each such interval proves both spanning and independence.

2. The quadratic cone in étale coefficients

The small Schubert surfaces for \(SL_2\) and \(GL_2\) have the same singularity. Their two-stratum geometry, coordinates and resolution are proved in Orbits and Schubert varieties, §§4–5. We first prove the two actual supported comparisons and then deduce the IC calculation in étale coefficients, including the characteristic-two quotient geometry.

The exact finite pushforward and geometric stalk formula are proved in Pushforward, pullback and finite morphisms, Theorem 4.1. Its Theorem 6.2 proves topological invariance. For the two supported comparisons, we use the actual smooth and section adjunctions proved in Semi-infinite orbits and weight functors, Theorems P.2.4.1 and P.3.4.1 and Exactness of weight functors, Appendix A.

Theorem 2.0 (actual supported orientations). Let \(k\) be a field with \(\ell\) invertible. Write \(\Lambda_m=\mathbf Z/\ell^m\) and let \(E/\mathbf Q_\ell\) be finite. The following are the actual counit-normalized supported comparisons at \(\Lambda_m\), at completed integral coefficients, and at \(E\). For the plane origin, \[ i_0^!E_{\mathbf A^2}=E(-2)[-4]. \tag{2.0} \] Its orientation is the composite of the two coordinate Kummer orientations. For the zero section \(i_D:D\hookrightarrow\mathbb V(\mathcal O_{\mathbf P^1}(-2))\), \[ i_D^!E=E_D(-1)[-2]. \] Restriction of its supported counit is cup product by \(c_1(\mathcal O_D(-2))\), with the negative sign. The maps retain coefficient reductions, rational realization, supported triangles, closed restriction and, for split models over \(\mathbf F_q\), their geometric Frobenius action. Over the geometric ground field, their algebraic scalar extensions are those of Appendix D in Lesson 11.

Proof. Factor the plane-origin immersion through the coordinate line: \[ \operatorname{Spec}k\xrightarrow{s_0}\mathbf A^1_v \xrightarrow{s_u}\mathbf A^2_{u,v}, \qquad s_u(v)=(0,v). \] Here \(s_u\) is the zero section of the trivial line projection \(q_u:\mathbf A^2\to\mathbf A^1_v\), and \(s_0\) is the zero section of \(q_v:\mathbf A^1_v\to\operatorname{Spec}k\). Lesson 6, Lemma A.1.1.1 identifies each actual section right adjoint on the constant coefficient with the inverse Tate line shifted by \([-2]\). Its normalization is the identity counit for \(q_us_u=1\), respectively \(q_vs_0=1\); on the trivial chart it is the positive coordinate Kummer class, not an unspecified nonzero orientation.

Actual exceptional transitivity and the invertible-twist comparison of Lesson 5, Theorems P.2.4.1 and P.3.4.1 consequently give \[ i_0^!E=s_0^!s_u^!E =s_0^!E_{\mathbf A^1_v}(-1)[-2] =E(-2)[-4]. \] The transitivity counit is the first section counit followed by the second. Thus its class is the composite of the two specified coordinate orientations. Each has degree two, so exchanging their order introduces the sign \((-1)^{2\cdot2}=1\). Their affine traces are each \(+1\); the composite trace is \(+1\) by counit transitivity. This verifies the coordinate normalization as well as the isomorphism of coefficient objects.

The exceptional curve in the cone resolution of Lesson 4, §§4–5 is exactly the zero section of \(L=\mathcal O_{\mathbf P^1}(-2)\). Its total space is smooth over every \(k\), including characteristic two; it is locally \(\mathbf A^1\times\mathbf A^1\), with the line transition functions. Apply the same section theorem to its line projection. It gives \(i_D^!E=E_D(-1)[-2]\) with its actual section counit. Lesson 6, Lemma A.1.2.1 identifies the restriction of that counit with the global Kummer class \(c_1(L)\). Its proof uses the relative divisor line \(\mathcal O_T(D)\) with its trivialization off \(D\), so includes the overlap units before forgetting support; equality on trivial line charts alone is not substituted for the global class. Since line-square and inverse transitions double and negate their Kummer cocycles, \[ c_1(\mathcal O(-2))=-2c_1(\mathcal O(1)). \] The positive projective-line trace sends \(c_1(\mathcal O(1))\) to \(1\), hence sends the displayed Euler class to \(-2\).

Lemma A.1.1.1 and Lemma A.1.2.1 are stated at every finite \(\Lambda_m\) and retain coefficient transitions. Lesson 6, Theorem A.2.2.1 constructs the actual completed supported triangle and its rationalization, with closed restriction and the same Euler map. The plane is the composite of two of those compatible section comparisons. Consequently both plane and exceptional-line comparisons retain their actual derived reductions and their completed/rational counits. This uses the already proved complete coefficient comparisons; it does not infer equality of maps solely from equality modulo \(\ell\).

For a split model over \(\mathbf F_q\), every coordinate section, line transition, relative divisor, Kummer connecting map and counit above is defined over \(\mathbf F_q\). The particular base comparisons of Lemma A.1.1.1 identify the section counit with its pullback after a field automorphism. The relative Kummer construction is natural under those same pullbacks. Thus passage to \(\overline{\mathbf F}_q\) carries the entire diagram with its descent action, and every displayed isomorphism intertwines geometric Frobenius. In particular the plane supported line \(E(-2)\) has eigenvalue \(q^2\), and the divisor line \(E(-1)\) has eigenvalue \(q\). These eigenvalues are the retained inverse Kummer lines, not a purity assertion about an unknown IC.

Finally, over the geometric ground field, Lesson 11, Theorem D.3.1.1 identifies scalar extension of the supported functors, counits and Tate lines with their actual algebraic coefficient versions. This proves the actual supported orientations and all their specified compatibilities. \(\square\)

Theorem 2.0 composes the actual section adjunctions; each coordinate orientation has trace \(+1\). The two degree-two classes commute with sign \(+1\), and their inverse Tate lines give \(E(-2)[-4]\). Editable SVG source.

Let

\[ C=\operatorname{Spec}k[x,y,z]/(xy-z^2), \qquad \pi:\mathbf A^2\longrightarrow C, \quad (u,v)\longmapsto(u^2,v^2,uv), \tag{2.1} \]

where \(k\) is algebraically closed of any characteristic different from \(\ell\). The map is finite: its source algebra is generated by the integral elements \(u,v\), satisfying the displayed square equations. The complement of the vertex in \(C\) is smooth. If \(\operatorname{char}k\ne2\), this follows from the derivatives \(y,x,-2z\). If \(\operatorname{char}k=2\), the only point with \(x=y=0\) is the vertex, because \(z^2=xy\).

Proposition 2.1 (the cone IC). For rational étale coefficients,

\[ \mathrm{IC}_C=E_C[2]. \tag{2.2} \]

The equality also respects the constant Weil structure over \(\mathbf F_q\).

Proof. First suppose the characteristic is odd. The involution \(\sigma(u,v)=(-u,-v)\) has quotient (2.1). To see the invariant algebra directly, a monomial \(u^iv^j\) is invariant exactly when \(i+j\) is even. If both exponents are even it is a monomial in \(x,y\); if both are odd, factor out \(z\). The sole relation is \(z^2=xy\), as the resulting even and odd monomial lists are linearly independent in \(k[u,v]\).

The natural map

\[ E_C\longrightarrow(\pi_*E_{\mathbf A^2})^{\sigma} \tag{2.3} \]

is an isomorphism on geometric stalks. Off the vertex, the two points of a fibre are exchanged; at the vertex, its reduced fibre is a single point. In both cases the invariant locally constant functions are one copy of \(E\), and (2.3) sends a scalar to that constant function. Since \(2\) is invertible in \(E\), taking invariants is the direct summand given by \((1+\sigma)/2\).

Let \(i\) be the vertex and \(i'\) its closed inverse image. The latter is the thickened origin \(\operatorname{Spec}k[u,v]/(u^2,v^2,uv)\). Its étale topos is that of its reduced point, by topological invariance, and sections with support depend only on that closed complement. Thus its exceptional restriction can be computed at the reduced origin. Finite pushforward is exact. Its costalk comparison is

\[ i^!\pi_*E\simeq\pi_{0*}i'^!E. \tag{2.4} \]

Here \(\pi_0\) is the finite map of the fibres at the origin, whose étale site is that of a point. For completeness, test (2.4) by an arbitrary complex \(M\) on that point. Closed extension and inverse image satisfy \(\pi^*i_*M=i'_*\pi_0^*M\), checked on geometric stalks. The two adjunctions then give

\[ \operatorname{Hom}(M,i^!\pi_*E) =\operatorname{Hom}(\pi^*i_*M,E) =\operatorname{Hom}(M,\pi_{0*}i'^!E). \]

Yoneda proves the comparison, including its equivariance.

Theorem 2.0 gives \(i'^!E=E(-2)[-4]\). The involution acts as the identity on this one-dimensional orientation space. One can check this without a degree convention: the origin's orientation is the composite of the two supported Kummer orientations of the coordinate divisors. Replacing a coordinate \(u\) by \(-u\) changes its Kummer class by that of the constant \(-1\); the latter has an \(\ell^n\)-th root in \(k\) and hence zero Kummer class. Each orientation, and therefore their composite, is fixed. Taking the summand in (2.3) and (2.4) gives

\[ i^*E_C=E, \qquad i^!E_C=E(-2)[-4]. \tag{2.5} \]

In characteristic two, (2.1) is a universal homeomorphism. For every algebraically closed extension field a point of \(C\) has a unique lift: take the unique square roots of \(x,y\); their product is the unique square root of \(z^2\), hence equals \(z\). The map is finite and universally bijective, so it is integral, radicial and surjective. Topological invariance makes \(\pi^*\) and \(\pi_*\) inverse exact equivalences. Thus \(E_C=\pi_*E\), and (2.4) again gives (2.5).

Now put \(K=E_C[2]\). On the smooth open surface it is perverse. At the vertex its stalk has only degree \(-2\), and its costalk has only degree \(2\). These satisfy the perverse conditions, with strict inequalities at the boundary. They also exclude any perverse quotient or subobject supported at the vertex, respectively. The intermediate-extension characterization proved in Intermediate extensions and intersection complexes, §1 therefore identifies \(K\) with \(j_{!*}E[2]\). All maps and the averaging projector are defined over \(\mathbf F_q\), and the isomorphism is the identity on the open stratum. Uniqueness of intermediate extension makes it Frobenius compatible. \(\square\)

This proof uses characteristic-zero coefficients even when the ground field has characteristic two. With coefficients of characteristic two, the averaging projector in the odd-characteristic quotient calculation is unavailable; the assertion is not an integral IC calculation.

3. Point counts and the two smallest IC functions

The point counts and all three small IC functions below are proved. The quadratic-cone IC function uses Proposition 2.1; the minuscule projective-line and central-point functions use the curve and point calculations.

For \(SL_2\), the quasi-minuscule Schubert variety has one vertex and an open stratum which is an affine-line bundle over \(\mathbf P^1\). An affine-line bundle over an \(\mathbf F_q\)-point has \(q\) rational points: its fibre is an affine line, since the one-dimensional vector space and its translation torsor over a field are trivial. Thus

\[ \#\mathrm{Gr}_{SL_2}^{\leq\alpha^\vee}(\mathbf F_q) =1+q(q+1)=q^2+q+1. \tag{3.1} \]

Equivalently the three cells in Orbits and Schubert varieties, §4 have \(q^2,q,1\) points. Counting the whole closure includes the vertex.

For \(GL_2\), write \(c_{ab}\) for the indicator of the double coset with dominant coweight \((a,b)\). Normalize Haar measure by \(\operatorname{vol}GL_2(\mathbf F_q[[t]])=1\). The Satake transform \(\mathcal S\) is the one proved and computed in Spherical Hecke algebras as functions on the affine Grassmannian, §§4–5:

\[ \begin{aligned} \mathcal S(c_{10})&=a(x_1+x_2),\\ \mathcal S(c_{20})&=q(x_1^2+x_2^2)+(q-1)x_1x_2,\\ \mathcal S(c_{11})&=x_1x_2. \end{aligned} \tag{3.2} \]

The first closure is \(\mathbf P^1\), so \(\mathcal I_{10}=E_{\mathbf P^1}[1](1/2)\). The second closure has open orbit \((2,0)\) and closed central point \((1,1)\), with the cone chart (2.1) at that point. Proposition 2.1 and the smooth open chart identify its IC with the shifted constant sheaf. The central point has dimension zero. Consequently

\[ \begin{aligned} t_{\mathcal I_{10}}&=-a^{-1}c_{10},\\ t_{\mathcal I_{20}}&=q^{-1}(c_{20}+c_{11}),\\ t_{\mathcal I_{11}}&=c_{11}. \end{aligned} \tag{3.3} \]

Applying (3.2) proves

\[ \begin{aligned} \mathcal S(t_{\mathcal I_{10}})&=-(x_1+x_2),\\ \mathcal S(t_{\mathcal I_{20}})&=x_1^2+x_1x_2+x_2^2,\\ \mathcal S(t_{\mathcal I_{11}})&=x_1x_2. \end{aligned} \tag{3.4} \]

These are the negative standard character, the symmetric-square character, and the determinant character. Their representation interpretation is also proved by the rank-one and minuscule calculations in Identifying the dual group, §§3–4. The minus sign in the first line comes from \([1]\).

4. Recovering the lattice-chain product

Consider the two-step modification with both relative positions \((1,0)\). Its source \(\widetilde Y\) is a smooth projective surface, an iterated \(\mathbf P^1\)-bundle, and the convolution map

\[ m:\widetilde Y\longrightarrow Y=\mathrm{Gr}_{GL_2}^{\leq(2,0)} \tag{4.1} \]

is an isomorphism over the open orbit. Its fibre over the central point is \(\mathbf P^1\). Indeed an index-two quotient lattice of type \((2,0)\) has a unique intermediate colength-one lattice; for type \((1,1)\), the quotient is a two-dimensional \(\mathbf F_q\)-space, and intermediate lattices are its lines. These are the actual chain fibres computed in Convolution and rigidity, §§2–3.

The normalized convolution is

\[ \mathcal I_{10}*\mathcal I_{10} =Rm_*E_{\widetilde Y}[2](1). \tag{4.2} \]

At an open-orbit point its stalk has trace \(q^{-1}\). At the central point the stalk is \(R\Gamma(\mathbf P^1,E)[2](1)\). The projective-line computation, with its Kummer generator, gives

\[ H^0(\mathbf P^1,E)=E, \qquad H^2(\mathbf P^1,E)=E(-1), \qquad H^i=0\quad(i\ne0,2). \tag{4.3} \]

The two Frobenius eigenvalues before the twist are \(1,q\). After the shift and twist their trace sum is \(q^{-1}+1\). Formula (4.3) is proved by the curve computation used in Torsion sheaves on curves, §4, and its Frobenius convention in Frobenius morphisms and their action on cohomology, §6. Thus direct fibre computation proves

\[ t_{\mathcal I_{10}*\mathcal I_{10}} =q^{-1}c_{20}+(q^{-1}+1)c_{11} =t_{\mathcal I_{20}}+t_{\mathcal I_{11}}. \tag{4.4} \]

It also proves trace compatibility with function convolution here. The trace on the source object is constantly \(q^{-1}\). Summing it over the unique rational intermediate lattice in the first case or the \(q+1\) rational lines in the second case gives exactly the two stalk traces just calculated. The chain model is the function convolution model because every right \(K\)-coset has measure one, as proved in Lesson 1.

There is a corresponding sheaf decomposition, whose Frobenius normalization matters:

\[ \mathrm{IC}^{\mathrm{raw}}_{10}* \mathrm{IC}^{\mathrm{raw}}_{10} \simeq\mathrm{IC}^{\mathrm{raw}}_{20} \oplus E_{(1,1)}(-1). \tag{4.5} \]

To verify (4.5) without a general decomposition theorem, use the exceptional curve \(D\). Its normal line bundle in the cone resolution is \(\mathcal O_{\mathbf P^1}(-2)\), computed from the lattice-chain charts in Convolution and rigidity, §10. On the inverse image of the cone chart, \(D\) is the zero section of that line bundle. Exceptional restriction is local in this open neighbourhood of \(D\), so Theorem 2.0 supplies the actual comparison \(i_D^!E=E_D(-1)[-2]\) with its specified orientation. Pushing the exceptional-support localization triangle along \(m\), which is an isomorphism off \(D\), gives, for \(A=Rm_*E[2]\) and the vertex inclusion \(i\),

\[ i^*A=R\Gamma(D,E)[2], \qquad i^!A=R\Gamma(D,E(-1)). \tag{4.8} \]

The first comparison is proper base change. For the second, the pushed localization triangle is the localization triangle at the vertex: the open terms agree because \(m\) is an isomorphism there, and composition of direct image identifies their maps. The closed term is therefore the stated supported complex.

Let \(P=E_{(1,1)}(-1)\). The inclusion of the degree-zero group in \(i^!A\) and the projection to the degree-zero group of \(i^*A\) define adjunction maps \(v:P\to A\) and \(u:A\to P\). Their composite is multiplication by \(-2\). To check it, the supported divisor orientation restricts to the Kummer class of its normal line bundle. In local divisor parameters the ratios of the parameters on overlaps are the transition functions of \(\mathcal O(D)\); their Kummer boundary is exactly that Chern class. Restricting to \(D\) gives \(c_1(\mathcal O(-2))\). The projective-line trace sends \(c_1(\mathcal O(1))\) to \(+1\), so it sends this class to \(-2\). Theorem 2.0 identifies this global supported counit and its Euler class, so the scalar is the actual composite \(uv=-2\).

Replace \(u\) by \(-u/2\) to get a retraction. Its complement agrees with \(E[2]\) on the open orbit. Equations (4.3) and (4.8) show that \(i^*A\) has groups \(E\) in degree \(-2\) and \(E(-1)\) in degree zero; \(i^!A\) has groups \(E(-1)\) in degree zero and \(E(-2)\) in degree two. The nonzero composite removes exactly the degree-zero group on each side. The complementary stalk and costalk therefore have only degrees \(-2\) and \(2\), respectively. The strict boundary inequalities make the complement perverse and exclude a subobject or quotient on the closed point. It is \(\mathrm{IC}^{\mathrm{raw}}_{20}\), by intermediate extension. This proves (4.5). All coordinates, orientations, trace maps and the scalar \(-2\) are defined over \(\mathbf F_q\), so the decomposition is Frobenius compatible. The two half twists then give the total twist \((1)\).

Theorem 2.0 and equation (4.8) give \(uv=-2\); the rescaled map \(r=-u/2\) satisfies \(rv=1\). The raw central summand is \(E_{(1,1)}(-1)\), with geometric Frobenius \(q\). The total twist \((1)\) gives the normalized central eigenvalue \(1\). Editable SVG source.

Twisting (4.5) by \((1)\) yields

\[ \mathcal I_{10}*\mathcal I_{10} \simeq\mathcal I_{20}\oplus\mathcal I_{11}. \tag{4.6} \]

The twist on the central raw summand is essential. Dropping it would incorrectly replace its Frobenius eigenvalue \(q\) by \(1\).

Finally insert (3.3) in (4.4), or take traces in (4.6):

\[ q^{-1}(c_{10}*c_{10}) =q^{-1}(c_{20}+c_{11})+c_{11}. \]

Multiplication by \(q\) proves the indicator relation

\[ \boxed{\ c_{10}*c_{10}=c_{20}+(q+1)c_{11}.\ } \tag{4.7} \]

Thus one copy of the determinant representation corresponds to the normalized point sheaf, while the indicator coefficient remains \(q+1\).

5. The general character formula and its arithmetic input

The passage from weights to functions uses actual compact traces, compact concentration and scalar Frobenius. Appendix B proves the trace input and the full spherical algebra comparison.

Theorem 5.1 (Satake character formula). Fix a dominant \(\lambda\). The following three proved statements hold for every \(\nu\):

1. The compact-support trace identity holds on the supported finite-type intersection \(S_\nu\cap\mathrm{Gr}^{\leq\lambda}\).
2. Its compact cohomology is zero except in degree \(h_\nu\).
3. Frobenius acts on that remaining group as the scalar \(a^{h_\nu}\), and its dimension is \(\dim V_\lambda(\nu)\), the \(\nu\)-weight multiplicity of the dual-group representation.

With the Satake normalization of Lesson 1,

\[ \mathcal S(t_{\mathcal I_\lambda}) =(-1)^{d_\lambda}\operatorname{ch}V_\lambda. \tag{5.1} \]

Proof. The compact trace assertion is Theorem B.1.1.1; concentration is Lesson 5, Theorem Q.4.2.1; the full weight identification is Lesson 11, Theorem D.4.2.1; and scalar Frobenius is Theorem A.3.1.1. Lemma B.0.1.1 identifies the rational semi-infinite points with the cosets of Lesson 1. Thus the coefficient of \(e^\nu\) in the Satake transform is

\[ a^{-h_\nu}\sum_{x\in S_\nu(\mathbf F_q)}t_{\mathcal I_\lambda}(x). \tag{5.2} \]

There are finitely many nonzero terms. By these three assertions the sum is \((-1)^{h_\nu}a^{h_\nu}\dim V_\lambda(\nu)\). Every nonzero weight satisfies \(\lambda-\nu\in\mathbf Z\Phi^\vee\); each simple coroot pairs to \(2\) with \(2\rho\). Therefore \(h_\nu\equiv d_\lambda\pmod2\). Substitute in (5.2) to obtain \((-1)^{d_\lambda}\dim V_\lambda(\nu)\). Summing over the weights proves (5.1). \(\square\)

All the assertions used here concern the actual rational-adic compact complexes and their descended geometric Frobenius maps. Appendix B also proves multiplicativity of the normalized transform directly from Iwasawa and identifies its image with the whole invariant group algebra.

Equation (5.1) gives the usual geometric interpretation of the classical Satake character basis. This normalization is discussed in Edward Frenkel's free Lectures on the Langlands program and conformal field theory, §5.3. If the half twist is instead defined using \(-a\), the trace of \(\mathcal I_\lambda\) is multiplied by \((-1)^{d_\lambda}\), and (5.1) has no sign. Equivalently one may multiply functions by the parity character of \(L/\mathbf Z\Phi^\vee\). This is well-defined because the coroot pairings just computed are even; under convolution the component parities add.

For a split torus, all the premises can be checked directly. A supported constructible sheaf ignores the nilpotent directions of its Grassmannian; its reduced support is a finite subset of \(L\), by Loop groups and the affine Grassmannian, §6. Every IC is the untwisted point object, all \(h_\nu\) and \(d_\lambda\) are zero, and its sole weight space is \(E\) with Frobenius one. Point convolution adds labels. Thus \(c_\lambda*c_\mu=c_{\lambda+\mu}\) and \(\mathcal S(c_\lambda)=e^\lambda\) follow directly.

6. A polynomial that remembers stalk degrees

Use the dual root system: its positive roots are the positive coroots of \(G\). Define \(P_u(\gamma)\in\mathbf Z[u]\) by the formal expansion

\[ \prod_{\beta\in\widehat\Phi^+}(1-u e^\beta)^{-1} =\sum_\gamma P_u(\gamma)e^\gamma. \tag{6.1} \]

For a fixed \(\gamma\), its coefficient is a finite sum. Indeed every positive root has positive height in the simple-root basis, so the sum of the multiplicities in a partition of \(\gamma\) is bounded by its height. A term with \(r\) roots contributes \(u^r\). Put \(\widehat\rho=\frac12\sum_{\beta\in\widehat\Phi^+}\beta\) and define

\[ m^\mu_\lambda(u) =\sum_{w\in W}(-1)^{\ell(w)} P_u\bigl(w(\lambda+\widehat\rho)-(\mu+\widehat\rho)\bigr). \tag{6.2} \]

The variable \(u\) in this definition is formal. It will be evaluated at the cardinality \(q\) only after the grading convention is fixed. The stalk theorem concerns dominant \(\lambda,\mu\) with \(\mu\leq\lambda\). It is not a positivity assertion about (6.2) for every nondonominant \(\mu\).

In the arithmetic setting of §1, the correctly normalized stalk identity is

\[ m^\mu_\lambda(u) =u^{\langle\rho,\lambda-\mu\rangle} \sum_{j\geq0} \dim_E\mathcal H^{-d_\lambda+2j} (i_\mu^*\mathrm{IC}^{\mathrm{raw}}_\lambda)\,u^{-j}. \tag{6.3} \]

Here \(i_\mu\) is the geometric point \(t^\mu\); the sum uses stalk cohomological degrees, not the global IC degrees. The theorem also asserts the relevant parity and vanishing, so the right side is a polynomial with nonnegative exponents. Theorem D.6.1.1 proves (6.3) from the actual stalk operators, the full orbit-indicator formula and infinitely many finite-field extensions.

Equivalently the right side is \(\sum_{r\geq0}\dim\mathcal H^{-h_\mu-2r}(i_\mu^*\mathrm{IC}^{\mathrm{raw}}_\lambda)u^r\): set \(r=\langle\rho,\lambda-\mu\rangle-j\). The integer prefactor in (6.3) is integral because \(\lambda-\mu\) is a sum of coroots. George Lusztig's freely accessible author copy of Singularities, character formulas, and a q-analogue of weight multiplicities, equations (9.3)–(9.4) and §11(c), printed pp. 225–227, gives the polynomial and IC conventions. The equation in (9.4) is presented there as a conjecture, followed by the note that Kato proved it. Dmitri Panyushev's free On Lusztig's q-analogues of all weight multiplicities of a representation, §1, equation (1.1), gives (6.1)–(6.2) with the positive powers used here. These free comparison locators specify the stalk conventions. The algebraic transition is proved below, and Appendix D proves the geometric equality (6.3) in the arithmetic setting of §1.

6.1. A finite Hall–Littlewood polynomial

We now prove an algebraic interpretation of (6.2). Keep the dual positive roots \(\widehat\Phi^+\), the coweight lattice \(L\), and its Weyl group \(W\). Thus the roots in this calculation are the coroots of \(G\). The lattice can contain nonprimitive roots, and \(\widehat\rho\) need not belong to it. Central directions, on which \(W\) is trivial, are retained.

Work over \(F=\mathbf Q(u)\). Set \[ Q_u=\prod_{\beta\in\widehat\Phi^+} \frac{1-e^{-\beta}}{1-u e^{-\beta}},\qquad W_\nu(u)=\sum_{w\nu=\nu}u^{\ell(w)}, \] and, for dominant \(\nu\), define initially in the fraction field \[ H_\nu(u)=\frac1{W_\nu(u)} \sum_{w\in W}w\left(\frac{e^\nu}{Q_u}\right). \tag{6.5} \] The stabilizer polynomial has constant term one, so is nonzero. We shall prove that \(H_\nu(u)\) is a finite \(W\)-invariant Laurent polynomial, supported in weights at most \(\nu\), with coefficient one on its top orbit. These are the Hall–Littlewood polynomials in the normalization (6.5).

The root and chamber arguments are proved in Root systems and their Weyl groups. The finite highest-weight characters and their support are proved in Weights, Verma modules and the theorem of the highest weight. We use the complete Weyl character proof in Weyl's character formula and the multiplicity formulas, Theorem 2.1 and Corollary 2.2.

Temporarily enlarge \(L\) to \(L'=L+\mathbf Z\widehat\rho\). It is a \(W\)-stable free lattice, because \(w\widehat\rho-\widehat\rho\) is in the root lattice. Put \[ D=e^{\widehat\rho}\prod_{\beta>0}(1-e^{-\beta}),\qquad A_\eta=\sum_w(-1)^{\ell(w)}e^{w\eta}. \] For a simple reflection, its changed root factor and changed \(\widehat\rho\) monomial give \(wD=(-1)^{\ell(w)}D\). Consequently \[ W_\nu(u)D H_\nu(u) =\sum_{S\subset\widehat\Phi^+}(-u)^{|S|} A_{\nu+\widehat\rho-\beta_S}, \qquad \beta_S=\sum_{\beta\in S}\beta. \tag{6.6} \] This is a finite expansion.

Each alternant divided by \(D\) is a Laurent polynomial in the original lattice. If \(\eta\) lies on a reflection wall, pair \(w\) with \(ws\), where \(s\eta=\eta\); their signs cancel, so \(A_\eta=0\). Otherwise choose \(v\) taking \(\eta\) into the open dominant chamber, with image \(\eta^+\). Its simple-coroot pairings are positive integers. Thus \(\eta^+-\widehat\rho\) is dominant integral and \[ \eta^+-\widehat\rho =v\nu+(v\widehat\rho-\widehat\rho)-v\beta_S\in L. \] The proved Weyl character formula gives \[ \frac{A_\eta}{D} =(-1)^{\ell(v)}\operatorname{ch} V_{\eta^+-\widehat\rho}. \tag{6.7} \] For a reductive lattice, apply that formula to the semisimple restriction and retain the central monomial. All weight differences are in the root lattice, so its character remains in \(\mathbf Z[L]\). Equation (6.6) now proves \(H_\nu(u)\in F[L]^W\). In particular no divisibility argument requiring a primitive root in \(L\) is needed.

To prove triangular support, expand downward, in translates of the negative cone generated by the simple dual roots. A fixed coefficient of any expansion has finitely many terms, because positive height bounds all root exponents. If \(w\) sends a root to a positive root, its factor in \(w(Q_u^{-1})\) is a downward geometric series. If it sends it to \(-\beta\), the factor is \[ \frac{1-u e^{\beta}}{1-e^{\beta}} =u+(u-1)\frac{e^{-\beta}}{1-e^{-\beta}}. \tag{6.8} \] Thus each \(w\)-summand is supported in \(w\nu-\widehat Q^+\), where \(\widehat Q^+\) is the nonnegative integral simple-root cone. For dominant \(\nu\), the reduced-word calculation gives \(\nu-w\nu\in\widehat Q^+\): telescope along a reduced expression for \(w\); the resulting inversion roots are positive and their coefficients are the nonnegative pairings of \(\nu\) with simple coroots. It follows that every weight of \(H_\nu\) is at most \(\nu\). Its dominant orbit representative also has this property, by \(W\)-invariance.

6.2. The exact dual coefficient

For dominant \(\mu\) and a \(W\)-invariant Laurent polynomial \(f\), define \[ \mathfrak l_\mu(f)=[e^\mu](fQ_u). \] The product is read in the downward completion just specified. We prove \[ \mathfrak l_\mu(H_\nu)=\delta_{\mu\nu}. \tag{6.9} \]

Let \(N(w)=\{\beta>0:w^{-1}\beta<0\}\). Factors that remain positive cancel in \(Q_u/wQ_u\). A changed factor, with \(x=e^{-\beta}\), is \((u-x)/(1-ux)\). Hence \[ \frac{Q_u}{wQ_u} =\prod_{\beta\in N(w)} \frac{u-e^{-\beta}}{1-u e^{-\beta}}. \tag{6.10} \] It is a downward series with constant term \(u^{\ell(w)}\). A contribution of the \(w\)-summand of \(H_\nu Q_u\) at \(\mu\) can therefore occur only if \[ \mu=w\nu-\sum_{\beta\in N(w)}n_\beta\beta, \qquad n_\beta\ge0. \] Applying \(w^{-1}\) gives \[ w^{-1}\mu=\nu+ \sum_{\beta\in N(w)}n_\beta(-w^{-1}\beta)\ge\nu. \] Dominance of \(\mu\) also gives \(\mu\ge w^{-1}\mu\). On the other hand, polynomial triangularity from §6.1 and the negative support of \(Q_u\) imply \(\mu\le\nu\) for every nonzero coefficient of their product. The simple-root cone is pointed. Thus \(\mu=\nu\), and every \(n_\beta\) is zero. The remaining terms have \(w\nu=\nu\), and their constant terms sum to \(W_\nu(u)\). Division by that sum proves (6.9). Using the triangularity of the sum here is legitimate: its polynomiality was established first, so a nonzero coefficient has that bound irrespective of cancellations between summands.

Only the coefficient of \(e^\nu\) in \(H_\nu\) can contribute to the coefficient of \(e^\nu\) in \(H_\nu Q_u\). Equation (6.9) therefore proves that this coefficient is one. Weyl invariance proves the same statement on the entire top orbit.

6.3. The character transition formula

Theorem 6.1. For every dominant coweight \(\lambda\), \[ \operatorname{ch}V_\lambda =\sum_{\substack{\mu\text{ dominant}\\\mu\le\lambda}} m^\mu_\lambda(u)H_\mu(u). \tag{6.11} \] This is a finite algebraic identity. Its coefficients are exactly the polynomials (6.2), for every reduced root datum and its coweight lattice.

Proof. There are finitely many dominant \(\mu\le\lambda\). Their central components are fixed. On the root span take a positive definite \(W\)-invariant inner product. Dominance gives \[ \|\lambda\|^2-\|\mu\|^2 =(\lambda+\mu,\lambda-\mu)\ge0, \] since \(\lambda-\mu\) is a nonnegative integral sum of simple roots, each having nonnegative inner product with \(\lambda+\mu\). Thus their root components lie in a bounded ball of a discrete lattice, which has finitely many points: bound their integer coordinates in any lattice basis.

The orbit sums \(\sum_{\eta\in W\mu}e^\eta\) are a basis of the invariant Laurent polynomials supported on these orbits. The proved triangularity and top coefficient one show that the \(H_\mu\) are another basis. Indeed, order this finite set by a linear extension of dominance; its change matrix is triangular with diagonal one and can be inverted by finitely many subtractions. Equation (6.9) supplies the exact dual basis. The highest-weight support theorem places \(\operatorname{ch}V_\lambda\) in this space, so its coefficient at \(H_\mu\) is \(\mathfrak l_\mu(\operatorname{ch}V_\lambda)\).

Multiply the proved Weyl character formula by \(Q_u\). It gives \[ \operatorname{ch}V_\lambda Q_u =\sum_{w\in W}(-1)^{\ell(w)} e^{w(\lambda+\widehat\rho)-\widehat\rho} \prod_{\beta>0}(1-u e^{-\beta})^{-1}. \tag{6.12} \] The negative-exponent version of (6.1) has coefficient \(P_u(\gamma)\) at \(e^{-\gamma}\), by the same finite partition count. Extraction of \(e^\mu\) in (6.12) is therefore exactly (6.2). This proves (6.11), including its finite support. Although the \(H_\mu\) were defined over \(\mathbf Q(u)\), every transition coefficient is in \(\mathbf Z[u]\). \(\square\)

At \(u=1\), \(Q_u=1\) and \(W_\mu(1)\) is the size of the stabilizer, so \(H_\mu(1)\) is its orbit sum. Thus (6.11) recovers the ordinary weight multiplicities. At \(u=0\), the alternant calculation makes \(H_\mu(0)=\operatorname{ch}V_\mu\). These specializations are defined: the only denominator in the finite polynomial expression is \(W_\mu(u)\), which is nonzero at both values.

Theorem 6.1 proves the algebraic transition underlying the weight polynomial. Theorem D.5.1.1 supplies the all-root spherical-Hecke comparison \[ \mathcal S(c_\mu)=q^{\langle\rho,\mu\rangle}H_\mu(q^{-1}) \tag{6.13} \] Theorem D.6.1.1 then proves the graded IC stalk comparison. Equation (6.3) is a polynomial identity in the formal variable \(u\). The Frobenius trace evaluates that same polynomial at \(q^{-1}\), accounting for the inverse powers in (6.13).

6.4. The rank-one check

For \(SL_2\), let \(\beta=\alpha^\vee\), the positive root of its dual \(PGL_2\). Then \(\widehat\rho=\beta/2\). Since the positive root system consists of \(\beta\),

\[ P_u(r\beta)=u^r\quad(r\geq0), \qquad P_u(r\beta)=0\quad(r<0). \]

For \(\lambda=\beta\) and \(\mu=0\), the identity Weyl element contributes \(P_u(\beta)=u\); the reflection contributes \(P_u(-2\beta)=0\). Thus

\[ m^0_{\alpha^\vee}(u)=u. \tag{6.4} \]

In the same rank-one lattice, \[ H_\beta(u)=e^\beta+e^{-\beta}+1-u,\qquad H_0(u)=1. \tag{6.14} \] Indeed put \(x=e^{-\beta}\) in (6.5). Its two summands are \(x^{-1}(1-ux)/(1-x)\) and \(x(x-u)/(x-1)\); their sum is \(x^{-1}+x+1-u\). For label zero their sum is \(1+u\), which is divided by \(W_0(u)=1+u\). Hence \(\operatorname{ch}V_\beta=H_\beta+uH_0\). The explicit \(SL_2\) Satake calculation in The Satake isomorphism and spherical representations, §6 verifies (6.13) in this example: multiplying (6.14) at \(u=q^{-1}\) by \(q\) gives \(q(e^\beta+e^{-\beta})+q-1\).

The cone calculation proves the same value directly: \(d_\lambda=2\), \(\langle\rho,\lambda\rangle=1\), and the stalk of \(E[2]\) has only \(\mathcal H^{-2}=E\). Hence the sum in (6.3) is \(1\), but its required prefactor is \(u\). At \(u=1\) the multiplicity is one; at \(u=q\) the polynomial is \(q\). These are different questions.

7. The Whittaker character and the Casselman–Shalika statement

We first specify the finite-characteristic statement. Work over an algebraic closure of a finite field, with \(\overline{\mathbf Q}_\ell\)-coefficients and \(\ell\ne p\). Choose a pinning and its simple-root quotient maps \(u_i:N\to\mathbf G_a\). Root multiplication adds these coordinates; higher-root terms do not affect them. Choose a nontrivial additive character \(\psi:\mathbf F_p\to\overline{\mathbf Q}_\ell^\times\) and its Artin–Schreier rank-one sheaf \(\mathcal L_\psi\) on \(\mathbf A^1\). Define the conductor-zero loop character

\[ \chi_0(n)=\sum_i\operatorname{Res}(u_i(n)\,dt), \qquad \chi_\mu(n)=\chi_0(t^\mu n t^{-\mu}). \tag{7.1} \]

The simple-root coordinates of a commutator vanish, so (7.1) is an additive group homomorphism. Its nonzero coefficient on every simple-root factor is the genericity condition.

On \(S_\nu=N((t))t^\nu\), the expression \(\chi_\mu(n)\), normalized to zero at \(t^\nu\), descends to a function \(\chi_\mu^\nu\) exactly when \(\mu+\nu\) is dominant. Indeed the simple-root coordinate of the stabilizer is \(t^{\langle\alpha_i,\nu\rangle}O\); conjugation by \(t^\mu\) multiplies it by \(t^{\langle\alpha_i,\mu\rangle}\). Residue is identically zero on that ideal exactly when \(\langle\alpha_i,\mu+\nu\rangle\geq0\). If the inequality fails, its \(t^{-1}\) coefficient can be nonzero, so descent fails. Coordinates for nonsimple roots do not affect the character. This proves the descent criterion.

Appendix E proves the nondominant-conductor branch, the upper degree bound, the top-degree component criterion and every highest-slice case. Appendix F proves the general top-character obstruction for conductor \(-\nu\), including its dominant simple-Levi projections and component-basis comparison. Appendix G constructs the actual relevant Whittaker strata, residue evaluation and finite Hecke coefficient comparisons. Theorems P.9.1.1–P.10.1.1 prove the general geometric Casselman–Shalika cohomology statement, including lower-degree concentration and the full multiplicity comparison: For dominant \(\lambda\) and for \(\mu,\nu\) such that \(\mu+\nu\) is dominant,

\[ H_c^r\left(\mathrm{Gr}^{\leq\lambda}\cap S_\nu, \mathrm{IC}^{\mathrm{raw}}_\lambda\otimes (\chi_\mu^\nu)^*\mathcal L_\psi\right) \cong \begin{cases} \operatorname{Hom}_{\widehat G} (V_\lambda\otimes V_\mu,V_{\mu+\nu}), &r=h_\nu\text{ and }\mu\text{ dominant},\\ 0,&\text{otherwise}. \end{cases} \tag{7.2} \]

The displayed isomorphism concerns geometric coefficient spaces; it does not assign a Weil twist to its right side. For example \(E[d]\) on \(\mathbf A^d\) has compact cohomology \(E(-d)[-d]\), so omission of the twist would change its Frobenius trace.

The complete proof of (7.2), with its raw Tate twist and actual Weil operator, is in Theorems P.9.1.1–P.10.1.1. The free Frenkel–Gaitsgory–Vilonen, Whittaker patterns in the geometry of moduli spaces of bundles on curves, Theorem 1, equation (1.7) states the formula in its original notation. The support is the closed Schubert variety. With \(\mu=0\) and dominant \(\nu\), the right side is one-dimensional only for \(\nu=\lambda\), and otherwise zero. Thus the complex has its surviving group in degree \(h_\lambda\), or is zero. In derived notation the nonzero geometric complex is \(E[-h_\lambda]\). This negative shift is consistent with the cohomological convention in §1.

There is also a categorical version. Its characteristic-zero incarnation requires exponential D-modules rather than ordinary complex local systems. On complex \(\mathbf A^1\), a rank-one local system is constant because the line is simply connected; it cannot provide the nontrivial additive character needed for Whittaker equivariance.

Here is the precise derived statement used in the 2024 geometric Langlands work. For a connected reductive group over an algebraically closed field of characteristic zero and a smooth curve \(X\), use the \(\rho(\omega_X)\)-twisted Grassmannian. The residue character is nonzero on every simple-root factor. Form the cocomplete derived category

\[ \operatorname{Whit}^{!}(G) =D\text{-}\mathrm{mod}_{1/2} (\mathrm{Gr}_{G,\rho(\omega_X)})^ {L(N)_{\rho(\omega_X)},\chi}. \tag{7.3} \]

The superscript denotes character-twisted invariants, not ordinary invariant sheaves. With the half-twist and factorization structures specified there, there is a canonical equivalence of factorization categories

\[ \operatorname{CS}_G: \operatorname{Whit}^{!}(G)\xrightarrow{\sim} \operatorname{Rep}(\widehat G), \qquad \Delta_\lambda\longmapsto V_\lambda. \tag{7.4} \]

In (7.4), the representation category is the derived cocomplete category in the source's convention. This is Arinkin and collaborators, Proof of the geometric Langlands conjecture II, Theorem 1.4.2, §1.4.4 and Remark 1.4.6. Sections 1.3–1.4 define the invariants and coinvariants and distinguish the local Grassmannian theorem from the global statement on \(\mathrm{Bun}_G\). The equivalence (7.4) is stated here; no proof of it is inferred from the classical Satake equivalence of Lesson 11.

For a torus the categorical assertion in the perverse setting has a complete elementary proof. There is no \(N\), no residue character and no shift. For \(T=\mathbf G_m\), a bounded supported perverse sheaf is a finite family \((V_n)_{n\in\mathbf Z}\) of finite-dimensional coefficient spaces. Multiplication of lattices gives

\[ (V*W)_r=\bigoplus_{m+n=r}V_m\otimes W_n. \tag{7.5} \]

The unit is \(E\) at zero. The dual places \(V_n^*\) at \(-n\); evaluation and coevaluation are the usual vector-space maps. Associativity and symmetry are those of vector spaces, and their coherence identities hold on each summand.

To identify this category with \(\operatorname{Rep}_E(\mathbf G_m)\), assign \(v\in V_n\) the coaction \(v\mapsto v\otimes z^n\). Conversely, expand the coaction of a finite-dimensional representation as \(\delta(v)=\sum_n p_n(v)\otimes z^n\). Coassociativity and counit give

\[ p_np_m=\delta_{nm}p_n,\qquad\sum_np_n=1. \tag{7.6} \]

A finite basis makes the sum finite. Therefore \(V=\bigoplus_n\operatorname{im}p_n\), and the two constructions are inverse, including on morphisms. The product coaction gives precisely (7.5). This proves the torus equivalence directly. The proof for a higher-dimensional split torus replaces \(\mathbf Z\) by its character lattice.

8. Factorization over finite sets of points

Let \(X\) be a smooth curve, and let \(I\) be a finite set. Its Beilinson–Drinfeld Grassmannian classifies a \(G\)-bundle together with a trivialization away from the graphs of \(I\) labeled points. Over the open subset \(X^{I,\circ}\) where all points are distinct, modifications at different discs are independent. The torsor-gluing proof and its compatibility with arbitrary parameter rings are in Beauville–Laszlo gluing and Fusion and the commutativity constraint, §§1–3. For a partition of \(I\) into disjoint blocks, the corresponding block-disjoint open has the analogous product description.

Along a diagonal the labels merge. Already for two points, the restriction of the canonical fusion extension to the diagonal gives the convolution of the two objects. Under a tensor equivalence with dual-group representations, this corresponds to

\[ \operatorname{Rep}(\widehat G\times\widehat G) \xrightarrow{\operatorname{Res}_{\mathrm{diag}}} \operatorname{Rep}(\widehat G), \qquad V\boxtimes W\longmapsto V\otimes W. \tag{8.1} \]

Indeed the diagonal element \(g\) acts as \(g\) on both factors. Fusion and the commutativity constraint, Theorems C.3.1.1 and C.6.1.1 prove the actual rational comparisons on every partial diagonal and their full refinement, block, permutation and unit coherence. For a merge from \(r\) labels to \(b\) blocks, ordinary restriction removes the common shift \([r-b]\); exceptional restriction has the opposite shift and the retained Tate line \(E(-(r-b))\). The component adjustment of Theorem C.5.1.1 gives the ordinary tensor regrouping used under the dual-group equivalence. These are statements about the specified equivariant, universally locally acyclic relative sheaves and their canonical extensions. They do not identify every constructible sheaf on \(X^I\) with a representation of a fixed product group.

The corresponding arithmetic theorem has further hypotheses. Fargues–Scholze, Geometrization of the local Langlands correspondence, Definition VI.9.1 and VI.9–VI.11 works on the local Hecke stack over \((\operatorname{Div}_X^1)^I\) for a nonarchimedean local field \(E_0\). Its coefficient ring \(\Lambda\) is killed by an integer prime to the residue characteristic. The Satake objects are ULA, flat, relative perverse sheaves. Proposition VI.9.2 retains the continuous \(W_{E_0}^I\)-action; VI.9.3–VI.9.4 give restriction and coherent fusion; VI.10.2–VI.10.3 reconstruct the product Hopf algebra internal to Weil representations. The dual-group identification and its cyclotomic pinning action are in VI.11.1. This arithmetic statement is not used as a proof of a classical equivalence on arbitrary \(X^I\).

The Ran space organizes points with varying finite label sets. Its finite-set data consist of the spaces \(X^I\), maps of diagonals induced by surjections of label sets, and product identifications over disjoint configurations. The preceding collision rule describes how Satake objects fit those maps. Constructing a full sheaf category on that colimit, with descent and an equivalence of categories, requires an additional theorem. The finite-set fusion statements above specify the part needed for the convolution category in this course.

9. The Hecke correspondence on bundles

The same modification geometry supplies the kernels of Hecke functors. Fix \(x\in X\). The Hecke correspondence at \(x\) classifies

\[ (\mathcal E,\mathcal E',\beta), \qquad \beta:\mathcal E|_{X-\{x\}} \xrightarrow{\sim}\mathcal E'|_{X-\{x\}}. \tag{9.1} \]

Its projections \(h_1,h_2\) forget \(\mathcal E'\) and \(\mathcal E\), respectively. Trivializing \(\mathcal E\) on the formal disc identifies the possible modifications with \(\mathrm{Gr}_G\). A change of disc trivialization acts by \(L^+G\), so an equivariant Satake object \(A\) descends to a kernel \(\mathcal K_{A,x}\) on this correspondence. The independence of such choices is exactly the torsor descent of Lesson 3 and the coordinate comparison of Lesson 8.

In a sheaf or D-module theory on \(\mathrm{Bun}_G\) with the indicated operations, the fixed-point Hecke transform is written

\[ \mathsf H_{A,x}(\mathcal F) =h_{2!}(h_1^*\mathcal F\otimes\mathcal K_{A,x}). \tag{9.2} \]

One fixes the shifts and, for D-modules, the choices of \(*\) and \(!\) according to that theory. Formula (9.2) records the geometric kernel convention; it does not assert that a six-functor theory on these stacks has been constructed in this lesson.

If those operations satisfy base change, the projection formula and composition, two successive transforms are represented by the stack of two successive modifications. Pushforward under composition of the modifications is the kernel convolution of Lesson 7. This explains the action relation \(\mathsf H_{A,x}\mathsf H_{B,x}\simeq\mathsf H_{A*B,x}\) in that formalism: the two sides integrate the same tensor product on the same chain correspondence. At distinct points the modifications are independent, as in §8. A Hecke eigensheaf for a dual-group local system has, for each representation \(V\), the associated local system as its parameter in these transforms. The global existence theorem and the categorical geometric Langlands equivalence are separate from this local kernel construction.

For \(\mathbf G_m\) the underlying operation is explicit even without a derived formalism. The bundle is a line bundle \(\mathcal L\). With the lattice convention \(t^nO\), a modification of index \(n\) produces \(\mathcal L(-nx)\): its local sections vanish to order \(n\). Applying index \(m\) next gives \(\mathcal L(-(n+m)x)\). This is the geometric counterpart of the point convolution (7.5).

10. Exercises with solutions

Exercise 12.1 (easy). Count the rational points of the \(SL_2\) quasi-minuscule Schubert closure, including its singular point.

Solution. Its open orbit is an affine-line bundle over \(\mathbf P^1\). There are \(q+1\) rational base points and \(q\) points over each one, so the open orbit has \(q(q+1)\) points. The remaining stratum consists of the vertex. The result is \(q^2+q+1\). This also agrees with the cell sizes \(q^2,q,1\).

Exercise 12.2 (easy). For the positive square root \(a\), compute the trace of the minuscule \(GL_2\) object and its Satake transform. Explain what changes if the negative square root defines the half twist.

Solution. The minuscule closure is \(\mathbf P^1\), and \(\mathcal I_{10}=E[1](1/2)\). Its sole stalk group is in degree \(-1\), with Frobenius \(a^{-1}\). Thus \(t_{\mathcal I_{10}}=-a^{-1}c_{10}\). Since \(\mathcal S(c_{10})=a(x_1+x_2)\), its transform is \(-(x_1+x_2)\). Replacing the half-twist eigenvalue by \((-a)^{-1}\) multiplies this trace by \(-1\), and the transform becomes \(x_1+x_2\).

Exercise 12.3 (medium). Recover \(c_{10}*c_{10}=c_{20}+(q+1)c_{11}\) from the normalized IC convolution. Give the raw central summand and its Frobenius eigenvalue.

Solution. The normalized decomposition is \(\mathcal I_{10}*\mathcal I_{10}=\mathcal I_{20}\oplus\mathcal I_{11}\). Here \(t_{\mathcal I_{10}}=-a^{-1}c_{10}\), \(t_{\mathcal I_{20}}=q^{-1}(c_{20}+c_{11})\), and \(t_{\mathcal I_{11}}=c_{11}\). Trace compatibility for this map was proved by the point and projective-line fibre calculations in §4. Hence

\[ q^{-1}c_{10}*c_{10}=q^{-1}c_{20}+(q^{-1}+1)c_{11}. \]

Multiply by \(q\). Before half twists, the central summand is \(E_{(1,1)}(-1)\), with eigenvalue \(q\), supplied by \(H^2(\mathbf P^1,E)\). The total twist \((1)\) cancels that \((-1)\), giving eigenvalue one in the normalized decomposition.

Exercise 12.4 (medium). Using Proposition 2.1, compute \(m^0_{\alpha^\vee}(u)\) for \(SL_2\) from (6.2), the transition (6.11), and the IC stalk. Explain why the stalk's dimension is not itself the polynomial.

Solution. The dual positive root is \(\beta=\alpha^\vee\), and \(\widehat\rho=\beta/2\). The identity term in (6.2) is \(P_u(\beta)=u\); the reflection term has argument \(-2\beta\) and is zero. Therefore \(m^0_\beta(u)=u\). Equation (6.14) gives \(\operatorname{ch}V_\beta=e^\beta+1+e^{-\beta}=H_\beta+uH_0\), so Theorem 6.1 gives the same coefficient. The raw IC stalk at the cone vertex is \(E[2]\), with only its degree \(-2\) group nonzero. In (6.3) it contributes \(1\) to the sum, multiplied by \(u^{\langle\rho,\beta\rangle}=u\). Its dimension and the ordinary weight multiplicity are one, the value at \(u=1\); its cohomological position gives the polynomial \(u\).

Exercise 12.5 (hard). State the character and support needed for (7.2), and prove the cohomology formula and the perverse Whittaker equivalence for \(\mathbf G_m\).

Solution. The character is the conductor-zero sum of residues in the pinned simple-root coordinates, conjugated by \(t^\mu\). It descends to \(S_\nu\) when \(\mu+\nu\) is dominant. The complex is restricted to the closed support \(\mathrm{Gr}^{\leq\lambda}\cap S_\nu\) and tensor-multiplied by the pullback of the chosen nontrivial Artin–Schreier character sheaf. Formula (7.2) places its surviving cohomology in degree \(h_\nu\) and identifies that space with \(\operatorname{Hom}(V_\lambda\otimes V_\mu,V_{\mu+\nu})\) if \(\mu\) is dominant; otherwise all groups vanish.

For a torus there are no roots, so every integer is dominant, the character is zero, and \(h_\nu=0\). The intersection is a point when \(\lambda=\nu\) and empty otherwise. Its compact cohomology is consequently \(E\) in degree zero in the first case and zero in the second. The representation Hom space is

\[ \operatorname{Hom}_{\mathbf G_m} (E_\lambda\otimes E_\mu,E_{\mu+\nu}) =\begin{cases}E,&\lambda=\nu,\\0,&\lambda\ne\nu,\end{cases} \]

because a morphism between two one-dimensional weight representations is zero unless their weights agree. Thus the full cohomology statement holds here. Whittaker equivariance imposes no condition because \(N=1\). Its perverse category is the category of finite supported families \((V_n)\), with convolution (7.5). The coaction construction and the mutually orthogonal projections (7.6) give inverse tensor equivalences with \(\operatorname{Rep}_E(\mathbf G_m)\). Evaluation and coevaluation, associativity and symmetry were checked summandwise in §7, completing the categorical assertion in this case.

Appendix A. Scalar Frobenius on every Satake weight

Throughout this appendix \(G/\mathbf F_q\) is split connected reductive, \(k=\overline{\mathbf F}_q\), and \(E/\mathbf Q_\ell\) is finite with \(\ell\nmid q\). Enlarge \(E\) to contain \(a\), where \(a^2=q\), when the half twist is used. The actual genuine rational Satake heart \(\mathcal C_G\), its convolution and adjusted symmetry, and its total cohomology fibre functor \(H_G\), are those of Lessons 5–11. Lesson 11, Theorem B.4.1.1 identifies the entire reconstructed group with the split dual group, and proves full-heart semisimplicity; Lesson 11, Theorem D.4.2.1 extends that result to algebraic coefficients. We use the finite-\(E\) version first.

Let \(\sigma^*\) denote pullback by the semilinear geometric Frobenius, using the \(\mathbf F_q\)-model to identify its translated support with the original one. This is the inverse arithmetic-field pullback. Its coefficient correspondence is the one of Frobenius morphisms and their action on cohomology, Theorem 3.2 and Proposition 3.3; in particular the resulting action on \(E(1)\) is \(q^{-1}\). We do not use the absolute Frobenius correspondence, whose operator with its specified coefficient identification is the identity.

A.1. The geometric Frobenius acts on the reconstructed group

Lemma A.1.1.1. The functor \(\sigma^*:\mathcal C_G\to\mathcal C_G\) is an \(E\)-linear exact symmetric tensor autoequivalence. There is a natural monoidal transport \[ u_P:H_G(\sigma^*P)\xrightarrow{\sim}H_G(P). \] For the canonical raw IC Weil map \(\alpha_\lambda:\sigma^*I_\lambda\to I_\lambda\), put \[ F_\lambda=H_G(\alpha_\lambda)u_{I_\lambda}^{-1}. \tag{A.1.1} \] There is an actual group-scheme automorphism \(\Phi_G\) of \(\widehat G_E\) such that on every IC representation \[ \rho_\lambda(\Phi_G(g))=F_\lambda\rho_\lambda(g)F_\lambda^{-1}. \tag{A.1.2} \] It fixes the specified weight torus pointwise. The canonical raw IC Weil structures are not asserted to form a tensor family.

Proof. A field automorphism is an isomorphism of the geometric schemes and of their coefficient topoi. Pullback preserves the constant \(E\)-ring, actual constructibility and the stratum support and costalk tests. Transport the finite-jet group, action, identity and cocycle along that same isomorphism. The split jet group and orbit models are defined over \(\mathbf F_q\), so this is the original genuine heart and its original stratum labels.

Convolution uses the modification correspondence, its descended external product and its proper-support image, all defined over \(\mathbf F_q\). The actual inverse-image, external product and proper-support comparisons of Lesson 5, Appendices O–P and Lesson 7, Appendix A consequently identify pullback of convolution with convolution of the two pullbacks. These maps retain their units and associative pasting, so give a tensor autoequivalence, with inverse the arithmetic pullback. The two-point modification family and its strict intermediate extension are also defined over \(\mathbf F_q\). Pullback identifies their off-diagonal maps; intermediate-extension full faithfulness identifies their collision maps. The signs in Lesson 8, Theorem A.4.2.1 depend only on the unchanged component labels. This proves symmetry with the adjusted constraint.

Transport of the structural map to the geometric point gives \(u\). In a resolution calculation it is the inverse of the coefficient pullback comparison on global sections; composition with a Weil map therefore gives (A.1.1). The proper moving cohomology and its localization inclusions that define the \(H_G\)-tensor map are transported by the same field isomorphism. Their diagrams commute before taking cohomology. Thus \(u\) is monoidal. This is the actual tensor map of Lesson 10, Theorem A.2.2.1, not a choice of identifications of vector spaces of equal dimension.

For every coefficient algebra \(R\) and \(g\in\operatorname{Aut}^{\otimes}(H_G)(R)\), define \[ \Theta_G(g)_P=u_Pg_{\sigma^*P}u_P^{-1}. \tag{A.1.3} \] The tensor structure of \(\sigma^*\) and the monoidality of \(u\) show that this is a natural tensor automorphism. The inverse autoequivalence supplies its inverse. These assignments are natural in \(R\). The all-test-algebra reconstruction of Lesson 9 therefore makes \(\Theta_G\) an actual group-scheme automorphism, not just a transformation of \(E\)-points.

The raw IC has a canonical Weil map: on its open orbit use the identity on the constant coefficient, then extend by full faithfulness of intermediate extension. Naturality of \(g\) for this map gives \[ g_{\sigma^*I_\lambda} =H_G(\alpha_\lambda)^{-1}g_{I_\lambda}H_G(\alpha_\lambda). \] Substitution in (A.1.3) gives \(\Theta_G(g)_{I_\lambda}=F_\lambda^{-1}g_{I_\lambda}F_\lambda\). Define \(\Phi_G=\Theta_G^{-1}\); this gives (A.1.2), with the direction fixed.

The actual weight decomposition is made of the localization projectors of Lesson 5, Theorem Q.4.3.1. Each semi-infinite correspondence, its coweight label and its projector is defined over \(\mathbf F_q\), so \(u\) carries each weight summand of \(H_G(\sigma^*P)\) to the same labelled summand of \(H_G(P)\). A torus element acts on it by the character \(\nu\) on both sides. Formula (A.1.3) therefore fixes every torus element, over every \(R\). So does its inverse \(\Phi_G\). \(\square\)

Lemmas A.1.1.1–A.2.1.1 retain the pullback direction and the raw eigenvalues \(q,1\). The group automorphism is conjugation by this operator. Compatibility with actual Levi constant term then gives the scalars \(q,q^{-1}\) on every simple root pair. Editable SVG source.

A.2. The root-line scalar is \(q\)

Lemma A.2.1.1. For every positive simple root \(\beta_i=\alpha_i^\vee\) of the dual group, \(d\Phi_G\) acts on its root line by \(q\), and on the negative root line by \(q^{-1}\). It is the identity on the torus Lie algebra.

Proof. First take the geometric group \(PGL_2\). Its minuscule IC is \(E_{\mathbf P^1}[1]\); its reconstructed dual group is \(SL_{2,E}\), by the full equivalence of Lesson 11, Theorem B.4.1.1. The highest and lowest weight spaces have characters \(\omega^\vee\) and \(-\omega^\vee\), respectively. The highest is the compact cohomology of the split open affine line and equals \(E(-1)\), with geometric Frobenius \(q\); the lowest is the point line \(E\), with Frobenius \(1\). These are the actual affine-line trace and projective-line Kummer comparisons, with the raw \([1]\) retained. In the basis ordered highest, lowest, \(F=\operatorname{diag}(q,1)\).

This two-dimensional representation is faithful for \(SL_2\). Lemma A.1.1.1 consequently identifies \(\Phi_{PGL_2}\) with conjugation by \(\operatorname{diag}(q,1)\) on that group. Direct multiplication scales \(E_{12}\) by \(q\) and \(E_{21}\) by \(q^{-1}\), and fixes diagonal matrices.

For any split rank-one geometric group \(M\), use its adjoint map \(M\to PGL_2\). Lesson 11, Theorem A.2.1.1 proves that the resulting proper one-point-fibre map on the Schubert support of \(\lambda=\alpha^\vee\) identifies its raw IC and its actual weight compact complexes with those of \(I_{2\omega^\vee}\). This map and the root coordinates in that proof are defined over \(\mathbf F_q\): the adjoint quotient preserves the pinned split root parameters, the integer loop exponents and their multiplication formulas. The unit, counit, proper-support composition and intermediate-extension comparisons therefore intertwine Frobenius. An inseparable comparison can multiply an orientation generator by a nonzero power of the ground characteristic; it still intertwines the operator, so does not change an eigenvalue.

The target \(I_{2\omega^\vee}\) has three one-dimensional weights, by A.2.1.1. Its positive raising operator joins both adjacent lines nontrivially, by the characteristic-zero rank-one module calculation. Equation (A.1.2) and the already computed target automorphism therefore make the ratios of consecutive eigenvalues \(q\). Its highest line is \(E(-2)\), with eigenvalue \(q^2\), by the split extremal affine chart in Lesson 11, Lemma B.1.1.1. Its three raw eigenvalues are thus \(q^2,q,1\).

The comparison transfers these eigenvalues to \(I_{\alpha^\vee}^M\). Its three labels are \(\alpha^\vee,0,-\alpha^\vee\), inside the original lattice. Its positive and negative root operators are nonzero; conjugation by the displayed Frobenius therefore scales them by \(q,q^{-1}\). Since the derivative of this representation is injective on each nonzero one-dimensional root line, equation (A.1.2) proves the same scalars for \(d\Phi_M\). This step retains a nonprimitive coroot and does not require lifting projective integral frames.

Now let \(M_i\) be the standard rank-one Levi of \(G\). The actual constant-term functor and the fibre-functor comparison \(\gamma:H_{M_i}\operatorname{CT}_{M_i}^G\to H_G\) are exact symmetric monoidal by Lesson 11, Theorems A.3.2.1 and A.4.1.1. Each comparison there consists of the represented parabolic correspondence, its actual hyperbolic unit/counit, component shifts, strict fusion extension and the displayed integer sign. All are defined over \(\mathbf F_q\). Field pullback identifies those correspondences and those maps, and equality off the diagonal extends by intermediate-extension uniqueness. In particular the square comparing their cohomology transports \(u_G,u_{M_i}\) commutes.

The all-test-algebra group homomorphism \(j_i:\widehat M_i\to\widehat G\) supplied by \(\gamma\) therefore satisfies \[ \Phi_Gj_i=j_i\Phi_{M_i}. \tag{A.2.1} \] This follows directly by inserting (A.1.3) in the \(\gamma\)-square and then taking inverses; it does not assume that raw IC Weil structures commute with constant term. Its restriction to the common weight torus is the identity, by Lesson 11, Theorem A.4.2.1.

Consequently \(dj_i\) takes the \(\pm\alpha_i^\vee\) root lines into the corresponding lines of \(\widehat G\). Both images are nonzero: if either were zero, their bracket would be zero, whereas their bracket in \(\widehat M_i\) is its nonzero coroot tangent vector, which \(dj_i\) carries to that same tangent vector in the common torus. All Lie algebras here are over characteristic-zero \(E\). Each map on those one-dimensional lines is thus an isomorphism. Equation (A.2.1) transfers \(q,q^{-1}\) to those simple lines of \(\widehat G\). Lemma A.1.1.1 already fixes the torus. \(\square\)

A.3. All weight Frobenius operators are scalars

Theorem A.3.1.1. Put \(d_\lambda=\langle2\rho_G,\lambda\rangle\) and \(h_\nu=\langle2\rho_G,\nu\rangle\). On the actual raw IC weight space, \[ F_\lambda\bigm|F_\nu(I_\lambda) =a^{d_\lambda+h_\nu}\operatorname{id} =q^{(d_\lambda+h_\nu)/2}\operatorname{id}. \tag{A.3.1} \] On \(\mathcal I_\lambda=I_\lambda(d_\lambda/2)\) it is \(a^{h_\nu}\operatorname{id}\). This is an equality of operators, including the assertion that no nontrivial unipotent part occurs. The canonical normalized Weil structures form an exact symmetric tensor lift of the geometric Satake heart. The result extends by actual scalar extension to every algebraic coefficient field containing \(a\).

Proof. Since \(2\rho_G\) is an integral character of \(T\), it is a cocharacter of the dual torus. Define \(t=(2\rho_G)(a)\in\widehat T(E)\). Thus \[ \nu(t)=a^{h_\nu},\qquad \beta_i(t)=a^2=q. \tag{A.3.2} \] Lemma A.2.1.1 identifies the differentials of \(\Phi_G\) and \(\operatorname{Ad}(t)\) on the torus and all simple positive and negative root lines. We check the actual group maps.

Choose a faithful finite-dimensional linear representation of the split dual group, as supplied by its affine finite-type reconstruction in Lessons 9–11. For a root parametrization \(x_i:\mathbf G_a\to\widehat G\), let \(N\) be its derivative in that representation. Every polynomial group homomorphism \(A:\mathbf G_a\to GL(V)\) satisfies \(A(s+r)=A(s)A(r)\). Differentiate in \(r\) at zero to obtain \(A'(s)=A(s)N\), with \(A(0)=1\). Coefficient comparison in characteristic zero gives \(A(s)=\sum_{n\ge0}s^nN^n/n!\); since \(A\) is polynomial this sum is finite. Applied to \(\Phi_Gx_i\), whose derivative is \(qN\), it gives \(\Phi_Gx_i(s)=x_i(qs)\). For the negative simple parameter it gives \(x_{-i}(q^{-1}s)\). These are polynomial identities and hold over every coefficient algebra.

The two automorphisms therefore agree on the torus and all simple root groups. They agree on the rank-one Weyl words made from these groups. Conjugation by those words gives every root group from a simple positive or negative one, by the proved root and normalizer construction of Root data, Weyl chambers and the Bruhat decomposition, §§4–5 and The isomorphism theorem and construction of split groups, §§1–2. The open Bruhat cell is a product of these root groups and the torus. The fppf word-generation argument in The isomorphism theorem and construction of split groups, Lemma 4.1 shows that these cell points generate the entire group over arbitrary test algebras locally in the fppf topology. Equality there descends, since the target is a group scheme. Thus \[ \Phi_G=\operatorname{Ad}(t) \tag{A.3.3} \] as group-scheme automorphisms; no finite central subgroup has been discarded.

For \(V_\lambda=H_G(I_\lambda)\), equations (A.1.2) and (A.3.3) show that \(\rho_\lambda(t)^{-1}F_\lambda\) commutes with the full group. The module is absolutely simple by Lesson 11, Theorem B.4.1.1, so that commutant is \(E\): it is the endomorphism algebra of its actual simple IC under the full faithful equivalence. Hence \(F_\lambda=c_\lambda\rho_\lambda(t)\).

To determine the scalar, use the highest weight. The split extremal chart of Lesson 11, Lemma B.1.1.1 is \(\mathbf A^{d_\lambda}\) over \(\mathbf F_q\), not merely an affine space after an unspecified extension. Its ordered root coefficients and integer loop exponents are all defined over \(\mathbf F_q\). The raw IC restricts there to \(E[d_\lambda]\). The iterated actual affine-line compact trace gives the highest line \(E(-d_\lambda)\), whose geometric Frobenius is \(q^{d_\lambda}=a^{2d_\lambda}\). Meanwhile \(t\) acts on that line by \(a^{d_\lambda}\). Therefore \(c_\lambda=a^{d_\lambda}\), proving (A.3.1). Every weight differs from \(\lambda\) by integral coroots, so \(d_\lambda+h_\nu\) is even. Multiplying by the half-twist eigenvalue \(a^{-d_\lambda}\) gives the normalized assertion.

Full-heart semisimplicity gives the canonical evaluation decomposition \(P=\bigoplus_\lambda I_\lambda\otimes_E\operatorname{Hom}(I_\lambda,P)\), with finite support in \(\lambda\). Pullback is \(E\)-linear. Applying each normalized IC Weil map and the identity on its displayed multiplicity space consequently defines a natural normalized Weil map \(\alpha_P^0:\sigma^*P\to P\), without choosing bases. Its total cohomology operator is \(\rho_P(t)\) by the proved simple calculation.

For \(P*Q\), the operator supplied by the tensor product of these Weil maps is, under the actual monoidal \(H\)-comparison, \[ \rho_P(t)\otimes\rho_Q(t)=\rho_{P*Q}(t). \] This is also the operator of the canonical normalized map \(\alpha_{P*Q}^0\). Since \(u\) is monoidal, the two original sheaf maps have identical images under \(H\). Faithfulness detects their equality. The unit and symmetry comparisons are likewise the original ones transported by \(\sigma^*\), so are Weil morphisms. Thus the normalized family is a genuine symmetric tensor lift. This says nothing about an arbitrary additional Weil action on the multiplicity spaces of a general Weil object.

Finally every finite tensor diagram and its actual maps extend by Lesson 11, Theorems D.3.1.1 and D.4.2.1. Scalar extension carries each operator equality and each normalized Weil tensor map to the algebraic coefficient field. \(\square\)

Theorem A.3.1.1 first identifies the entire group automorphism, then uses the simple IC commutant and the highest split affine slice. The raw operator is \(a^{d_\lambda+h_\nu}\); its half twist gives \(a^{h_\nu}\). Faithful cohomology proves the normalized Weil maps are tensor maps before passing to functions. Editable SVG source.

A.4. Two normalization checks

Exercise A.4.1 (intermediate). Explain why the raw IC Weil structures are not already a tensor family in the \(GL_2\) product, and compute the normalized operators on its three IC summands.

Solution. The actual raw decomposition is \(I_{10}*I_{10}=I_{20}\oplus I_{11}(-1)\). Its central summand has Frobenius \(q\), whereas the canonical raw point \(I_{11}\) has Frobenius \(1\). They are the same geometric point object with different Weil structures. The two half twists give a total \((1)\), so the normalized decomposition is \(\mathcal I_{20}\oplus\mathcal I_{11}\). On \(\mathcal I_{10}\), its two weights have operators \(a,a^{-1}\). On \(\mathcal I_{20}\), the three weights have operators \(q,1,q^{-1}\); on the normalized point the operator is \(1\). Thus the tensor-square has \(q,1,1,q^{-1}\), exactly the normalized sum.

Exercise A.4.2 (advanced). For the geometric group \(PGL_2\), keep the nonprimitive dual root and check the group automorphism, the central subgroup and the normalized minuscule operators.

Solution. Its dual is \(SL_2\), with character lattice generated by \(\omega^\vee\), root \(2\omega^\vee\), and centre \(\mu_2=D(\mathbf Z\omega^\vee/\mathbf Z(2\omega^\vee))\). The element \(t\) is \(\operatorname{diag}(a,a^{-1})\); its root value is \(a^2=q\). Conjugation scales \(E_{12}\) by \(q\) and \(E_{21}\) by \(q^{-1}\), and fixes the entire torus, including \(\mu_2\). The raw minuscule operators on the highest and lowest lines are \(q,1\), and its half twist multiplies them by \(a^{-1}\), giving \(a,a^{-1}\). The scalar \(a\) between the raw operator and \(\rho(t)\) is essential. Replacing the root \(2\omega^\vee\) by its primitive ray would lose both this root value and the full centre.

Appendix B. Compact trace and the full spherical algebra

B.0. Rational semi-infinite points

Lemma B.0.1.1 (rational semi-infinite points). The rational points of the geometric semi-infinite orbit \(S_\nu\) are exactly \(N(\mathbf F_q((t)))t^\nu K/K\). On a bounded Schubert support this identifies the finite-field stalk sum with the coset sum in Lesson 1, Proposition 4.1.

Proof. Order the positive roots by increasing height, refining ties arbitrarily. The proved ordered root coordinates and commutator formulas make the height filtration triangular: a commutator of coordinates of heights \(r,s\) has height at least \(r+s\). Put \(k_\alpha=\langle\alpha,\nu\rangle\). The stabilizer is \[ H_\nu=t^\nu N(\overline{\mathbf F}_q[[t]])t^{-\nu}; \] its ordered coordinates are \(t^{k_\alpha}\overline{\mathbf F}_q[[t]]\). These thresholds respect commutators because \(k_{\alpha+\beta}=k_\alpha+k_\beta\). Right multiplication by its height-\(r\) coordinates changes those height-\(r\) coordinates additively and changes only larger heights otherwise. Successively remove the terms of degree at least \(k_\alpha\) in each coordinate. Every right coset consequently has a representative whose \(\alpha\)-coordinate is a finite Laurent polynomial of degrees strictly below \(k_\alpha\).

This representative is unique. At the smallest height where two such representatives differ, the coordinate of their quotient is the difference of the corresponding Laurent polynomials. Membership of that quotient in \(H_\nu\) puts the difference in \(t^{k_\alpha}\overline{\mathbf F}_q[[t]]\), while its degrees are strictly smaller; hence it is zero. Induction over the heights proves equality of every coordinate. All operations and thresholds are defined over \(\mathbf F_q\), so Frobenius fixes a coset precisely when it fixes every coefficient of this unique representative. Such coefficients lie in \(\mathbf F_q\). This proves the assertion on geometric points and therefore on rational points of the reduced orbit. A bounded Schubert intersection is finite type; its rational point set is finite. Applying the same normal form there gives the claimed equality of sums. \(\square\)

B.1. Compact support with actual rational coefficients

Theorem B.1.1.1. Let \(X_0/\mathbf F_q\) be separated of finite type, \(X=X_0\times\overline{\mathbf F}_q\), and let \(K_0\) be a bounded constructible actual rational-adic \(E\)-complex, with finite \(E/\mathbf Q_\ell\) and \(\ell\nmid q\). Use its descended geometric Frobenius correspondence. Then \[ \sum_i(-1)^i\operatorname{Tr} (F;H_c^i(X,K)) =\sum_{x\in X_0(\mathbf F_q)} \sum_i(-1)^i\operatorname{Tr} (F_x;H^i(K_{\bar x})). \tag{B.1.1} \] Every group in the cohomological sum is finite dimensional and only finitely many are nonzero. The compact image here is the actual rational proper-support functor of Lesson 5, Theorem O.4.2.1. In particular it applies to the supported semi-infinite IC intersection in Theorem 5.1.

Proof. We first close the inherited finite constructibility input in the all-dimensional trace argument. Constructible complexes on algebraic varieties, Proposition3.2 proves bounded proper-support constructibility, finite geometric stalks and the fibre bound for every finite ring killed by a power of \(\ell\). It explicitly proves the result from curve finiteness and finite root-free affine projections, without importing its companion's general constructibility assertion. This includes \(O_n=O/\varpi^n\), where \(O\) is the integer ring of \(E\). Thus the finite central constructibility premise declared in §1 of The trace formula in all dimensions is a proved programme input at precisely the finite rings needed here.

Here is the resulting finite trace induction, including its coefficient class. Suppose \(C_0\) is bounded constructible over \(O_n\), with Tor amplitude \([a,b]\). For a separated finite-type \(f_0:Y_0\to Z_0\), the just proved finite constructibility gives bounded constructible \(Rf_{0!}C_0\). For any right \(O_n\)-module \(M\), its geometric stalk tensor is \[ M\otimes_{O_n}^{\mathbf L} R\Gamma_c(Y_{\bar z},C_{\bar z}) =R\Gamma_c(Y_{\bar z}, M\otimes_{O_n}^{\mathbf L}C_{\bar z}). \tag{B.1.2} \] This is the actual coefficient projection map, proved for arbitrary \(M\) by the bounded acyclic resolution and finite cohomological-dimension argument in Traces of perfect complexes and Lefschetz numbers, Theorem4.3. Its finiteness premise is now Proposition3.2 above; its dimension bound for arbitrary prime-to-characteristic torsion is the compact-support dimension theorem used there. The spectral sequence on the right puts its degrees in \([a,b+2c]\), if the geometric fibre dimensions are at most \(c\). This proves finite Tor amplitude of \(Rf_{0!}C_0\); the perfectness criterion of that same theorem applies at a geometric point.

For dimension at most one, The trace formula for curves, Theorem1.1 proves the finite trace formula, including singular and nonreduced schemes and this perfect finite coefficient scope. Its smooth projective completion input has a programme proof in Constructible complexes on algebraic varieties, §§O.1, P.3 and AG.1–AG.4: normalize the projective closure of a smooth affine curve, using the proved finite normalization; normal curve local rings are DVRs and, over a perfect field, the proved Jacobian argument makes their points smooth. The original smooth open is unchanged. Pullback of the hyperplane bundle under the finite normalization is ample by its affine section opens, so a very ample power makes this proper completion projective. The finite boundary is retained. Over \(\mathbf F_q\), the perfect closure in §O.1 is already the ground field, so these completion maps are defined over \(\mathbf F_q\).

For higher dimension, partition the scheme into affine opens in successive closed remainders and use the actual Frobenius-preserved compact localization filtration. A dimension-\(d\) affine piece has a finite map to \(\mathbf A^d_{\mathbf F_q}\), by the finite-field monic-variable proof of Lemma3.2 in the all-dimensional trace lesson. Projection onto \(\mathbf A^{d-1}\) has geometric fibres of dimension at most one. Apply induction to its proper-support image, whose constructible finite-Tor scope was just checked. Compact base change and composition identify its fibre operator with the original fibre's geometric Frobenius operator; this actual operator comparison is Lemma2.1 of that lesson, constructed from the descended coefficient map. The curve formula on each rational fibre then gives \[ \operatorname{Tr}(F;R\Gamma_c(Y,C)) =\sum_{z\in\mathbf A^{d-1}(\mathbf F_q)} \sum_{y\in Y_{0,z}(\mathbf F_q)} \operatorname{Tr}(F_y;C_{\bar y}). \] The rational fibres partition the rational points. Trace additivity along the affine-piece filtration proves the formula for the original scheme. This reconstructs Theorem4.1 of the all-dimensional trace lesson with its finite constructibility premise supplied by a programme proof.

Next take an integral lisse lattice \(L_0\) with finite free \(O\)-stalks. Its finite quotients \(L_{0,n}\) are finite locally free over \(O_n\), so have Tor amplitude zero. Put \(C_n=R\Gamma_c(X,L_n)\). The preceding proof gives perfect \(O_n\)-complexes and their finite trace identities. Their actual coefficient projection maps give \[ C_{n+1}\otimes_{O_{n+1}}^{\mathbf L}O_n=C_n. \] These maps preserve the descended Frobenius operators. L-adic sheaves and their cohomology, Theorem6.1 constructs a single bounded finite free \(O\)-complex for this compatible system and its endomorphism. The completed sections and proper-support reduction comparisons of Lesson 5, Lemma M.4.1.1 and Theorem O.4.2.1 identify its derived limit with the actual integral compact-support complex; the coefficient comparison (O.4.2.2) in Lesson 5, Theorem O.4.2.1 and Lemma N.1.1.1 identify its rationalization with the actual \(E\)-compact image. No discrete constant-\(E\) replacement is made.

The perfect-complex trace on this finite free \(O\)-model reduces to the finite-level trace for every \(n\). The rational point set is finite, and each local trace is the reduction of its finite free \(O\)-stalk operator. Separatedness \(\bigcap_n\varpi^nO=0\) therefore gives equality of the two integral traces. Tensoring with \(E\) gives (B.1.1) for \(L_0\otimes_OE\); the finite free model also gives bounded finite-dimensional cohomology. The equality concerns traces of compatible matrices, not an interchange of an unbounded sum with a limit.

For general \(K_0\), refine its finite cohomology stratification to smooth connected locally closed schemes over \(\mathbf F_q\). The perfect ground field gives a dense smooth open in every reduced component; delete component intersections, then repeat on the smaller-dimensional complement. This is a finite process. Nilpotents do not change the coefficient sites. On such a smooth stratum, the actual rational lisse coefficient corresponds to a continuous representation of the profinite étale fundamental group by The pro-étale site and l-adic complexes, Theorem7.10 and Corollary7.11. Its image is compact. Choose a basis lattice \(\Lambda\subset E^r\). Compactness bounds every translated lattice inside \(\varpi^{-b}\Lambda\). The sum of its translates is consequently a finite \(O\)-module between \(\Lambda\) and \(\varpi^{-b}\Lambda\), invariant under the whole arithmetic fundamental group. It is torsion-free and spans \(E^r\), so is an integral lisse lattice defined on this stratum over \(\mathbf F_q\). In particular it retains Frobenius, not just the geometric local system.

Apply the lattice case to every cohomology local system on every stratum. The finite truncation filtration supplies the ordinary cohomological signs. Arrange the finite strata as opens in closed remainders and apply compact localization. All restrictions, truncation maps and support maps are defined over \(\mathbf F_q\), so their filtrations are Frobenius-preserved. Additivity of alternating traces from Lemma 1.1 proves (B.1.1) for their actual extension complex \(K_0\), without descending its attaching maps to a single integral lattice. The finite filtrations also give bounded finite cohomology.

For \(\mathcal I_\lambda\), the constant open \(E\)-coefficient, its intermediate extension and the supported intersection have \(\mathbf F_q\)-models. The half-twist line also does: \(a\) is an \(\ell\)-adic unit, and the homomorphism \(\mathbf Z\to O^\times\) sending \(1\) to \(a^{-1}\) extends continuously to \(\widehat{\mathbf Z}\). Indeed in every finite quotient of \(O^\times\) its image is finite cyclic, so the homomorphism factors through the corresponding finite cyclic quotient of \(\widehat{\mathbf Z}\); these maps are compatible. Thus the normalized Weil coefficient is a genuine continuous finite-field coefficient. Its supported restriction is bounded constructible by the actual pullback and closed-support operations. Theorem B.1.1.1 applies and proves the compact trace assertion in Theorem 5.1. \(\square\)

Theorems B.1.1.1–B.2.1.1 keep the complete coefficient passage, the sign from degree \(h_\nu\), and the measure factor \(a^{-h_\nu}\). Lemma B.0.1.1 identifies the finite-field points with the required cosets. The rank-one induced module proves multiplication before Corollary B.3.1.1 deduces the normalized IC convolution identity. Editable SVG source. The normalization can also be compared with Frenkel's free lectures, §5.3, cited in §5.

B.2. The general Satake character formula and the algebra map

Theorem B.2.1.1. For every dominant \(\lambda\), \[ \mathcal S(t_{\mathcal I_\lambda}) =(-1)^{d_\lambda}\operatorname{ch}V_\lambda. \tag{B.2.1} \] The transform of Lesson 1 is an algebra isomorphism from the spherical Hecke algebra to \(E[X_*(T)]^{W_G}\), with the positive square root \(a\) and its stated normalization.

Proof. The actual compact concentration theorem, the full Satake weight identification, Theorem A.3.1.1 and Theorem B.1.1.1 supply the three assertions used in Theorem 5.1. Its coefficient calculation gives (B.2.1). In particular the sign is retained; it comes from \(h_\nu\equiv d_\lambda\pmod2\).

We prove multiplicativity directly from the Iwasawa partition, before using a function-sheaf convolution comparison. Put \(\Gamma=G(\mathbf F_q((t)))\), \(K=G(\mathbf F_q[[t]])\), \(A=E[L]\), and write \(e^\nu\) for its group-algebra basis. The split Iwasawa decomposition and its unique coweight from Lesson 1, §§2–4 identify \(\Gamma=B(\mathbf F_q((t)))K\). If \(b=nt^\nu t_0\), with \(n\in N(\mathbf F_q((t)))\) and \(t_0\in T(\mathbf F_q[[t]])\), put \(\eta(b)=a^{-h_\nu}e^\nu\). This is multiplicative: the torus normalizes \(N\), and the two torus valuations add. It is trivial on \(B\cap K\).

Let \(\mathcal V\) be the \(A\)-module of functions \(\phi:\Gamma\to A\) satisfying \(\phi(bg)=\eta(b)\phi(g)\). Its right-\(K\)-invariant part is free of rank one. The normalized generator is \(\phi_0(bk)=\eta(b)\). It is well-defined because two such expressions differ by an element of \(B\cap K\); Iwasawa shows that every value of a right-\(K\)-invariant function is determined by its value at \(1\). No analytic representation theorem is used.

For a spherical function \(f\) and \(\phi\in\mathcal V^K\), set \[ (T_f\phi)(x)=\sum_{gK\in\Gamma/K} f(g)\phi(xg). \tag{B.2.2} \] The sum is finite by the finite-jet Cartan coset calculation of Lesson 1. It is independent of the representatives because \(\phi\) is right-\(K\)-invariant. Left \(B\)-equivariance follows by pulling \(\eta(b)\) outside the sum. Left \(K\)-invariance of \(f\), by replacing \(g\) with \(k^{-1}g\), makes the output right-\(K\)-invariant. Thus \(T_f\) acts by a scalar on this rank-one module. Evaluation at \(1\), followed by the Iwasawa partition, gives exactly \[ (T_f\phi_0)(1) =\sum_\nu a^{-h_\nu} \sum_{gK\in N(\mathbf F_q((t)))t^\nu K/K} f(g)\,e^\nu =\mathcal S(f). \tag{B.2.3} \] This is Lesson 1's normalized coefficient formula. Lemma B.0.1.1 identifies this sum with the finite-field semi-infinite sum used in (5.2).

For spherical \(f,h\), expand \(T_fT_h\) using the two finite sums. Regroup the products \(gzK\). The coefficient of a fixed coset \(yK\) is \(\sum_{gK}f(g)h(g^{-1}y)\), precisely the convolution with \(\operatorname{vol}K=1\). Bi-\(K\)-invariance makes each representative change harmless. Hence \(T_fT_h=T_{f*h}\), and their scalars give \(\mathcal S(f*h)=\mathcal S(f)\mathcal S(h)\).

The trace functions form a basis by the finite triangular calculation of Lesson 12 §1. The characters \(V_\lambda\) form a basis of \(E[L]^{W_G}\): orbit sums form a basis, and the highest-weight calculation gives \(\operatorname{ch}V_\lambda\) its orbit sum with coefficient1 and only smaller dominant labels. Below each dominant label there are finitely many labels in the same component, by the proved root-order bound of Lesson 4, §3. Finite triangular inversion gives both spanning and independence. Formula (B.2.1) sends one basis to the other, with invertible signs. It therefore proves that \(\mathcal S\) has exactly this invariant image and is bijective onto it. The multiplicative calculation completes the algebra isomorphism. \(\square\)

B.3. Convolution of normalized IC traces

Corollary B.3.1.1. For dominant \(\lambda,\mu\), \[ t_{\mathcal I_\lambda*\mathcal I_\mu} =t_{\mathcal I_\lambda}*t_{\mathcal I_\mu} =\sum_\tau \dim\operatorname{Hom}_{\widehat G} (V_\tau,V_\lambda\otimes V_\mu)\, t_{\mathcal I_\tau}. \tag{B.3.1} \] The decomposition in the first expression has the canonical normalized Weil structure. Its finite multiplicity spaces have Frobenius1. Raw IC multiplicity twists must still be retained separately.

Proof. Full geometric Satake and semisimplicity give the finite IC decomposition with these representation multiplicities. Theorem A.3.1.1 proves equality of the tensor Weil map with the canonical normalized map on that same underlying object. Its evaluation decomposition is therefore a Weil decomposition with identity on its multiplicity spaces. Taking stalk traces gives the first expression equal to the displayed sum.

Every nonzero constituent is in the component \(\lambda+\mu\), since convolution adds the component labels. Thus \(d_\tau\equiv d_\lambda+d_\mu\pmod2\), by the even pairing of \(2\rho_G\) with each coroot. Apply Theorem B.2.1.1 to the sum: its transform is \[ (-1)^{d_\lambda+d_\mu} \operatorname{ch}(V_\lambda\otimes V_\mu) =\mathcal S(t_{\mathcal I_\lambda}) \mathcal S(t_{\mathcal I_\mu}). \] The algebra homomorphism and injectivity just proved identify it with the convolution of the two functions. This establishes the function-sheaf convolution identity for these IC objects without assuming a general rational lifting theorem for chain frames. \(\square\)

B.4. Trace and convolution exercises

Exercise B.4.1 (intermediate). For the geometric group \(PGL_2\), compute the compact trace of the normalized minuscule \(\mathcal I_{\omega^\vee}=E_{\mathbf P^1}[1](1/2)\). Obtain its two Satake coefficients from the open affine line and the closed point, keeping their signs.

Solution. The raw projective-line compact cohomology has \(E\) in degree \(-1\) and \(E(-1)\) in degree \(1\). Its half twist gives geometric Frobenius \(a^{-1},a\), and both degrees are odd. The compact trace is therefore \(-a^{-1}-a\). At each of the \(q+1\) rational points the normalized stalk is \(E[1](1/2)\), with trace \(-a^{-1}\); their sum is \(-(q+1)a^{-1}=-a^{-1}-a\). The highest semi-infinite intersection is the affine line with \(q\) rational points and \(h_{\omega^\vee}=1\). Its normalized Satake coefficient is \(a^{-1}q(-a^{-1})=-1\). The lowest intersection is a point with \(h_{-\omega^\vee}=-1\), and its coefficient is \(a(-a^{-1})=-1\). Thus \(\mathcal S(t_{\mathcal I_{\omega^\vee}})=-(e^{\omega^\vee}+e^{-\omega^\vee})\), as the odd dimension requires.

Exercise B.4.2 (advanced). Give \(P=\mathcal I_\lambda\otimes_E M\) and \(Q=\mathcal I_\mu\otimes_E N\) Weil maps with respective multiplicity operators \(B\) and \(C\). Deduce their convolution trace identity from B.3.1.1 and determine the Weil operator on each IC multiplicity space.

Solution. Their trace functions are \(\operatorname{Tr}(B)t_{\mathcal I_\lambda}\) and \(\operatorname{Tr}(C)t_{\mathcal I_\mu}\), by Lemma1.1. In the canonical IC decomposition of \(\mathcal I_\lambda*\mathcal I_\mu\), its representation-theoretic multiplicity space has the identity Weil operator by B.3.1.1. The extra tensor multiplicity has operator \(B\otimes C\), so the full operator on the \(\tau\)-multiplicity space is \(1\otimes B\otimes C\). Its trace is \(N_{\lambda\mu}^\tau\operatorname{Tr}(B)\operatorname{Tr}(C)\), as a tensor product basis verifies even for nonsemisimple operators. B.3.1.1 and bilinearity of function convolution therefore give \(t_{P*Q}=t_P*t_Q\). Identity multiplicity Frobenius in the normalized IC theorem refers to the canonical IC tensor family; an additional given Weil operator is retained.

Appendix C. Split Tate Frobenius on IC stalks and costalks

C.1. Cycle classes remove the affine-paving extension ambiguity

Lemma C.1.1.1. Let \(Y_0/\mathbf F_q\) be proper of finite type with a finite closed filtration whose differences are split affine spaces \(\mathbf A^{e_r}_{\mathbf F_q}\). Nilpotents are allowed. For finite \(E/\mathbf Q_\ell\), \(\ell\nmid q\), \[ H^{2j+1}(Y,E)=0,\qquad H^{2j}(Y,E)=E(-j)^{\,c_j}, \tag{C.1.1} \] as actual Frobenius modules, where \(c_j\) is the number of cells of dimension \(j\). In particular geometric Frobenius is exactly \(q^j\operatorname{id}\), including when \(c_j>1\).

Proof. Use the actual structural dualizing complex \(\Omega_Z=a_Z^!E\) from Lesson 5, Theorems P.2.4.1 and P.3.4.1, and define \[ \operatorname{BM}_m(Z,E)=H^{-m}(Z,\Omega_Z). \tag{C.1.2} \] The actual dual exchange of Lesson 5, Theorem P.3.3.1, applied to the structure map and the constant coefficient, identifies this with the linear dual of \(H_c^m(Z,E)\), compatibly with the contragredient geometric Frobenius action. Lesson 5, Lemma Q.4.1.1 and its proof bound \(H_c^m(Z,E)\) by \(0\le m\le2\dim Z\). Thus \(\operatorname{BM}_m(Z,E)=0\) for \(m>2\dim Z\).

For an open affine cell \(U=\mathbf A^e\), the specified smooth orientation and iterated affine-line unit give \(\Omega_U=E(e)[2e]\) and \(R\Gamma(U,E)=E\). Its sole Borel–Moore group is consequently \[ \operatorname{BM}_{2e}(U,E)=E(e), \tag{C.1.3} \] with geometric Frobenius \(q^{-e}\). This retains the smooth counit normalization, not just the dimension of the group.

Let \(C\) be the reduced closure of this cell in the proper scheme containing it. It is integral of dimension \(e\), and \(D=C\setminus U\) has dimension at most \(e-1\), by the proved finite-type dimension formula. Dualizing the ordinary constant-coefficient localization triangle gives \[ Ri_*\Omega_D\longrightarrow\Omega_C \longrightarrow Rj_*\Omega_U. \tag{C.1.4} \] The two groups \(\operatorname{BM}_{2e}(D,E)\) and \(\operatorname{BM}_{2e-1}(D,E)\) vanish by the dimension bound. Restriction is therefore an isomorphism \[ \operatorname{BM}_{2e}(C,E) \xrightarrow{\sim}\operatorname{BM}_{2e}(U,E)=E(e). \tag{C.1.5} \] Define the cell-closure fundamental class as its inverse map from \(E(e)\). The closure, boundary, orientation, triangle and inverse are all defined over \(\mathbf F_q\); hence this is a Frobenius-compatible map. Its proper push to the ambient scheme is the actual counit \(Ri_*\Omega_C\to\Omega_Y\), obtained from exceptional composition for the closed immersion.

Apply (C.1.4) to each stage of the closed affine paving. Induction gives zero odd Borel–Moore groups and short exact sequences \[ 0\longrightarrow\operatorname{BM}_{2j}(Y_{r-1},E) \longrightarrow\operatorname{BM}_{2j}(Y_r,E) \longrightarrow\operatorname{BM}_{2j}(\mathbf A^{e_r},E) \longrightarrow0. \tag{C.1.6} \] When \(e_r=j\), the proper image of that cell's closure class restricts to the generator of the right-hand \(E(j)\). When \(e_r\ne j\), the right-hand group is zero. The classes for all dimension-\(j\) cells thus give a basis of \(\operatorname{BM}_{2j}(Y,E)\). Each is the image of the same Frobenius line \(E(j)\). Frobenius is consequently \(q^{-j}\operatorname{id}\) on the whole group, not merely on its associated graded. Duality gives (C.1.1), because \(Y\) is proper. Nilpotent invariance of the actual coefficient sites and proper-support images applies throughout. No purity theorem for a singular closed immersion is used. \(\square\)

Lemma C.1.1.1 constructs each class by the actual supported triangle and the dimension bound. The resulting basis consists of maps from \(E(j)\); it proves the scalar operator on the whole group. Editable SVG source. The actual duality and affine trace proofs are in Lesson 5, Appendices P–Q; the free parity framework is Juteau–Mautner–Williamson, cited below.

C.2. Compressing a scalar Frobenius operator

Lemma C.2.1.1. Let \(I=j_{!*}E[d]\) be the actual simple IC on an \(\mathbf F_q\)-support, and let \(B\) have its descended Weil map. Suppose underlying derived maps \[ I\xrightarrow{u}B\xrightarrow{v}I,\qquad vu=1 \tag{C.2.1} \] both restrict to the identity on the dense open, where \(B\) also restricts to \(E[d]\) with the identity coefficient Weil map. At an \(\mathbf F_q\)-point \(x\), if Frobenius on \(H^i(B_{\bar x})\) is the scalar \(b_i\), then its canonical raw action on \(H^i(I_{\bar x})\) is exactly \(b_i\operatorname{id}\). The maps \(u,v\) are not required to be Weil maps.

Proof. Write \(\sigma^*\) for the geometric-field pullback and \(\alpha_B:\sigma^*B\to B\) for its actual Weil map. Its restriction on the displayed open is the identity on the raw constant coefficient. The composite \[ v\alpha_B\sigma^*u:\sigma^*I\longrightarrow I \tag{C.2.2} \] has the same dense-open restriction as the canonical raw IC Weil map \(\alpha_I\). Both source and target are perverse intermediate extensions. A morphism between them with zero open restriction has boundary-supported image, which their intermediate-extension characterization excludes. Thus (C.2.2) is exactly \(\alpha_I\).

At the rational point, field pullback and the point restriction have their canonical cohomology transport \(\tau_P:H(i_x^*\sigma^*P)\to H(i_x^*P)\). To construct it, the rational-point square has \(\sigma_Xi_x=i_x\sigma_S\), with \(S=\operatorname{Spec}\overline{\mathbf F}_q\). Actual pullback composition gives \(i_x^*\sigma_X^*=\sigma_S^*i_x^*\). The coefficient category on this geometric field point is the actual coefficient-module category, and field pullback there is canonically the identity, preserving that coefficient action. This supplies the displayed transport, naturally for all underlying derived maps; it does not assert that a geometric map descends to the finite field. The stalk version of the transport in A.1.1.1 has \[ F_P=H(\alpha_P)\tau_P^{-1},\qquad \tau_BH(\sigma^*u)=H(u)\tau_I. \tag{C.2.3} \] Substitution of (C.2.2) therefore gives \[ F_I=H(v)F_BH(u). \tag{C.2.4} \] In degree \(i\), this is \(b_iH^i(v)H^i(u)=b_i\operatorname{id}\), by (C.2.1). Naturality of the field transport is essential; this is not an assertion that the two underlying maps intertwine Frobenius separately. The same argument works after a common Weil twist. \(\square\)

C.3. All IC stalks have split Tate Frobenius

Theorem C.3.1.1. For every dominant \(\lambda\) and every spherical label \(\mu\) in its support, the actual raw IC stalk at \(t^\mu\) has \[ H^i(i_\mu^*I_\lambda)= E\bigl(-(i+d_\lambda)/2\bigr)^{\,b_{\lambda\mu,i}}. \tag{C.3.1} \] It vanishes if \(i+d_\lambda\) is odd. The equality is one of actual Frobenius modules, so every nonzero operator is \(q^{(i+d_\lambda)/2}\operatorname{id}\). The normalized IC stalk has operator \(a^i\operatorname{id}\) in degree \(i\). The raw costalk has the same scalar formula in its own degree, and the normalized costalk has scalar \(a^i\). These statements extend by actual scalar extension to algebraic coefficient fields containing \(a\).

Proof. Use the rootwise resolution \(f_\lambda:\mathcal B_\lambda\to Z_\lambda\) and \(B_\lambda=Rf_{\lambda*}E[d_\lambda]\) of Lesson 11, Lemma C.2.1.1. Its wall charts, integer root exponents, Weyl representatives, root actions and boundary filtrations are defined over \(\mathbf F_q\). In that proof's fibre induction at the rational endpoint \(t^\mu\), the closed/open pieces are the earlier fibre and its product with a split affine line. The auxiliary flag paving uses the split Borel selected by the signs of \(\mu+\varepsilon\gamma\); all its root subgroups and representatives are defined over \(\mathbf F_q\). It therefore gives a closed paving of the reduced proper fibre by split \(\mathbf F_q\)-affine cells, including in characteristic two.

Lesson 5, Theorem O.4.1.1, with its descended Frobenius operator, identifies \[ H^i(i_\mu^*B_\lambda) =H^{i+d_\lambda}(f_\lambda^{-1}(t^\mu),E). \tag{C.3.2} \] Lemma C.1.1.1 makes its Frobenius exactly \(q^{(i+d_\lambda)/2}\) when this degree is even, and makes it zero when the degree is odd. This is where the cell-closure cycle classes eliminate a possible unipotent extension between equal-dimensional cells.

Full geometric IC parity is already proved in Lesson 11, Theorem C.4.1.1. Lesson 11, Corollary C.4.2.1 consequently supplies the actual underlying derived maps (C.2.1) from that same resolution image, with both dense restrictions the identity. Apply Lemma C.2.1.1. It transfers the displayed scalar in each stalk degree to the canonical raw IC Weil action. In an odd resolution degree the stalk is a retract of zero. Thus (C.3.1) follows. Continuity and density of geometric Frobenius in the finite-field Galois group identify the scalar module with the displayed Tate line; all exponents are integers. The half twist multiplies its eigenvalue by \(a^{-d_\lambda}\), yielding \(a^i\).

The actual smooth orientation on the open orbit identifies the raw dual there with \(E(d_\lambda)[d_\lambda]\), compatibly with its Frobenius correspondence. Dual exchange and intermediate-extension uniqueness give the Weil identity \(DI_\lambda=I_\lambda(d_\lambda)\), so \(D\mathcal I_\lambda=\mathcal I_\lambda\). Point dual exchange consequently identifies the normalized costalk group in degree \(i\) with the dual of the normalized stalk group in degree \(-i\). Its contragredient scalar is \((a^{-i})^{-1}=a^i\). Removing the half twist gives the raw costalk scalar \(a^{i+d_\lambda}=q^{(i+d_\lambda)/2}\). Finally Lesson 11, Theorem D.3.1.1 carries these actual stalk/costalk and operator maps through algebraic scalar extension. \(\square\)

Lemma C.2.1.1 uses the dense-open identity and natural field transport to prove the compression formula. Theorem C.3.1.1 then identifies every raw and normalized stalk and costalk operator, retaining the degree, Tate sign and half twist. Editable SVG source. The retraction comes from the actual parity proof in Lesson 11, Appendix C; compare the odd-Hom and lifting framework in the free Parity sheaves, Corollaries 2.8–2.9.

C.4. Two scalar checks

Exercise C.4.1 (intermediate). At the central point of the \(GL_2\) Schubert surface, compare the raw resolution image \(B_{20}\), the raw IC \(I_{20}\), and their normalized stalk operators.

Solution. Here \(d_{20}=2\), and the resolution fibre is \(\mathbf P^1\). Thus \(B_{20}\) has stalk \(E\) in degree \(-2\) and \(E(-1)\) in degree \(0\), with operators \(1,q\). The actual cone IC has only \(E\) in degree \(-2\), with operator \(1=q^{(-2+2)/2}\). The common half twist \((1)\) multiplies by \(q^{-1}\): the normalized resolution stalk has \(q^{-1},1\), and its normalized IC retract has \(q^{-1}=a^{-2}\) in degree \(-2\). The other degree-zero term is the normalized central point summand.

Exercise C.4.2 (advanced). Why does a filtration with every quotient \(E(-j)\) fail to prove scalar Frobenius, and what additional map in C.1.1.1 proves it?

Solution. On \(E^2\), the operator \(q^j\begin{pmatrix}1&1\\0&1\end{pmatrix}\) preserves a line and acts by \(q^j\) on that line and the quotient, yet is not scalar. C.1.1.1 constructs Frobenius-compatible maps from the actual line \(E(j)\) to the Borel–Moore group, one for each cell closure. The maps form a basis, so they identify the entire group with \(E(j)^{c_j}\) as an operator module. Contragredient duality then makes cohomology \(E(-j)^{c_j}\). The cycle maps, rather than just the filtered quotients, exclude this Jordan block.

Appendix D. Orbit indicators and the graded IC formula

Keep \(F=\mathbf F_q((t))\), \(O=\mathbf F_q[[t]]\), the positive Iwahori \(I\subset K=G(O)\), and \(\operatorname{vol}K=1\). Let \(\mathcal H=C_c(I\backslash G(F)/I)\), \(T_x=(\operatorname{vol}I)^{-1}1_{IxI}\), and \(M=C_c(T(O)N(F)\backslash G(F)/I)\). Write \(v_x\) for the quotient-point indicators and \(A=E[L]\). The following proof derives the full indicator normalization and then every graded IC multiplicity. The free Iwahori-Hecke Algebras, §§1.3–1.15, by Thomas Haines, Robert Kottwitz and Amritanshu Prasad, supplies comparison material for the universal module and the rank-one integral.

D.1. The unramified module and its finite basis

Lemma D.1.1.1. The map \(h\mapsto v_1h\) identifies the right regular Iwahori Hecke module with \(M\). The commuting normalized torus action embeds \(E[L]\) in \(\mathcal H\). Writing its monomials as \(\theta_\nu\), the elements \(\theta_\nu T_w\), \(\nu\in L,w\in W\), form a basis; for dominant \(\xi\), \(\theta_\xi=a^{-d_\xi}T_{t^\xi}\).

Proof. Work over \(F=\mathbf F_q((t))\), \(O=\mathbf F_q[[t]]\), with \(I\) the inverse image of the positive Borel in \(K=G(O)\). Keep \(\operatorname{vol}K=1\). The integrals here are finite coset counts, or finite sums of additive-root cosets with \(\operatorname{vol}O=1\). Their rational values embed in \(E\). Put \[ T_x=(\operatorname{vol}I)^{-1}1_{IxI},\qquad \mathcal H=C_c(I\backslash G(F)/I). \] Lesson 1, Theorem 2.C proves the affine double-coset decomposition, root-coordinate cells, one-wall product supports and extended alcove stabilizers. Thus \[ \int T_x=q^{\ell(x)}. \tag{D.1.1} \] There are \(q^{\ell(x)}\) right or left \(I\)-cosets in its root cell. For a simple wall \(s\), the zero/nonzero root parameter gives support of \(T_s^2\) in \(I\cup IsI\). Its value at the identity is \(q/(\operatorname{vol}I)\); its integral is \(q^2\). These determine \[ T_s^2=(q-1)T_s+qT_1. \tag{D.1.2} \] For a length-increasing wall product, its single-double-coset support and total mass determine coefficient one. For a descent, write \(T_x=T_sT_{sx}\) and apply (D.1.2). Right-wall statements follow by inversion. A length-zero representative normalizes \(I\) and acts without a scalar. This proves every multiplication rule used below in every characteristic.

The triangular regular module.

Let \(M=C_c(T(O)N(F)\backslash G(F)/I)\), where support is finite in the displayed discrete quotient. Refine Lesson 1, Lemma 4.0's rational Iwasawa decomposition by split finite-flag Bruhat cells: \[ T(O)N(F)\backslash G(F)/I =\{t^\nu w:\nu\in L,\ w\in W\}. \tag{D.1.3} \] The coweight is unique by the Iwasawa position. Once it is fixed, the integral representatives differ on the left by \(N(F)\cap K=N(O)\) and \(T(O)\). Reduction then gives the unique finite Bruhat label. The integral intersection is Lesson 1, Theorem 2.C's integral big-cell and root-coordinate test. This is a bijection over \(\mathbf F_q\), not just after algebraic closure. Write \(v_x\) for its point indicators and \(v_1=1_{T(O)N(F)I}\).

Right convolution makes \(M\) an \(\mathcal H\)-module. The map \[ \mathcal H\longrightarrow M,\qquad h\longmapsto v_1h \tag{D.1.4} \] is an isomorphism. Choose a reduced wall word for the affine part of \(y\), with its length-zero factor at the end. If \(v_1T_y(x)\ne0\), an element of \(T(O)N(F)xI\) belongs to \(IyI\). Absorb the unit torus into \(I\), obtaining \(nx\in IyI\). Choose integral strictly dominant \(\zeta\) large enough that \(t^\zeta nt^{-\zeta}\in I\). Every parameter of this one \(n\) has bounded pole order; multiples of the sum of positive coroots suffice. Thus \(It^\zeta xI\) occurs in \(It^\zeta IyI\).

Apply the right one-wall support rule along the word for \(y\). Every output is \(t^\zeta y'\), with \(y'\) a subword product followed by the same length-zero factor. Disjointness gives \(x=y'\). Omitting a wall leaves length less than \(\ell(y)\); retaining all walls gives \(x=y\). Only finitely many subwords occur, all in the same component as \(y\). The coefficient at \(y\) is nonzero because the left coset \(Iy\) contributes one. The other contributions are nonnegative rational numbers, so cannot cancel in \(E\).

Injectivity follows by the maximal length in a finite relation. Surjectivity follows by induction on length: subtract the diagonal multiple of \(v_1T_y\), then express the shorter indicators. At length zero there are no shorter labels. The central component group need not be finite.

Two exact special cases are \[ v_1T_w=v_w\ (w\in W),\qquad v_1T_{t^\xi}=v_{t^\xi}\ (\xi\text{ dominant}). \tag{D.1.5} \] For the first, \(I=N(O)T(O)N^-(tO)\). Move the last factor past the integral \(w\); it stays in the first congruence subgroup. Hence \(T(O)NIwI=T(O)NwI\). Also \(T(O)NI\cap K=I\), by the integral-coordinate test. At \(w\), exactly one left \(I\)-coset contributes, because the conjugated right factor must lie in that intersection.

For the second, dominance gives \(t^\xi N(O)t^{-\xi}\subset N(O)\) and \(t^{-\xi}N^-(tO)t^\xi\subset N^-(tO)\), so the support is \(T(O)Nt^\xi I\). In the ordered big-cell coordinates of \(t^\xi It^{-\xi}\), positive-root thresholds are \(\langle\alpha,\xi\rangle\ge0\), negative-root thresholds are \(1-\langle\alpha,\xi\rangle\). Membership in \(T(O)NI=N(F)T(O)N^-(tO)\) additionally forces every negative-root coordinate to have level at least one. Its intersection with \(t^\xi It^{-\xi}\) therefore lies in \(I\). As before exactly one left coset contributes at \(t^\xi\), giving coefficient one. A trailing right \(I\) does not alter this intersection test, since \(T(O)NI\) is right-\(I\)-stable.

The torus algebra.

Let \(A=E[L]\). Define a commuting left action on \(M\) by \[ e^\nu f(g)=a^{-h_\nu}f(t^{-\nu}g),\qquad e^\nu v_x=a^{-h_\nu}v_{t^\nu x}. \tag{D.1.6} \] Torus translation preserves the left quotient and commutes with right convolution. The regular-module isomorphism embeds \(A\) in \(\mathcal H=\operatorname{End}_{\mathcal H}(M)\); denote the image of \(e^\nu\) by \(\theta_\nu\). It is injective because the translated \(v_1\)'s are distinct. Equations (D.1.5) give \[ \theta_\xi=a^{-d_\xi}T_{t^\xi}\quad(\xi\text{ dominant}),\qquad v_1\theta_\nu T_w=a^{-h_\nu}v_{t^\nu w}. \tag{D.1.7} \] Thus the \(\theta_\nu T_w\), \(\nu\in L,w\in W\), are an \(E\)-basis; \(\mathcal H\) is free as a left \(A\)-module on the \(T_w\). The group law and inverses follow from the torus action. Writing \(\nu=\xi-\zeta\) with both dominant gives \(\theta_\nu=\theta_\xi\theta_\zeta^{-1}\), independently of the choice. No list of minuscule alcove stabilizers is needed. ∎

D.2. The entire simple-coroot calculation

Lemma D.2.1.1. For every simple positive \(\alpha\), every \(\nu\in L\), and the whole coroot \(\beta=\alpha^\vee\), the actual operators satisfy the Bernstein relation (D.2.6). Its quotient is the finite full-coroot string, including when \(\beta\) is nonprimitive in \(L\).

Proof. Fix simple positive \(\alpha\), put \(\beta=\alpha^\vee\), \(s=s_\alpha\), and choose its pinned \(\dot s\). Define \[ \mathscr I_\alpha f(g)=\int_F f(\dot s^{-1}u_\alpha(x)g)\,dx, \qquad \operatorname{vol}O=1. \tag{D.2.1} \] For \(v_1\) this integral is finite at every \(g\). First establish its quotient invariances. Positive roots other than \(\alpha\) form the unipotent radical \(R_\alpha\) of the minimal parabolic; \(s\) permutes them positively and \(R_\alpha\) is normal in \(N=U_\alpha R_\alpha\). For left \(R_\alpha\), commute its factor through \(u_\alpha(x)\) and \(\dot s^{-1}\), then absorb it in the left \(N\). For left \(u_\alpha(c)\), substitute \(x+c\); for \(T(O)\), rescale \(x\) by a root-character unit. These preserve measure and compact support. Right \(I\)-invariance is immediate. Evaluate now at (D.1.3).

If \(\dot s^{-1}u_\alpha(x)t^\nu w\in T(O)NI\), then \(w\in\{1,s\}\). Its first factor lies in \(P_\alpha(F)\); a parabolic is closed, so \(P_\alpha(F)\cap K=P_\alpha(O)\). Finite flag reduction gives those two labels. Project to \(M_\alpha\), killing \(R_\alpha\) and sending \(I\cap P_\alpha\) to the positive rank-one Iwahori. In its two-cell Bruhat normal form, take the torus valuation modulo \(\mathbf Z\beta\). This defines a group homomorphism: when a root factor changes cells, the rank-one Gauss identity changes the torus by \(\beta(z)\); when a Weyl representative passes a torus, its valuation changes by \(\nu-s\nu=\langle\alpha,\nu\rangle\beta\). Neither change affects the class. The two root subgroups and the unit torus therefore map to zero. The rank-one Iwahori is their product with negative parameters in \(tO\), so also maps to zero. Applying this homomorphism to the asserted membership gives \(\nu=0\) in \(L/\mathbf Z\beta\), or \(\nu=j\beta\).

On these labels the computation is in the root \(SL_2\). Its possible central kernel does not change the additive root parameters, whose pinned maps are coordinate isomorphisms. Nor does it change the unit-torus test: a coroot valuation \(j\beta\) is zero only when \(j=0\), since \(\beta\ne0\) in the free lattice \(L\). The unique ordered Gauss coordinates thus have exactly the same integral thresholds in the Levi. No surjectivity on points of a central isogeny is assumed. Up to a unit torus factor, \[ \dot s\,u_\alpha(x)t^{j\beta} =\begin{pmatrix}0&-t^{-j}\\t^j&xt^{-j}\end{pmatrix}. \tag{D.2.2} \] A determinant-one rank-one matrix with bottom row \((c,d)\) belongs to \(T(O)NI\) exactly when \(c\in tO\), \(d\in O^\times\): positive Gauss form proves necessity; left upper-root elimination gives a unit torus and lower parameter \(c/d\in tO\), proving sufficiency.

For \(w=1\) this gives \(j\ge1\) and \(x\in t^jO^\times\), of measure \(q^{-j}(1-q^{-1})\). For \(w=s\), the bottom row is \((xt^{-j},-t^j)\); the test gives \(j=0\), \(x\in tO\), of measure \(q^{-1}\). All other cases vanish. Every fixed-label support is therefore compact. We obtain \[ \mathscr I_\alpha v_1=q^{-1}v_s+ (1-q^{-1})\sum_{j\ge1}q^{-j}v_{t^{j\beta}} =q^{-1}v_s+(1-q^{-1})\sum_{j\ge1}e^{j\beta}v_1. \tag{D.2.3} \] The last equality uses \(h_\beta=2\), and retains the whole \(\beta\).

For \(f=v_1h\), right convolution is a finite coset sum. Interchanging it with the integral is valid because each summand has the compact support just proved. This defines \(\mathscr I_\alpha\) on all \(M\) and proves right-\(\mathcal H\)-linearity. Its values lie in the formal completion along \(\mathbf Z_{\ge0}\beta\): (D.2.3) has this support, and \(h\) acts \(A\)-linearly by a finite matrix in the finite \(W\)-basis. Every series coefficient is finite; no analytic continuation is used.

For \(n=\langle\alpha,\nu\rangle\), move \(t^{-\nu}\) past \(\dot s^{-1}\), then substitute \(y=t^nx\). The measure changes by \(q^n\); meanwhile \(h_{s\nu}=h_\nu-2n\), so \(a^{-h_\nu}q^n=a^{-h_{s\nu}}\). Hence \[ \mathscr I_\alpha e^\nu=e^{s\nu}\mathscr I_\alpha. \tag{D.2.4} \] Set \(J_\alpha=(1-e^\beta)\mathscr I_\alpha\). The geometric series cancels: \[ J_\alpha v_1=q^{-1}(1-e^\beta)v_s+(1-q^{-1})e^\beta v_1. \] Regularity and right linearity show that \(J_\alpha\) preserves \(M\) and corresponds to \[ J_\alpha=q^{-1}(1-\theta_\beta)T_s+ (1-q^{-1})\theta_\beta. \tag{D.2.5} \] Equation (D.2.4) gives \(J_\alpha\theta_\nu=\theta_{s\nu}J_\alpha\). Substitute (D.2.5), rearrange, and cancel \(1-\theta_\beta\). Cancellation is valid in the free left \(A\)-module \(\mathcal H\), since \(A\) is a Laurent-polynomial domain. This proves \[ T_s\theta_\nu-\theta_{s\nu}T_s =(q-1)\frac{\theta_\nu-\theta_{s\nu}}{1-\theta_{-\beta}}. \tag{D.2.6} \] For \(n>0\) the quotient is \(\sum_{j=0}^{n-1}\theta_{\nu-j\beta}\); for \(n<0\) it is \(-\sum_{j=1}^{-n}\theta_{\nu+j\beta}\); for \(n=0\), zero. The relation is in the actual group algebra, including nonprimitive coroots, all central factors and every finite-field characteristic. ∎

Lemma D.2.1.1 gives the exact additive measures and the normalized torus factor. The displayed \(PGL_2\) quotient has \(\beta=2\gamma\); its two summands are the full-coroot steps. Editable SVG source. The rank-one matrix identities are proved in Lesson 1; compare the free unramified-module exposition cited below.

D.3. The finite spherical symmetrizer

Lemma D.3.1.1. Under \(\mathcal H e_K=E[L]\), the spherical averaging operator is (D.3.2), with \(N_+=|\widehat\Phi^+|\) and \(W(q)=\sum_wq^{\ell(w)}\). It preserves the full lattice group algebra.

Proof. Put \(W(z)=\sum_{w\in W}z^{\ell(w)}\), \(N_+=|\widehat\Phi^+|\). Split finite flag cells give \([K:I]=W(q)\), so \[ e_K=1_K=\frac1{W(q)}\sum_{w\in W}T_w. \] The left module \(\mathcal H e_K\) is freely generated over \(A\) by \(e_K\). Spanning follows from \(T_we_K=q^{\ell(w)}e_K\). Independence follows in \(M\): \(v_1e_K=W(q)^{-1}\sum_wv_w\) is a nonzero vector in a free \(A\)-module. Identify \(\theta_\nu e_K\) with \(e^\nu\). The proved Bernstein relation gives \[ T_sf=q\,s(f)+(q-1)\frac{f-s(f)}{1-e^{-\beta}}. \tag{D.3.1} \] In the fraction field of \(A\), write \(x=e^{-\beta}\). This operator is \(A_s+B_ss\), where \[ A_s=\frac{q-1}{1-x},\qquad B_s=\frac{1-qx}{1-x}. \] For \(\mathsf M=\sum_wT_w\), pair \(w\) with \(sw\); the quadratic and length rules give \(T_s\mathsf M=q\mathsf M\). Since \(q-A_s=B_s\ne0\), this implies \(s\mathsf M=\mathsf M\). All coefficients in its finite crossed-product expression are therefore Weyl translates of the identity coefficient.

Distinct field automorphisms are linearly independent as operators. Otherwise, in a shortest nonzero relation \(\sum c_\sigma\sigma=0\), choose an element separating two automorphisms, apply the relation to its product with an arbitrary argument, and subtract one of its scalar multiples. A shorter nonzero relation results. The same proof applies to their restrictions to \(A\), using a separating monomial and an arbitrary argument in \(A\). The Weyl action on \(L\) is faithful, because \(L\) contains the full coroot lattice spanning the root space. Consequently operator identities known on \(A\) are identities in the finite crossed product over its fraction field; no prior localization of the Hecke action is assumed.

The coefficient of \(w_0\) in \(\mathsf M\) comes only from \(T_{w_0}\): omitting a reflection from a reduced word gives length less than \(N_+\). Its reduced product gives coefficient \[ \prod_{\beta>0}\frac{1-qe^{-\beta}}{1-e^{-\beta}}, \] because the successive inversion roots are exactly the positive roots. Applying \(w_0\) gives the identity coefficient \(\prod_{\beta>0}(q-e^{-\beta})/(1-e^{-\beta})\). Thus \[ e_Kf=\frac{q^{N_+}}{W(q)} \sum_{w\in W}w\left( f\prod_{\beta>0}\frac{1-q^{-1}e^{-\beta}}{1-e^{-\beta}}\right). \tag{D.3.2} \] Its left side preserves \(A\). This is proved from actual finite Hecke operators; neither root primitivity nor a separate formal braid claim is assumed. ∎

D.4. The normalized Satake functional

Lemma D.4.1.1. The finite sum \(\mathcal L(f)=\sum_{zK}f(z)\phi_0(z)\) identifies \(\theta_\nu e_K\) with \(e^\nu\) for every \(\nu\in L\). On spherical functions it is the Satake transform of Appendix B.

Proof. Let \(\phi_0(nt^\nu k)=a^{-h_\nu}e^\nu\), Appendix B's rational induced generator. For compact left-\(I\), right-\(K\) functions put \[ \mathcal L(f)=\sum_{zK}f(z)\phi_0(z). \] For dominant \(\xi\), \(It^\xi K\subset Nt^\xi K\), by negative-root contraction as in D.1.5. All its support has \(\phi_0=a^{-d_\xi}e^\xi\); its integral is \(a^{d_\xi}\). Therefore \[ \mathcal L(\theta_\xi e_K)=e^\xi. \tag{D.4.1} \] To extend this to arbitrary \(\xi\), let \(z\) range over finitely many representatives of the support of a compact \(f\). Choose a congruence subgroup \(K_r\subset zKz^{-1}\) for all of them. A finite pole bound for \(z,z^{-1}\) in a faithful affine matrix realization proves existence: \(z^{-1}(1+t^rA)z\) is integral once \(r\) exceeds the sum of the pole bounds. Closedness of \(G\) in the matrix group then makes it a \(G(O)\)-point. Choose strictly dominant \(\zeta\) large enough that \(t^{-\zeta}N^-(tO)t^\zeta\subset K_r\).

For \(i=n_+t_0n_-\in I\), absorb its contracted last factor on the right of \(t^\zeta z\). The first factors lie in \(N(F)T(O)\), on which \(\phi_0\)'s character is one. Thus \[ \phi_0(it^\zeta z)=a^{-d_\zeta}e^\zeta\phi_0(z). \] Interchange the finite coset sum defining \(\theta_\zeta f\) with the finite sum over \(zK\). Its right \(I\)-factor is absorbed by the left \(I\)-invariance of \(f\). The preceding identity and \(\int\theta_\zeta=a^{d_\zeta}\) give \[ \mathcal L(\theta_\zeta f)=e^\zeta\mathcal L(f). \tag{D.4.2} \] Apply this to \(f=\theta_\xi e_K\), also making \(\zeta+\xi\) dominant. The translation group law and (D.4.1) show \(\mathcal L(\theta_\xi e_K)=e^\xi\) for every \(\xi\). Hence \(\mathcal L\) is exactly the polynomial-module identification. On spherical functions it is the Satake transform, by Appendix B's finite coset formula. ∎

D.5. Every orbit indicator

Theorem D.5.1.1. For every dominant \(\lambda\), \(\mathcal S(c_\lambda)=a^{d_\lambda}H_\lambda(q^{-1})\), where \(H_\lambda\) is the exact dual-root, full-lattice Hall–Littlewood polynomial (6.5). The formula retains every central direction and holds over every finite-field extension.

Proof. For dominant \(\lambda\), the function \(e_K\theta_\lambda e_K\) is bi-\(K\)-invariant and supported exactly on \(Kt^\lambda K\). Positivity under the chosen embedding \(a=\sqrt q>0\) rules out cancellation. Its integral is \(a^{d_\lambda}\). Consequently \[ e_K\theta_\lambda e_K =\frac{a^{d_\lambda}}{\#\mathrm{Gr}^\lambda(\mathbf F_q)}c_\lambda. \tag{D.5.1} \] Let \(W_\lambda\) be the stabilizer, and \(N_\lambda\) the number of positive roots pairing to zero with \(\lambda\). The root subsystem is generated by the simple roots pairing to zero. Finite flag cells and Lesson 1, Theorem 3.1 give \[ \#\mathrm{Gr}^\lambda(\mathbf F_q) =q^{d_\lambda-(N_+-N_\lambda)}\frac{W(q)}{W_\lambda(q)} =q^{d_\lambda}\frac{W(q^{-1})}{W_\lambda(q^{-1})}. \tag{D.5.2} \] The longest-element bijection sends length \(r\) to \(N_+-r\): its action reverses the sign of every positive root, complementing the inversion set. Apply the same argument to \(W_\lambda\) for the second equality. The stabilizer and its coset-length decomposition are the finite chamber argument: walls containing \(\lambda\) generate its stabilizer; the minimal coset representative sends the subsystem's positive roots positively, so multiplication by an element of \(W_\lambda\) adds precisely that element's inversions. Thus the flag-cell count is the displayed quotient, with no divisibility assumption.

Apply \(\mathcal L\) to (D.5.1). The left side is (D.3.2) applied to \(e^\lambda\). Using \(W(q)=q^{N_+}W(q^{-1})\) and (D.5.2), cancel the nonzero rational scalars. This proves \[ \mathcal S(c_\lambda) =a^{d_\lambda}\frac1{W_\lambda(q^{-1})} \sum_{w\in W}w\left( e^\lambda\prod_{\beta>0} \frac{1-q^{-1}e^{-\beta}}{1-e^{-\beta}}\right) =a^{d_\lambda}H_\lambda(q^{-1}). \tag{D.5.3} \] Here \(H_\lambda\) is exactly (6.5). All steps persist over \(\mathbf F_{q^n}\), replacing \(q,a\) by \(q^n,a^n\). ∎

D.6. The graded IC stalk polynomial

Theorem D.6.1.1. In the arithmetic setting of §1, for dominant \(\mu\le\lambda\), the raw IC stalks satisfy (6.3), equivalently (D.6.1). The relevant degrees have the stated parity and vanishing, and the coefficients are nonnegative integers.

Proof. Write \(I_\lambda=\mathrm{IC}^{\mathrm{raw}}_\lambda\). In the arithmetic setting of §1, take dominant \(\mu\le\lambda\) and put \[ g_{\lambda\mu}(u)=\sum_{r\ge0} \dim_E H^{-d_\mu-2r}(i_\mu^*I_\lambda)\,u^r. \] The perverse stalk bound is degree at most \(-d_\mu\); the delivered parity and resolution retraction give the lower bound \(-d_\lambda\). Since \(d_\lambda-d_\mu\) is even in a component, this is a finite polynomial with \(0\le r\le(d_\lambda-d_\mu)/2\). On a strict boundary orbit, intermediate extension gives the strict upper stalk bound, hence \(r\ge1\); on the open orbit \(g_{\lambda\lambda}=1\).

The actual normalized scalar in degree \(-d_\mu-2r\), proved in Theorem C.3.1.1, is \(a^{-d_\mu}q^{-r}\). Every degree has sign \((-1)^{d_\lambda}\), so \[ t_{\mathcal I_\lambda}(t^\mu) =(-1)^{d_\lambda}a^{-d_\mu}g_{\lambda\mu}(q^{-1}). \] Expand the finite-support trace function in the indicators \(c_\mu\), apply (D.5.3), and use Theorem B.2.1.1. The signs cancel, giving \[ \operatorname{ch}V_\lambda =\sum_{\mu\le\lambda}g_{\lambda\mu}(q^{-1})H_\mu(q^{-1}). \] The triangular polynomials \(H_\mu\) remain a basis after specialization: their top coefficient is one and \(W_\mu(q^{-1})\) is a nonzero positive rational number. The proved character transition (6.11) therefore makes \(g_{\lambda\mu}(q^{-1})=m^\mu_\lambda(q^{-1})\).

Repeat over every \(\mathbf F_{q^n}\), with half twist \(a^n\). The geometric IC and stalk dimensions are unchanged, while Frobenius is the \(n\)-th power of the actual scalar operator. Hence equality holds at every \(u=q^{-n}\). These are infinitely many distinct characteristic-zero numbers. A nonzero polynomial has at most its degree many roots, by linear-factor division and induction. It follows that \[ m^\mu_\lambda(u) =\sum_{r\ge0}\dim_E H^{-d_\mu-2r}(i_\mu^*I_\lambda)\,u^r =u^{(d_\lambda-d_\mu)/2} \sum_{j\ge0}\dim_E H^{-d_\lambda+2j}(i_\mu^*I_\lambda)\,u^{-j}. \tag{D.6.1} \] The second equality changes index by \(r=(d_\lambda-d_\mu)/2-j\). This proves (6.3), including parity, vanishing and nonnegative coefficients. The actual algebraic coefficient-extension theorem in Lesson 11, Theorem D.3.1.1 retains these stalks and their dimensions after extending the coefficient field algebraically. ∎

Theorems D.5.1.1–D.6.1.1 retain the full root datum, degree bounds, shift sign and formal variable. The value \(q^{-n}\) comes from geometric Frobenius over \(\mathbf F_{q^n}\); the dimensions stay fixed. Editable SVG source. The actual stalk and compact-trace providers are Appendices B–C; the Hall–Littlewood transition is proved in §6.

D.7. Exercises

Exercise D.7.1 (medium). In \(G=PGL_2\), let \(\gamma\) be the class of \(\operatorname{diag}(t,1)\); then \(\beta=\alpha^\vee=2\gamma\). Compute the Bernstein quotient for \(\nu=\gamma,-\gamma,2\gamma\), and the matrix of \(T_s\) on the span of \(e^\gamma,e^{-\gamma}\). Verify its quadratic relation.

Solution. The root pairings are \(1,-1,2\). The three quotients in D.2.6 are respectively \(e^\gamma,-e^\gamma,e^{2\gamma}+1\). These are obtained by dividing the differences by \(1-e^{-2\gamma}\), using the finite geometric sums. With \(z=e^\gamma\), the polynomial-module formula gives \(T_sz=(q-1)z+qz^{-1}\) and \(T_sz^{-1}=z\). In the ordered basis \((z,z^{-1})\) the matrix is \(\begin{pmatrix}q-1&1\\q&0\end{pmatrix}\). Multiplication gives \(\begin{pmatrix}(q-1)^2+q&q-1\\q(q-1)&q\end{pmatrix}=(q-1)T_s+q\,1\). Replacing the denominator by \(1-e^{-\gamma}\) would give extra odd steps and would change this actual operator. The root-coordinate integral dictates the whole coroot.

Exercise D.7.2 (advanced). For \(GL_2\), \(\lambda=(2,0)\), \(\mu=(1,1)\), compute the Hall–Littlewood transition, the graded central IC stalk and the two spherical normalizations in D.5.1 and D.5.3.

Solution. Write \(x=e^{(1,0)}\), \(y=e^{(0,1)}\). The two Weyl summands in (6.5), with stabilizer denominator one for \((2,0)\), give \(H_{(2,0)}(u)=x^2+y^2+(1-u)xy\). The central polynomial is \(H_{(1,1)}(u)=xy\). Since \(\operatorname{ch}\operatorname{Sym}^2=x^2+xy+y^2\), its transition is \(H_{(2,0)}(u)+uH_{(1,1)}(u)\). Thus \(m^{(1,1)}_{(2,0)}(u)=u\). Here \(d_\lambda=2\), \(d_\mu=0\), so the stalk has one raw group in degree \(-2\), no others. Its raw Frobenius is one and its normalized Frobenius is \(a^{-2}=q^{-1}\). This agrees with the direct quadratic-cone IC proof of §2.

The open orbit has \(q(q+1)\) points by Lesson 1, Theorem 3.1. Hence \(e_K\theta_\lambda e_K=(q+1)^{-1}c_\lambda\), while \(\mathcal S(c_\lambda)=qH_\lambda(q^{-1})\). The polynomial-module image of the first expression is \(q(q+1)^{-1}H_\lambda(q^{-1})\), equal to the finite symmetrizer. Finally \(t_{\mathcal I_\lambda}=q^{-1}(c_\lambda+c_\mu)\) has transform \(H_\lambda(q^{-1})+q^{-1}xy=\operatorname{ch}\operatorname{Sym}^2\), retaining its positive shift sign.

Appendix E. Geometric character vanishing and top Whittaker cohomology

Work over an algebraically closed field of characteristic \(p>0\), with \(\ell\ne p\), and an algebraic characteristic-zero coefficient field containing the values of \(\psi:\mathbf F_p\to E^\times\). Keep \(I_\lambda=\mathrm{IC}^{\mathrm{raw}}_\lambda\), \(Z_\lambda=\mathrm{Gr}^{\le\lambda}\), \(h_\nu=\langle2\rho,\nu\rangle\), and the characters of (7.1). The free Whittaker patterns in the geometry of moduli spaces of bundles on curves, Theorem 1 and §7, by Frenkel, Gaitsgory and Vilonen, is comparison material for the conductor and top-degree statements. The proofs below supply the character vanishing and the actual cohomology maps.

Proper-support base change, projection, structural duality and affine trace are proved in Lesson 5, Theorems O.4.2.1, O.4.3.1, P.3.3.1, P.3.4.1 and Lemma Q.1.2.1. Algebraic coefficient extension retains them by Lesson 11, Theorem D.3.1.1.

E.1. The Artin–Schreier line and its affine vanishing

Lemma E.1.1.1. The unit-normalized Artin–Schreier line has the additive comparison (E.1.1), the affine vanishing (E.1.2), and the vector-space calculation (E.1.3).

Proof. The polynomial map \(a:\mathbf A^1_z\to\mathbf A^1_x\), \(x=z^p-z\), is a finite étale \(\mathbf F_p\)-torsor. Its algebra is free on \(1,z,\ldots,z^{p-1}\), its derivative is \(-1\), and two roots differ by an element of \(\mathbf F_p\). The translations \(z\mapsto z+c\) are its deck transformations. The character projectors \[ \frac1p\sum_{c\in\mathbf F_p}\eta(c)^{-1}[c] \] split \(a_*E\) into rank-one lisse summands. Let the deck operator [c] act by pullback under z↦z+c. Denote its ψ-eigensummand by \(\mathcal L_\psi\); its fibre sections satisfy f(z+c)=ψ(c)f(z). At zero choose the unit with f(0)=1. This fixes the trace convention: over \(\mathbf F_q\), a root over b satisfies z^q=z+Tr_{q/p}(b). Geometric Frobenius acts on fibre sections by precomposition with arithmetic Frobenius, so its scalar is \(\psi(\operatorname{Tr}_{q/p}b)\). The geometric cover is connected, since its total space is an affine line. Its monodromy therefore surjects onto the deck group, so this nontrivial summand is not constant.

The addition map has the actual character-line comparison \[ m^*\mathcal L_\psi \simeq \mathcal L_\psi\boxtimes\mathcal L_\psi. \tag{E.1.1} \] Indeed add roots \(z_1,z_2\) of \(z_i^p-z_i=x_i\). Their sum is a root over \(x_1+x_2\). Quotienting the product torsor by \((c,-c)\) makes this map an isomorphism of \(\mathbf F_p\)-torsors: both sides are finite étale of degree \(p\), and on each geometric fibre it is the bijection of root sums. The product of the two characters is \(\psi(c_1+c_2)\). Taking its associated line gives (E.1.1). Associativity, the zero unit and the two unit identities are induced by the corresponding identities for root addition, so are comparisons of actual maps.

There is an actual vanishing \[ R\Gamma_c(\mathbf A^1,\mathcal L_\psi)=0. \tag{E.1.2} \] Finite proper-support composition identifies \[ R\Gamma_c(\mathbf A^1_x,a_*E) =R\Gamma_c(\mathbf A^1_z,E)=E(-1)[-2]. \] The last equality, including its trace orientation, is Lesson 5, Lemma Q.1.2.1. Every deck translation acts as the identity on it. To check the map, compactify to \(\mathbf P^1\). Translation fixes infinity and pulls the degree-one line bundle to itself, so preserves the positive Kummer class and the infinity-section localization triangle used in that lemma. It therefore preserves its compact trace generator. Only the trivial character projector can act nontrivially on this one-dimensional group. The nontrivial projector's derived summand is zero, proving (E.1.2).

For an \(m\)-dimensional vector space and a linear form \(b\), it follows that \[ R\Gamma_c(\mathbf A^m,b^*\mathcal L_\psi)= \begin{cases}E(-m)[-2m],&b=0,\\0,&b\ne0.\end{cases} \tag{E.1.3} \] For \(b\ne0\), choose coordinates making \(b\) the first projection. The product calculation follows from the actual proper-support base-change and projection maps, not a dimension count. For \(b=0\), the chosen zero unit makes the line constant and one iterates the affine trace. The same argument after a local splitting proves relative vanishing for a vector bundle with a fibrewise nonzero linear form. ∎

E.2. A character-equivariant compact complex vanishes

Lemma E.2.1.1. If a separated finite-type \(\mathbf G_a\)-scheme carries a bounded complex with nontrivial additive-character equivariance, its entire compact complex vanishes.

Proof. Let \(X\) be separated of finite type and let \(\mathbf G_a\) act by \(b:\mathbf A^1\times X\to X\). If a bounded constructible \(F\) has an actual comparison \[ b^*F\simeq\mathcal L_\psi\boxtimes F, \tag{E.2.1} \] then \(R\Gamma_c(X,F)=0\).

Here is a proof retaining the map and all degrees. The morphism \[ (c,x)\longmapsto(c,c\cdot x) \] is an automorphism of \(\mathbf A^1\times X\), with inverse \((c,y)\mapsto(c,(-c)\cdot y)\); its second projection is \(b\). Thus the compact complex of \(b^*F\) equals that of the pullback of \(F\) under the ordinary projection. The latter is \[ E(-1)[-2]\otimes R\Gamma_c(X,F). \] On the other hand (E.2.1) and (E.1.2) make it zero. Both product identities are the counit-defined projection comparison of Lesson 5, Theorem O.4.3.1 and the product square in Lesson 5, Theorem O.4.2.1. Tensoring with the invertible Tate line and reversing the shift proves the assertion. No assertion that a connected group acts trivially on arbitrary cohomology is needed.

In particular a complex on a fixed point with nontrivial character equivariance is zero: apply this argument to the trivial action on that point. ∎

E.3. Every nondominant conductor in (7.2)

Proposition E.3.1.1. For \(\mu+\nu\) dominant and \(\mu\) nondominant, every group in (7.2) vanishes. A conductor-zero twisted-equivariant object also vanishes on every nondominant semi-infinite orbit.

Proof. Fix dominant \(\lambda\), and \(\mu,\nu\) with \(\mu+\nu\) dominant. Put \[ X_{\lambda\nu}=Z_\lambda\cap S_\nu,\qquad F_{\lambda\mu\nu} =i^*I_\lambda\otimes(\chi_\mu^\nu)^*\mathcal L_\psi . \] All schemes and operations are on this bounded finite-type intersection. If \(\mu\) is not dominant, every compact group of \(F_{\lambda\mu\nu}\) is zero.

Choose a simple root \(\alpha_i\) with \(n=\langle\alpha_i,\mu\rangle<0\), and put \(m=-n-1\ge0\). The morphism \[ c\longmapsto u_i(c\,t^m) \tag{E.3.1} \] is an additive subgroup of \(N(O)\). It preserves \(Z_\lambda\), by the \(G(O)\)-action, and preserves \(S_\nu\), by the \(N(F)\)-action. Its action on the bounded support is algebraic: the displayed root coordinate is polynomial in \(c\), and the positive-loop action factors through a finite jet on that support. Its character is exactly \[ \chi_\mu(u_i(c\,t^m)) =\operatorname{Res}(c\,t^{m+n}dt)=c. \] The other simple-root coordinates are zero. The descended character consequently satisfies \(\chi_\mu^\nu(c\cdot x)=c+\chi_\mu^\nu(x)\). The actual \(G(O)\)-equivariance of \(I_\lambda\), proved in Lesson 6, Theorem B.2.3.1, together with (E.1.1), gives (E.2.1) on this subgroup. Lemma E.2.1.1 proves vanishing in all degrees. This proves the entire nondominant-\(\mu\) branch of (7.2), without a semisimplicity or cleanness premise.

There is also the required local relevance obstruction for a character-equivariant object on the semi-infinite orbit \(S_\eta\) with conductor zero. If \(\eta\) is nondominant, choose \(i\) with \(\langle\alpha_i,\eta\rangle\le-1\). The subgroup \(u_i(c\,t^{-1})\) fixes \(t^\eta\), since conjugation by \(t^{-\eta}\) gives exponent \(-1-\langle\alpha_i,\eta\rangle\ge0\), but \(\chi_0=c\) on it. Restricting twisted equivariance to this fixed point and using Lemma E.2.1.1 forces the stalk to be zero. Translation to other points of the orbit forces every stalk there to vanish. ∎

Lemmas E.1.1.1–E.2.1.1 and Proposition E.3.1.1 retain the actual character line, subgroup, shifts and compact-support maps. Editable SVG source. The affine trace and proper-support comparisons are proved in Lesson 5, Appendices O–Q.

E.4. The full upper-degree bound and its actual top group

Theorem E.4.1.1. For \(X_{\lambda\nu}=Z_\lambda\cap S_\nu\), raw \(I_\lambda\), and the descended character line, compact degrees above \(h_\nu\) vanish. Its top group has the canonical open-orbit comparison (E.4.2).

Proof. For arbitrary \(\mu,\nu\) with descent, one has \[ H_c^r(X_{\lambda\nu},F_{\lambda\mu\nu})=0 \quad(r>h_\nu), \tag{E.4.1} \] and a canonical comparison \[ H_c^{h_\nu}(X_{\lambda\nu},F_{\lambda\mu\nu}) \simeq H_c^{d_\lambda+h_\nu} (\operatorname{Gr}^{\lambda}\cap S_\nu, (\chi_\mu^\nu)^*\mathcal L_\psi). \tag{E.4.2} \]

Stratify \(X_{\lambda\nu}\) by the finitely many \(\operatorname{Gr}^{\kappa}\cap S_\nu\), \(\kappa\le\lambda\). The dimension proof in Lesson 5, §§8–10 gives dimension at most \((d_\kappa+h_\nu)/2\). The perverse stalk bound gives ordinary degrees at most \(-d_\kappa\) for \(I_\lambda\) on this orbit. Twisting by a lisse line does not change that bound. Lesson 5, Lemma Q.4.1.1 applied to its ordinary cohomology sheaves, with the finite hypercohomology filtration, therefore bounds the total compact degree by \(h_\nu\). A finite sequence of open–closed triangles proves (E.4.1).

On a strict boundary orbit the intermediate extension bound is strictly less than \(-d_\kappa\). The actual parity theorem of Lesson 11, Theorem C.4.1.1 makes it at most \(-d_\kappa-2\): \(d_\lambda\) and \(d_\kappa\) have the same component parity. The boundary compact complex is thus zero above \(h_\nu-2\). The open–closed triangle for \(\operatorname{Gr}^{\lambda}\cap S_\nu\subset X_{\lambda\nu}\) consequently induces an isomorphism in degree \(h_\nu\). On that open orbit \(I_\lambda=E[d_\lambda]\), giving (E.4.2). This checks the possible connecting group as well as the upper-degree vanishing. ∎

E.5. Which top components survive the character

Theorem E.5.1.1. The top Whittaker group is (E.5.2), with one trace-oriented line for each top component on which the character is zero. Its raw Frobenius is \(q^{(d_\lambda+h_\nu)/2}\operatorname{id}\); after the half twist it is \(a^{h_\nu}\operatorname{id}\).

Proof. Let \(Y=\operatorname{Gr}^{\lambda}\cap S_\nu\), and \(D=(d_\lambda+h_\nu)/2\). If \(Y\) is nonempty, its top components have dimension \(D\), by Lesson 5's dimension proof. Each is stable under the constant torus: the closure of the image of the irreducible variety \(T\times K\) is irreducible and contains the component \(K\), so stays in that maximal irreducible component. Write \(f=\chi_\mu^\nu=\sum_i f_i\). On the orbit these regular functions have the distinct simple-root characters \[ f_i(t\cdot x)=\alpha_i(t)f_i(x). \tag{E.5.1} \] The coweight conjugation commutes with this constant torus. Each \(f_i\) descends by the same stabilizer test as (7.1).

On a connected smooth dense torus-stable open \(U\) of a reduced component \(K\), the line \(f^*\mathcal L_\psi\) is geometrically trivial if and only if \(f|_K=0\).

The nontrivial implication deserves a proof in every characteristic. The nontrivial character of \(\mathbf F_p\) is faithful. Trivial monodromy of its associated line therefore makes the pulled-back \(\mathbf F_p\)-torsor trivial. A section supplies a regular function \(g\) on \(U\) with \(g^p-g=f\). The action makes every regular function a finite sum of torus weights. To verify this also for nonaffine \(U\), use a finite affine cover. Since \(U\) is separated its pairwise intersections are affine. Flat tensoring with \(k[T]\) commutes with the equalizer computing global sections, so the action pullback of \(g\) lies in \(k[T]\otimes\Gamma(U,\mathcal O_U)\). Expansion in the torus characters, followed by the action's associativity, gives the finite weight decomposition.

In any nonzero \(p\)-chain of weights of \(g\), let \(p^a\gamma\) and \(p^b\gamma\) be the lowest and highest weights occurring. In \(g^p-g\) the first has nonzero coefficient \(-g_{p^a\gamma}\), and \(p^{b+1}\gamma\) has nonzero coefficient \(g_{p^b\gamma}^{p}\). The latter is nonzero because \(U\) is integral. Thus \(f\) would have two distinct weights related by a positive power of \(p\). But its nonzero weights are simple roots, and two distinct simple roots are not rational multiples of each other. A multiple \(p^r\alpha_i\), \(r>0\), is not a simple root either. This is a contradiction. All nonzero weights of \(g\) vanish; its zero-weight contribution gives \(f=0\), since \(f\) has no zero weight. Vanishing on the dense open is vanishing on the reduced component. Conversely \(f=0\) gives the unit-trivialized line. This argument retains nonprimitive roots in the full character lattice; it never divides a root character by \(p\).

Structural duality on the smooth \(D\)-dimensional open gives \[ H_c^{2D}(U,L) =H^0(U,L^{-1})^*\otimes E(-D). \] For a rank-one lisse line the right side vanishes unless its monodromy is trivial. The complement has dimension at most \(D-1\), so Lesson 5, Lemma Q.4.1.1 and localization identify this group with the top group of the component. Remove intersections between components and their singular loci; these are torus-stable lower-dimensional sets. Consequently \[ H_c^{h_\nu}(X_{\lambda\nu},F_{\lambda\mu\nu}) \simeq\bigoplus_{\substack{K\text{ a top component of }Y\\f|_K=0}} E(-D) \tag{E.5.2} \] as geometric coefficient lines, with their positive trace orientations. Smaller-dimensional components contribute no top group. If \(Y\) is empty, that group is zero. This assertion concerns the top degree only.

When the data are split over \(\mathbf F_q\), these selected zero-character component lines inject into the untwisted top group by their fundamental-class basis. The injection is independent of the dense open: deleting a lower-dimensional subset leaves the same top group and trace orientations. It is Frobenius-compatible, because the character-zero condition, its unit trivialization and the component permutation are all defined by the same data. Theorem A.3.1.1 says that the actual untwisted raw weight operator is \(q^D\operatorname{id}\). Its restriction to the selected subspace is therefore also \(q^D\operatorname{id}\), with no Jordan part. For normalized IC the operator is \(a^{h_\nu}\operatorname{id}\). A numerical trace vanishing in lower degrees is not used. ∎

E.6. Every highest-slice case, including its Weil line

Theorem E.6.1.1. For every dominant \(\mu,\lambda\) and \(\nu=\lambda\), the raw compact complex is \(E(-d_\lambda)[-d_\lambda]\). Its only degree is \(h_\lambda=d_\lambda\), and its multiplicity in (7.2) is one.

Proof. For dominant \(\mu\) and \(\nu=\lambda\), the whole intersection is the split extremal chart of Lesson 11, Lemma B.1.1.1: \[ Z_\lambda\cap S_\lambda\simeq\mathbf A^{d_\lambda}. \] Its representatives are ordered \(u_\alpha(\sum_{0\le j<\langle\alpha,\lambda\rangle}x_{\alpha,j}t^j)t^\lambda\) with positive \(\alpha\), all in \(N(O)t^\lambda\). The simple-root residues of \(\chi_\mu\) vanish because \(\langle\alpha_i,\mu\rangle\ge0\); nonsimple root factors have no abelianized coordinate. Thus \(f=0\). The raw IC is constant with shift \([d_\lambda]\) on this entire chart, and \[ R\Gamma_c(X_{\lambda,\lambda},F_{\lambda,\mu,\lambda}) =E(-d_\lambda)[-d_\lambda]. \tag{E.6.1} \] Its one geometric group lies in degree \(h_\lambda=d_\lambda\). Its raw Frobenius is \(q^{d_\lambda}\), and the normalized half twist changes it to \(a^{d_\lambda}\). This agrees with (7.2): the largest weight in \(V_\lambda\otimes V_\mu\) is \(\lambda+\mu\), and its line is the tensor of the two highest lines. All other weights are lower by a nonzero positive-root combination. Semisimplicity and the highest-weight classification already proved in Lesson 11, Theorem B.4.1.1 therefore give \(\dim\operatorname{Hom}(V_\lambda\otimes V_\mu,V_{\lambda+\mu})=1\). Choosing generators of those highest lines identifies the geometric line with this multiplicity line; no unspecified raw Tate line is discarded. ∎

Theorems E.4.1.1–E.6.1.1 determine the top degree and every highest-slice case. Lower-degree concentration for all dominant conductors is a further assertion. Editable SVG source. The compact dimension, parity and extremal-chart providers are proved in Lessons 5 and 11.

E.7. Exercises

Exercise E.7.1 (medium). For \(GL_2\), \(\lambda=\nu=(1,0)\), compare \(\mu=0\) with \(\mu=(-1,0)\). Verify character descent, compute the character on the highest affine chart, and calculate the entire raw compact complex.

Solution. The simple root pairs as the difference of the two entries. Both \(\nu\) and \(\mu+\nu\) are dominant; for the second choice \(\mu+\nu=(0,0)\). The chart is \(u_\alpha(x)t^\lambda\), \(x\in k\), by the coefficient range \(0\le j<1\) in Lesson 11, Lemma B.1.1.1. A lattice in this minuscule orbit satisfies \(tO^2\subset L\subset O^2\), and \(L/tO^2\) is a line in \(O^2/tO^2\). Conversely the inverse image of such a line is a lattice of this position. The same construction over parameter schemes is the rank-one subbundle functor, represented by \(\mathbf P^1\). The integral group acts transitively locally on the base, and this proper line functor is already the orbit closure. Thus the raw IC is \(E[1]\).

For \(\mu=0\), the residue of \(x\,dt\) is zero. Thus the raw compact complex is \(E(-1)[-1]\); its one group is in degree one with Frobenius \(q\). The half twist changes its scalar to \(a\). For \(\mu=(-1,0)\), conjugation changes the simple-root coefficient to \(x t^{-1}\), so the residue character is \(x\). Lemma E.1.1.1 gives \(R\Gamma_c(\mathbf A^1,\mathcal L_\psi[1])=0\). Equivalently (E.3.1) has \(n=-1,m=0\). This checks the complete nondominant-conductor vanishing on an actual Schubert slice.

Exercise E.7.2 (advanced). In characteristic two let \(T=\mathbf G_m\) act on \(U=\mathbf G_m\) with coordinate \(z\) of weight \(\gamma\). Let \(f=z^2\), of nonprimitive weight \(2\gamma\), and let \(\psi:\mathbf F_2\to E^\times\) be nontrivial. Show that \(f^*\mathcal L_\psi\) is geometrically nonconstant. Compute its entire compact complex on \(U\), and its Frobenius when the data are over \(\mathbf F_2\).

Solution. A trivial character line would give \(f=g^2-g\) with \(g\) a finite Laurent polynomial. Any nonzero weight chain of \(g\) produces both its lowest weight and twice its highest weight in \(g^2-g\). These cannot both equal the sole weight \(2\gamma\) of \(f\). The zero-weight part cannot produce \(z^2\). Thus the line is nonconstant. In particular \(g=z\) gives \(z^2-z\), retaining the extra weight \(\gamma\); dividing the character \(2\gamma\) by the characteristic would change the problem.

There is nevertheless a cover comparison with the usual Artin–Schreier line on \(\mathbf A^1_z\). If \(u^2-u=z^2\), set \(v=u-z\). Then \(v^2-v=z\). This is an isomorphism of \(\mathbf F_2\)-torsors, equivariant for the same deck translations and preserving the chosen zero unit. Its affine compact complex is therefore zero by Lemma E.1.1.1. Localization at zero gives \[ R\Gamma_c(\mathbf G_m,f^*\mathcal L_\psi) \longrightarrow0\longrightarrow E \longrightarrow R\Gamma_c(\mathbf G_m,f^*\mathcal L_\psi)[1]. \] Hence that compact complex is \(E[-1]\): degree one is \(E\), while degree two vanishes. Over \(\mathbf F_2\) all maps and the zero unit are defined over the base field, so its Frobenius scalar is one. Its alternating trace is \(-1\), agreeing with the sum over the single nonzero field point. The same scalar persists over finite extensions. Vanishing of the top group alone therefore cannot prove lower-degree vanishing; (7.2)'s general concentration needs its geometric Whittaker argument.

Appendix F. Levi projections and the last-residue obstruction

Keep raw \(I_\lambda\), \(d_\lambda=\langle2\rho_G,\lambda\rangle\), \(h_\nu=\langle2\rho_G,\nu\rangle\), and \(Y_{\lambda\nu}=\mathrm{Gr}_G^\lambda\cap S^G_\nu\). A top component has dimension \(D=(d_\lambda+h_\nu)/2\). Theorems E.4.1.1–E.5.1.1 with the trivial coefficient line identify its positive trace class \(v_K\) with a member of a basis of the actual weight space \(F_\nu(I_\lambda)\). We prove the required compatibility of these axes with simple Levi restriction.

Free comparison material is Braverman–Gaitsgory's Crystals via the affine Grassmannian, §§3–4, and Frenkel–Gaitsgory–Vilonen's Whittaker patterns, §6. The first is written over \(\mathbf C\). Our proofs use the actual coefficient operations and root charts of Lessons 5 and 11 over every algebraically closed geometric field with \(\ell\) invertible. For character lines assume \(\operatorname{char}k=p>0\) and the coefficient field contains a nontrivial \(\psi:\mathbf F_p\to E^\times\). The algebraic coefficient extension of Lesson 11, Theorem D.3.1.1 retains these comparisons.

F.1. The relative fibre bound

Proposition F.1.1.1. For a standard parabolic \(P=MU\) and a component \(\theta\) of \(\mathrm{Gr}_M\), set \(\delta=\langle2\rho_G-2\rho_M,\theta\rangle\). Let \(q:\mathrm{Gr}_P\to\mathrm{Gr}_M\) be the Levi projection. For an \(M\)-dominant \(\eta\) in that component, every nonempty fibre \(F_y\) of \(\mathrm{Gr}_G^\lambda\cap q^{-1}(\mathrm{Gr}_M^\eta)\to\mathrm{Gr}_M^\eta\) satisfies \[ 2\dim F_y\le d_\lambda+\delta-d_\eta^M. \tag{F.1.1} \]

Proof. All maps are on the represented bounded supports of Lesson 11, Lemma A.3.1.1. Put \(P_\lambda=j_!E[d_\lambda]\) for the open immersion \(j:\mathrm{Gr}_G^\lambda\to Z_\lambda\). This bounded genuine equivariant complex has perverse degrees at most zero. Indeed \(j^*\) is perverse exact; adjunction gives \(\operatorname{Hom}(j_!E[d_\lambda],A)=0\) for every \(A\) of perverse degrees at least one. This is the defining orthogonality condition. The proper-support base-change squares retain finite-jet equivariance.

The actual \(C=Rq_!i^*P_\lambda[\delta]\) also has perverse degrees at most zero. Lesson 11, Theorem A.3.2.1 proves heart exactness; induction through its bounded perverse truncation filtration proves this bound. On the \(M\)-orbit of dimension \(d_\eta^M\), its ordinary degree is at most \(-d_\eta^M\). To see the bound directly, equivariance makes any nonzero cohomology support contain the entire orbit, and the perverse support condition bounds that support dimension by the negative of its degree; apply this to each truncation piece.

Actual proper-support base change gives \(C_y=R\Gamma_c(F_y,E)[d_\lambda+\delta]\), by Lesson 5, Theorem O.4.2.1. If \(f=\dim F_y\), then \(H_c^{2f}(F_y,E)\ne0\): delete intersections and singular loci of its maximal components and use their structural trace lines and the compact dimension bound, exactly as in E.5.1.1. Thus \(2f-d_\lambda-\delta\le-d_\eta^M\), proving (F.1.1), also for nonpure fibres. ∎

F.2. Rank-one coordinates and the last residue

Lemma F.2.1.1. Let \(M\) have roots \(\{\pm\alpha\}\), and retain its original torus lattice. For dominant \(\eta\) put \(m=\langle\alpha,\eta\rangle\), \(n=\langle\alpha,\nu\rangle\). In the component \(\eta+\mathbf Z\alpha^\vee\), the reduced slice is nonempty exactly when \(m\ge|n|\). Its coordinates are \[ s={n-m\over2},\quad r={m+n\over2},\qquad u_\alpha\!\left(\sum_{j=s}^{n-1}a_jt^j\right)t^\nu, \qquad Z_\eta^M\cap S_\nu^M=\mathbf A^r. \tag{F.2.1} \] The open-orbit slice is \(\mathbf G_m\times\mathbf A^{r-1}\) if \(|n|<m\), all \(\mathbf A^m\) if \(n=m\), and a point if \(n=-m\). The character with conductor \(-\nu\) is the last coefficient \(a_{n-1}\) when \(r>0\), and zero when \(r=0\). It is nonconstant on every positive-dimensional open-orbit slice.

Proof. Root Laurent parameters modulo \(t^nO\) represent \(S_\nu^M\), since conjugation gives that exact stabilizer. Lesson 11, Theorem A.2.1.1 preserves these reduced coordinates under the adjoint rank-one map, with the original component and integral coroot step. In its \(PGL_2\)-coordinates the lattice matrix is \[ \begin{pmatrix}t^n&x(t)\\0&1\end{pmatrix}, \qquad a=\min(n,0,\operatorname{val}x),\qquad m'=n-2a. \tag{F.2.2} \] The minimum valuation of its entries is its smaller elementary divisor; determinant \(t^n\) then gives the other. These are the two-by-two integral row and column reductions of Lessons 2–4. Membership in \(Z_m\) is \(m'\le m\), equivalently \(s\le\min(n,0)\) and all coefficients below \(s\) zero. The remaining coefficients give (F.2.1). Membership in the open orbit is \(m'=m\). For \(|n|<m\) this requires \(a_s\ne0\); at the two endpoints it gives the stated affine space or point.

The calculation identifies reduced finite schemes. The polynomial root map lands in the original \(M\)-support: the proper universal homeomorphism and orbit-label calculation just cited identify its geometric points, so the support ideal pulls back to zero on the reduced polynomial scheme. Conversely the reduced slice lies in a bounded Laurent-root chart. The matrix inequalities kill its excluded coefficients on every geometric point and hence in its reduced coordinate ring. Its included coefficients give the inverse root map. The adjoint map preserves root parameters, so both compositions are the identity. No integral torus frame is lifted through an inseparable centre.

Conjugation by \(t^{-\nu}\) multiplies \(x\) by \(t^{-n}\). Its \(t^{-1}\) residue is \(a_{n-1}\). For \(r>1\) that last coordinate is unrestricted; for \(r=1\) it is the nonconstant coordinate of \(\mathbf A^1\) or \(\mathbf G_m\); for \(r=0\) the root sum is empty. This retains nonprimitive roots and coroots in the full lattices. ∎

F.3. Dominant simple-Levi projections

Proposition F.3.1.1. Let \(M_i\) be the rank-one Levi for \(\alpha_i\), with projection \(q_i\). Every top \(K\subset Y_{\lambda\nu}\) has a unique generic \(M_i\)-dominant orbit label \(\eta_i\). A dense open of \(K\) projects dominantly onto \(\mathrm{Gr}_{M_i}^{\eta_i}\cap S^{M_i}_\nu\).

Proof. There are finitely many orbits on the bounded target; irreducibility singles out the unique label meeting \(K\) densely. Let \(B\) be the closure of that projected image inside its orbit slice. The Cartesian Borel reduction of Lesson 11, Theorem A.3.2.1 places it in \(S^{M_i}_\nu\). Write \(m=d_{\eta_i}^{M_i}\), \(n=h_{M_i}(\nu)\), \(\delta=h_G(\nu)-h_{M_i}(\nu)\). The finite-type dimension formula and (F.1.1) give \[ \dim B\ge {d_\lambda+h_G(\nu)\over2} -{d_\lambda+\delta-m\over2}={m+n\over2}. \tag{F.3.1} \] Lemma F.2.1.1 makes the target irreducible of exactly this dimension. A proper closed subset has smaller dimension by Krull dimension and Noether normalization, §6. Therefore \(B\) is the entire target. This proves dominance for each \(i\); positivity for at least one \(i\) follows next. ∎

Proposition F.1.1.1 supplies the fibre bound, Lemma F.2.1.1 the complete reduced chart, and Proposition F.3.1.1 the dominant projection. Editable SVG source.

F.4. Upper opens and raising-operator images

Theorem F.4.1.1. Put \(n_i=\langle\alpha_i,\nu\rangle\le0\), \(\eta_0=s_i\nu\), and let \(U_i\) be the complement of \(Z_{\eta_0}^{M_i}\) in the bounded Levi component. Under the actual Satake weight comparison, the image \[ H_c^{h_G(\nu)}(Y_{\lambda\nu}\cap q_i^{-1}U_i,E[d_\lambda]) \longrightarrow F_\nu(I_\lambda) \tag{F.4.1} \] is \(e_i(V_\lambda)_{\nu-\alpha_i^\vee}\). It is the span of precisely the component axes \(v_K\) with \(d_{\eta_i}^{M_i}+n_i>0\). The image is independent of rescaling the positive simple-root operator \(e_i\) of the dual group.

Proof. The subset in (F.4.1) is open in \(Y\). Its top classes are exactly the trace classes of top components meeting it. Delete component intersections and singular loci in both spaces; their dimension is at most \(D-1\), so localization leaves top groups unchanged. The remaining disjoint smooth component opens have their structural trace lines. Extension by zero sends each to its corresponding nonzero line. This proves the asserted component-axis description of its image.

Put \(C_i=\operatorname{CT}_{M_i}^G(I_\lambda)\). Actual proper-support base change and composition identify the analogous image for the full IC coefficient with \[ \operatorname{im}\bigl( H_c^{n_i}(S^{M_i}_\nu\cap U_i,C_i) \longrightarrow F_\nu^{M_i}(C_i)\bigr). \tag{F.4.2} \] The shift is \(\delta=h_G(\nu)-n_i\), exactly as in Lesson 11, Theorem A.3.2.1. Passage from the open \(G\)-orbit coefficient commutes with this map. Indeed on its strict boundary the ordinary raw-IC degree is at most \(-d_\kappa-2\), by intermediate extension and Lesson 11, Theorem C.4.1.1. Restricting further to \(q_i^{-1}U_i\) leaves the compact dimension bound unchanged. Its compact degrees remain at most \(h_G(\nu)-2\), as in Theorem E.4.1.1. Localization thus gives the same top isomorphism with the open-orbit coefficient on both sides, compatibly with extension by zero.

The actual Satake equivalence and Levi tensor comparison of Lesson 11, Theorems A.4.1.1 and B.4.1.1 give a semisimple decomposition \(C_i=\bigoplus_\eta I_\eta^{M_i}\otimes M_\eta\). It corresponds to restriction of \(V_\lambda\) to the dual Levi with the specified weight torus and simple root groups. In this component the dominant labels are ordered by nonnegative integral multiples of \(\alpha_i^\vee\). A string contains \(\nu\) exactly when its pairing \(m\ge|n_i|\) and the integral coroot difference is allowed. Its smallest such label is \(\eta_0=s_i\nu\).

For \(\eta\le\eta_0\) the summand vanishes on \(U_i\). For \(\eta>\eta_0\), its boundary in \(Z_{\eta_0}\) has compact degrees at most \(n_i-2\), by the same strict perverse/parity calculation for \(M_i\). Localization is an isomorphism in degree \(n_i\) from its open restriction to its full weight group. Thus (F.4.2) is exactly the sum of the \(\nu\)-weight spaces in the \(\eta>\eta_0\) summands.

In an irreducible rank-one module of highest pairing \(m\), weights go from \(m\) to \(-m\) in steps of two. The image of \(e_i\) from weight \(n_i-2\) is zero exactly at \(n_i=-m\), and the entire line otherwise. The nonzero consecutive symmetric-power coefficients of Lesson 11, Theorem A.2.1.1 prove this over the characteristic-zero coefficient field. Precisely \(\eta>\eta_0\) contribute. Finally dominant projection and Lemma F.2.1.1 say that \(K\) meets \(U_i\) exactly when \(m+n_i>0\). This proves (F.4.1) without an assumed compatibility of crystal bases. ∎

F.5. Generation from the lowest line

Lemma F.5.1.1. For \(\nu\ne w_0\lambda\), \[ (V_\lambda)_\nu=\sum_i e_i(V_\lambda)_{\nu-\alpha_i^\vee}. \tag{F.5.1} \]

Proof. Choose \(v_-\ne0\) in the one-dimensional lowest line, proved in Lesson 11, Lemma B.1.1.1 and Theorem B.4.1.1. Let \(W\) span all words in the \(e_i\) applied to \(v_-\), including the empty word. It is torus-stable because the words have specified weights, and stable under every \(e_i\). It is stable under each negative simple \(f_i\) as well. Commute \(f_i\) through a word using \([f_i,e_j]=0\) for \(j\ne i\) and \([f_i,e_i]=-h_i\); then commute \(h_i\) through the remaining letters by their weights. The first identity follows since the difference of two distinct simple roots is not a root; the second is the rank-one matrix identity, with its root frames normalized. These root decompositions and rank-one groups are proved in Roots and reductive groups of rank one and Root data and Bruhat decomposition. Induction gives shorter words; the final \(f_iv_-\) is zero because no lower weight exists.

A characteristic-zero root-subgroup action is \(\exp(ce_i)\), or \(\exp(cf_i)\). Indeed its polynomial matrix \(A(c)\) satisfies \(A(c+d)=A(c)A(d)\). Differentiating in \(d\) gives \(A'(c)=A(c)A'(0)\), with \(A(0)=1\). Comparing coefficients gives \(A(c)=\sum_r c^rA'(0)^r/r!\); polynomiality forces nilpotence. Thus \(W\) is stable under both simple root subgroups. Their rank-one Weyl representatives generate the Weyl group. Every root is Weyl-conjugate to a simple root of its length, by the height-reduction proof in the root-data lesson. Hence these subgroups contain every root subgroup. Together with the torus they generate the dual group by Bruhat decomposition. The nonzero \(W\) is therefore the entire irreducible module.

Every nonempty word begins with an \(e_i\) and raises weight by a nonzero positive-coroot sum. Only the empty word has the lowest weight. Take the \(\nu\)-weight component to obtain (F.5.1). This asserts a sum of images. ∎

F.6. The general top-character obstruction

Theorem F.6.1.1. If \(\nu\ne w_0\lambda\), every top component of \(Y_{\lambda\nu}\) has a positive-dimensional simple-Levi image. Its character \(\chi_{-\nu}\) is nonconstant and its Artin–Schreier line has nontrivial monodromy. Consequently \[ H_c^{h_\nu}(Z_\lambda\cap S_\nu, I_\lambda\otimes(\chi_{-\nu}^{\nu})^*\mathcal L_\psi)=0. \tag{F.6.1} \] At \(\nu=w_0\lambda\) the entire raw compact complex is \(E[d_\lambda]\), with Frobenius \(1\); its normalized Frobenius is \(a^{-d_\lambda}\).

Proof. If some \(n_i=\langle\alpha_i,\nu\rangle>0\), every nonempty rank-one slice has \(m\ge|n_i|\), hence positive dimension \((m+n_i)/2\). Proposition F.3.1.1 supplies its dominant image.

Otherwise every \(n_i\le0\). Theorem F.4.1.1 identifies each \(e_i\)-image with the span of the independent component axes having positive-dimensional \(i\)-image. Their sum is the whole weight space by Lemma F.5.1.1. The union of these basis subsets must contain every \(v_K\); otherwise its span would omit that independent vector. Thus every \(K\) has the required positive image.

Ordered roots give \(N_G=U_i\rtimes N_{M_i}\). Projection keeps the simple \(\alpha_i\)-coordinate \(x_i(t)\); the other radical factors have no such term in the simple-root abelianization. Therefore the \(i\)-th summand \(f_i\) of \(\chi_{-\nu}\) is the last-residue function of Lemma F.2.1.1 composed with \(q_i\). Dominance makes it nonconstant on \(K\).

The constant torus preserves \(K\), since it is connected and the component set finite. The \(f_j\) have distinct simple-root weights \(\alpha_j\). Their sum cannot cancel its nonzero \(f_i\), or become a constant of weight zero. The finite-weight and integral \(p\)-chain argument of Theorem E.5.1.1 says that its character line on a dense smooth component open is trivial exactly when that sum is zero. Thus it is nontrivial here. Theorems E.4.1.1–E.5.1.1 identify the top group with zero-character component lines; none survive, proving (F.6.1).

At \(w_0\lambda\), Lesson 11, Lemma B.1.1.1 makes the whole extremal slice \(\mathbf A^0\) inside the open orbit. Its character is zero and raw IC is \(E[d_\lambda]\). This is its compact complex. No Tate twist occurs at the point, so raw Frobenius is \(1\); the half twist contributes \(a^{-d_\lambda}\), agreeing with \(h_\nu=-d_\lambda\). The argument retains every central direction and includes the torus case. ∎

Theorem F.4.1.1 and Lemma F.5.1.1 prove the basis argument used in Theorem F.6.1.1. The \(A_2\) example is computed in Exercise F.7.2. Editable SVG source. This top theorem does not alone prove vanishing of lower compact degrees.

F.7. Exercises

Exercise F.7.1 (medium). For \(GL_2\), \(\eta=(3,0)\), \(\nu=(1,2)\), find the open rank-one slice and its conductor \(-\nu\) character. Compute the entire compact complex of the raw open-orbit coefficient. Explain its relation to the top theorem.

Solution. Here \(m=3\), \(n=-1\), \(s=-2\), \(r=1\). Thus \(x(t)=a_{-2}t^{-2}\), with \(a_{-2}\ne0\), gives \(\mathbf G_m\). The conductor \(-\nu=(-1,-2)\) is dominant and its residue is \(a_{-2}\). The raw IC restriction on the orbit is \(E[3]\). Lemma E.1.1.1 gives \(R\Gamma_c(\mathbf A^1,\mathcal L_\psi)=0\), while its zero fibre is the unit \(E\). Localization yields \(R\Gamma_c(\mathbf G_m,\mathcal L_\psi)=E[-1]\). Hence the raw open-slice complex is \(E[2]\): its only group is in degree \(-2\), while the top degree \(h_\nu=-1\) vanishes. The full-IC top theorem does not determine the lower groups or their boundary attaching maps.

Exercise F.7.2 (advanced). In type \(A_2\), take \(\lambda=\alpha_1^\vee+\alpha_2^\vee\) and \(\nu=0\). Determine the number of top components, their generic simple-Levi images and fibre dimensions, and the consequence for \(\chi_0\).

Solution. The dual highest weight is its highest root, so its module is the adjoint module. To verify this identification, the root decomposition has highest-root line; simplicity of the type \(A_2\) Lie algebra follows directly from its matrix units. A nonzero ideal is stable under the diagonal adjoint operators, so their weight projections give either a matrix unit \(E_{ab}\) or a nonzero trace-zero diagonal matrix \(H\). In the latter case some diagonal entries differ, and \([H,E_{ab}]\) gives a matrix unit. From \(E_{ab}\), its bracket with \(E_{ba}\) gives a diagonal difference; brackets with the remaining matrix units then give every off-diagonal unit and both diagonal differences. Thus the ideal is the whole algebra. Its zero-weight space is \(E h_1\oplus E h_2\), and \(e_i\) from the negative-simple-root line has image \(E h_i\). These two lines are independent in characteristic-zero coefficients. Thus there are exactly two top component axes. Theorem F.4.1.1 assigns one to each raising image, and they are distinct. In the matching Levi the label is \(\eta_i=\alpha_i^\vee\), with \(m=2,n=0\), so its generic image is \(\mathbf G_m\). In the other Levi the label is zero and the image a point. There are no other labels in this component: the adjoint weights on its simple-coroot line are \(-\alpha_i^\vee,0,\alpha_i^\vee\).

Since \(d_\lambda=4\) and \(D=2\), the dimension formula gives generic fibres of dimensions one and two respectively. The trace vector of each component is a nonzero multiple of its \(h_i\); that multiple depends on the root frame. On each component its matching last residue is nonconstant. Distinct simple-root weights prevent cancellation in their sum \(\chi_0\), and Theorem F.6.1.1 gives zero top Whittaker group. No raw trace normalization is identified with a chosen Lie-algebra basis scalar.

Appendix G. Actual Whittaker strata and finite Hecke coefficients

Work on \(X=\mathbf P^1\), with \(x=0\), coordinate \(t\) there and \(u=t^{-1}\) on the complementary affine chart. This global model supplies the actual relevant strata, evaluation and finite Hecke coefficients used in the local Whittaker proof. We retain the original torus lattice and the coefficient operations of Lessons 5 and 11. The free Whittaker patterns, §§3.3, 4.1 and 6 is comparison material. Every construction and coefficient comparison used below is proved here or in the indicated earlier lesson.

G.1. Torus bundles, Laurent normal forms and represented strata

Theorem G.1.1.1. Set \(\gamma=2\rho^\vee=\sum_{\alpha>0}\alpha^\vee\in X_*(T)\). Let \(\mathcal T_0\) be the T-torsor induced from \(\mathcal O(-1)^\times\) by \(\gamma\), and \(\mathcal T_\mu=\mathcal T_0(-\mu x)\). Thus for a positive root \(\beta\), \[ a_\beta=\langle\beta,\gamma\rangle=2\operatorname{ht}(\beta),\quad b_\beta=\langle\beta,\mu\rangle,\quad \mathcal L_{\beta,\mu}=\mathcal O(-a_\beta-b_\beta). \tag{G.1.1} \] No half-coweight is required to belong to the original lattice. For dominant \(\mu\), the functor \(\mathcal B_\mu\) of B-bundles with a specified identification of their T-quotient with \(\mathcal T_\mu\) is an affine scheme \[ \mathcal B_\mu\simeq M_\mu\simeq\mathbf A^{m_\mu},\qquad m_\mu=\sum_{\beta>0}(a_\beta+b_\beta-1)=m_0+h_\mu. \tag{G.1.2} \] Here \(M_\mu\) is the ordered root-parameter space with coefficient range \(-a_\beta<j<b_\beta\) in its \(\beta\)-coordinate.

Proof. The root products and their triangular multiplication are those of Root data, Weyl chambers and the Bruhat decomposition, §5. Order positive roots by height. On the two affine opens the T-torsor has fixed frames. The outside frame in the local \(\mathcal T_0\)-frame gives root coefficients \(t^{-a_\beta}R[t^{-1}]\); the disc frame for \(\mathcal T_\mu\) gives \(t^{b_\beta}R[[t]]\). Define \[ H_-(R)=\operatorname{Ad}(t^{-\gamma})N(R[t^{-1}]),\qquad H_{+,\mu}(R)=\operatorname{Ad}(t^\mu)N(R[[t]]). \tag{G.1.3} \] For every ring \(R\), multiplication gives the natural bijection \[ H_-(R)\times M_\mu(R)\times H_{+,\mu}(R) \xrightarrow{\sim}N(R((t))). \tag{G.1.4} \] To prove it, begin at height one. Each Laurent series splits uniquely into powers \(j\le-a_\beta\), \(-a_\beta<j<b_\beta\), and \(j\ge b_\beta\). The ranges are disjoint since \(a_\beta\ge2\), \(b_\beta\ge0\). At each later root the product coordinate is its three unknown coordinates plus a polynomial in the already determined lower-height coordinates. Subtract that known polynomial and split by the same three ranges. This proves existence and uniqueness inductively. The global part is a finite Laurent polynomial and the local part a power series. Every output coefficient depends on finitely many input coefficients on any specified pole bound. Thus these are actual morphisms of the bounded root-coordinate schemes and their inverse limits. No commutator constant is divided out, so the argument retains every geometric characteristic and parameter nilpotent.

The identical argument with \(R[t]\) on the right gives algebraic Laurent-polynomial gluing on \(U_0\cap U_\infty\). A torsor for a vector-group bundle on an affine scheme is trivial: its additive transition cocycle glues an extension \(0\to L\to V\to\mathcal O\to0\) by upper triangular matrices, and projectivity of the quotient \(\mathcal O\) splits that extension. A central root-height series then proves that every torsor for the twisted N is trivial on each of these affine opens. Indeed trivialize its quotient by the last central vector subgroup by induction; the inverse image of that trivialization is a vector-group torsor and has a section. This proof applies over arbitrary affine parameter rings and commutes with flat descent.

Such B-bundles with fixed T-quotient are the torsors for the N twisted by that T-bundle: twisting the split \(T\subset B\) frame gives this equivalence directly on gluing functions and their isomorphisms. Their transition functions therefore reduce uniquely to \(M_\mu\) by the polynomial version of (G.1.4). An automorphism would give two different global/local factorizations of the same middle representative, so it is the identity. This proves the groupoid equivalence with the represented affine scheme on every affine parameter base; descent proves it for all bases. Its dimension is the coefficient count (G.1.2). Central characters of \(\mathcal T_\mu\) are retained even though they act trivially on N.

Finally the formal quotient \(S_\mu=N((t))/H_{+,\mu}\) is \(H_-\times M_\mu\) by (G.1.4), and its map to \(\mathcal B_\mu\) is projection. It is the actual forgetful map from a formal modification to its global B-bundle: the middle representative algebraizes that bundle, and global and disc changes of frames are precisely its two factors. For finite approximations take the root coefficients of \(H_-\) in \(-c\operatorname{ht}(\beta)\le j\le-2\operatorname{ht}(\beta)\), \(c\ge2\). These form a finite-dimensional unipotent group \(H_{-,c}\): commutator exponents add within the bounds. Their union is \(H_-\). Thus each approximation is the actual trivial smooth \(H_{-,c}\)-torsor \(H_{-,c}\times M_\mu\to M_\mu\). ∎

G.2. The global evaluation is the actual residue

Lemma G.2.1.1. The inclusion \(\mathcal L_{\alpha_i,\mu}=\Omega_X(-b_i x)\hookrightarrow\Omega_X\) and the simple-root quotient define a global evaluation \(e_\mu:\mathcal B_\mu\to\mathbf A^1\). In the preceding coordinates, \[ e_\mu(m)=\sum_i a_{i,-1}=\chi_0(m). \tag{G.2.1} \] Its pullback to \(S_\mu\) is the descended conductor-zero character, normalized to zero at \(t^\mu\).

Proof. The simple-root coordinates add under root multiplication: every nontrivial commutator term has higher height. Hence they give a homomorphism \(N\to\prod_i\mathbf G_a\), without needing a statement about the full abelianization in small characteristic. A twisted N-torsor gives an additive cocycle for each \(\mathcal L_{\alpha_i,\mu}\). Its first-cohomology class is its Laurent coordinate modulo outside and local sections. In the \(\mathcal T_0\)-frame these are powers \(j\le-2\) and \(j\ge b_i\). Inclusion into \(\Omega_X\) changes the latter bound to \(j\ge0\). Its remaining class is exactly \(a_{i,-1}t^{-1}dt\), so its trace is \(a_{i,-1}\). Explicitly the two-open Čech quotient is \[ R[t,t^{-1}]dt\big/(R[t]dt+t^{-2}R[t^{-1}]dt)=R\,t^{-1}dt. \tag{G.2.2} \] Take the positive trace that sends that generator to one. The frames \(dt\) on \(U_0\) and \(-du\) on \(U_\infty\) give the displayed outside bound and this sign.

Residue is coordinate independent. For \(u=t a(t)\), with \(a\) a unit, the residue of \(u^rdu\) is one at \(r=-1\) and zero otherwise. At \(r=-1\) use \(du/u=dt/t+da/a\); at \(r\ge0\) it is regular. For \(r\le-2\) the required coefficient is a polynomial over \(\mathbf Z[a_0^{\pm1},a_1,\ldots]\); over its characteristic-zero fraction field it is the zero residue of \(d(u^{r+1})/(r+1)\). The integer polynomial is therefore zero and remains zero under every specialization. This proves the assertion also when the actual characteristic divides \(r+1\), without dividing by it there.

Summing the simple classes gives (G.2.1), an equality of morphisms on parameter rings. The character vanishes on \(H_-\) since its simple coefficients have powers \(\le-2\); it vanishes on \(H_{+,\mu}\) since \(b_i\ge0\). Thus its descent is exactly \(e_\mu\) under the normal form. ∎

Theorem G.1.1.1 and Lemma G.2.1.1 prove the factorization on all parameter rings and its positive residue sign. Proposition G.3.1.1 retains the actual geometric endomorphism complex. Editable SVG source.

G.3. The stratum character object and its self-maps

Proposition G.3.1.1. Assume characteristic \(p>0\) and the same coefficient field and additive character as Appendix E. Define \[ \Psi_\mu=e_\mu^*\mathcal L_\psi[m_\mu],\qquad \Psi_\mu^{\mathrm n}=\Psi_\mu(m_\mu/2). \tag{G.3.1} \] These are simple perverse objects on \(\mathcal B_\mu\), and their geometric derived endomorphism complex is \(E\) concentrated in degree zero, with its actual identity Frobenius. If G has positive semisimple rank, their entire compact complexes on the stratum vanish.

Proof. The affine scheme is smooth and connected of dimension \(m_\mu\). Its rank-one line is lisse by the finite étale Artin–Schreier construction in E.1.1.1. A shifted rank-one local system on a connected smooth scheme is simple perverse. On any proper smooth closed stratum of codimension \(c>0\), its ordinary restriction has perverse degree \(-c\). Structural duality gives exceptional restriction in perverse degree \(c\). The strict closed restriction criterion therefore excludes a closed-supported subobject or quotient. On a dense smooth open a nonzero sub-local system of a rank-one line is the entire line. Stratifying a proposed proper support and applying these two statements proves simplicity, using the actual duality and perverse localization of Lesson 5, Theorems P.3.3.1 and P.3.4.1 and Lesson 6, Appendix A. The half twist does not change its geometric simplicity.

The tensor dual of that line times the line is the unit with its actual evaluation. Thus its derived endomorphisms are \(R\Gamma(\mathbf A^{m_\mu},E)\). Iterating the affine compact trace of Lesson 5, Lemma Q.1.2.1 gives \(R\Gamma_c(\mathbf A^{m_\mu},E)=E(-m_\mu)[-2m_\mu]\); structural duality and the smooth dualizing comparison give \(R\Gamma(\mathbf A^{m_\mu},E)=E\). Its generator is the identity, and Frobenius fixes it. Shifts and twists cancel in this endomorphism calculation. This is an equality of actual complexes, not merely an Ext-dimension count.

When the root set is nonempty, \(e_\mu\) is a nonzero linear form in the affine coordinates: every simple root has its -1 coefficient. Lemma E.1.1.1 computes compact cohomology of that linear-character pullback as zero in all degrees. For a torus there are no root coordinates, the stratum is a point and its character object is E. These are assertions on the represented stratum; cleanness across its compactified boundary remains a further theorem. ∎

G.4. The actual finite Hecke chart and evaluation comparison

Theorem G.4.1.1. Let \(\mu\) and \(\eta\) be dominant, put \(\nu=\eta-\mu\), and fix dominant \(\lambda\). Set \[ X_{\lambda\nu}=Z_\lambda\cap S_\nu,\qquad Z_{\mu,\eta}^{\lambda}=\mathcal B_\mu\times X_{\lambda\nu},\qquad l:Z_{\mu,\eta}^{\lambda}\to\mathcal B_\mu. \tag{G.4.1} \] There is an actual target morphism \(r\) to \(\mathcal B_\eta\) classifying the compatible B-reduction after the bounded G-modification, and \[ e_\eta\circ r=e_\mu\circ l+\chi_\mu^\nu\circ\operatorname{pr}_X. \tag{G.4.2} \]

Proof. In the universal middle representative \(m\) over \(\mathcal B_\mu\), write a point of \(S_\nu\) as \(n t^\nu G(O)\). Its modified lattice relative to the background bundle is \[ m t^\mu n t^\nu=p t^\eta, \qquad p=m\,\operatorname{Ad}(t^\mu)n\in N((t)). \tag{G.4.3} \] Normalize \(p\) by (G.1.4) for \(\eta\), writing \(p=h_-m'h_{+,\eta}\). Its middle part \(m'\) defines \(r\). Changing \(n\) by the stabilizer of \(t^\nu\) changes \(p\) on the right by \(H_{+,\eta}\), so \(m'\) is unchanged. On a fixed pole bound the normal-form coefficients are finite polynomial expressions. Therefore \(r\) is a scheme morphism on the actual finite intersection, with all its defining coefficient ideals; reduction and its actual coefficient equivalence can be used if that intersection has nilpotents.

This also proves its moduli interpretation. The universal B-bundle is trivialized on the disc by its Čech presentation. Extending its structure group and applying the displayed finite modification gives the G-bundle lattice in (G.4.3). The same meromorphic N-reduction away from x acquires T-quotient \(\mathcal T_\eta\) because its semi-infinite label is \(\eta\). The normal form supplies precisely its global B-bundle. A change of disc \(N\)-frame produces the usual \(N(O)\)-twisted correspondence and the same descent; our universal trivialization gives the actual product chart (G.4.1). All constructions recover the modification and both reductions, so there is no replacement of a correspondence by its dimensions.

Since \(\chi_0\) is additive on products of \(N\) and zero on both outside and \(\eta\)-disc factors, applying it to (G.4.3) gives \(e_\eta(m')=e_\mu(m)+\chi_0(\operatorname{Ad}(t^\mu)n)\). This is (G.4.2), with the positive residue sign. Its function descends since \(\mu+\nu=\eta\) is dominant. All maps are defined over the split finite field, when the data are, so this equality and the additive character-line comparison retain Frobenius. ∎

G.5. Actual compact images, shifts and normalized Weil lines

Theorem G.5.1.1. Put \(C_{\lambda\mu\nu}=R\Gamma_c(X_{\lambda\nu},I_\lambda\otimes(\chi_\mu^\nu)^*\mathcal L_\psi)\). On the finite chart one has the actual comparisons \[ Rl_!(r^*\Psi_\eta\otimes\operatorname{pr}_X^*I_\lambda) =\Psi_\mu\otimes C_{\lambda\mu\nu}[h_\nu], \tag{G.5.1} \] \[ Rl_!(r^*\Psi_\eta^{\mathrm n}\otimes\operatorname{pr}_X^*I_\lambda(d_\lambda/2)) =\Psi_\mu^{\mathrm n}\otimes C_{\lambda\mu\nu}(D)[h_\nu], \qquad D={d_\lambda+h_\nu\over2}. \tag{G.5.2} \] Both retain the actual coefficient transitions, base change, projection, units and Frobenius. The degree-zero coefficient in the normalized formula has Frobenius the identity.

Proof. Pull back the additive character-line comparison of E.1.1.1 along (G.4.2). It identifies \(r^*e_\eta^*\mathcal L_\psi\) with the product of \(l^*e_\mu^*\mathcal L_\psi\) and the \(X\) character line, preserving the zero units. The stratum shifts differ by \(m_\eta-m_\mu=h_\eta-h_\mu=h_\nu\). Apply the actual proper-support projection formula Lesson 5, Theorems O.4.3.1 and O.4.2.1. They give (G.5.1) as a comparison of complexes and specified maps on all parameter bases.

The half twists on the left of (G.5.2) add to \((m_\eta+d_\lambda)/2\). Removing the source half twist \(m_\mu/2\) leaves exactly \((d_\lambda+h_\nu)/2=D\). This gives (G.5.2); negative or odd shifts are kept, with the previously chosen square root \(a\) of \(q\). The top group of the raw \(C\) has actual scalar \(q^D\) by E.5.1.1. Its extra twist \((D)\) changes that operator to the identity, and its shift \([h_\nu]\) places it in degree zero. No assertion about lower coefficient groups is used here. ∎

G.6. The relevant-stratum Hecke bound for the zero label

Theorem G.6.1.1. For \(\eta=0\) and dominant \(\mu\), the normalized complex in (G.5.2) has perverse degrees \(\le0\) on \(\mathcal B_\mu\). They are strictly negative unless \(\mu=-w_0\lambda\). In that extremal case the entire complex is \(\Psi_\mu^{\mathrm n}\), by its actual unit/trace comparison.

Proof. Here \(\nu=-\mu\). E.4.1.1 says that \(C\) has ordinary degrees \(\le h_\nu\). Hence \(C(D)[h_\nu]\) has ordinary degrees \(\le0\). Tensoring that point complex with the lisse perverse \(\Psi_\mu^{\mathrm n}\) converts its ordinary degrees into the same perverse degrees on the smooth stratum. If \(\nu\ne w_0\lambda\), F.6.1.1 kills its degree zero, so the bound is strictly negative. If \(\nu=w_0\lambda\), the entire slice is a point in the open orbit and \(C=E[d_\lambda]\). Now \(h_\nu=-d_\lambda\) and \(D=0\), so the coefficient complex in (G.5.2) is exactly \(E\) in degree zero with the actual zero-character unit. This identifies the compact image with \(\Psi_\mu^{\mathrm n}\), including Frobenius and the central label. These are the actual relevant-stratum bounds needed in the Hecke calculation. They do not yet construct the proper compactified Hecke morphism or prove its exceptional boundary bounds. ∎

Theorems G.4.1.1–G.6.1.1 prove the finite kernel comparisons and the relevant-stratum bounds. Proper compactification and exceptional boundary bounds are further requirements. Editable SVG source.

G.7. Exercises

Exercise G.7.1 (medium). For \(GL_2\), take \(\mu=0\), \(\eta=\lambda=(1,0)\). Compute the torus bundles, the two relevant strata, the actual target map of (G.4.1), and both compact images in (G.5.1)–(G.5.2).

Solution. Here \(\gamma=(1,-1)\), so \(\mathcal T_0\) is the pair \((\mathcal O(-1),\mathcal O(1))\), and \(\mathcal T_\eta\) is \((\mathcal O(-2),\mathcal O(1))\). Their root lines are \(\mathcal O(-2)\) and \(\mathcal O(-3)\). The middle representatives are \(u_\alpha(a t^{-1})\) and \(u_\alpha(a t^{-1}+b)\), giving \(\mathcal B_0=\mathbf A^1\), \(\mathcal B_\eta=\mathbf A^2\), with evaluations \(a\) in both. The highest slice has representative \(u_\alpha(z)t^\lambda\), by Lesson 11, Lemma B.1.1.1. Thus (G.4.3) has \(p=u_\alpha(a t^{-1}+z)\), already in its target middle form. The actual morphism is \(r(a,z)=(a,z)\); its conductor-zero character on \(z\) is zero.

The raw IC restriction is \(E[1]\), with compact complex \(E(-1)[-1]\). Here \(h_\nu=1\) and \(D=1\), so (G.5.1) gives \(\Psi_0\otimes E(-1)\), with the actual affine trace. In (G.5.2) the extra twist \((1)\) and shift \([1]\) turn that coefficient into the unit. The normalized compact image is exactly \(\Psi_0^{\mathrm n}\). This verifies the target morphism, positive evaluation comparison, shifts and Weil normalization.

Exercise G.7.2 (advanced). For \(SL_3\) and \(\mu=0\), calculate the full middle scheme, its evaluation and its character-object self-maps. Explain why a model keeping only simple-root coordinates loses data, and check characteristic two.

Solution. The positive root heights are \(1,1,2\), so \(m_0=1+1+3=5\). An ordered representative is \[ u_{12}(a t^{-1})u_{23}(b t^{-1}) u_{13}(c_{-3}t^{-3}+c_{-2}t^{-2}+c_{-1}t^{-1}). \] Its matrix entry \((1,3)\) is \(c_{-3}t^{-3}+(c_{-2}+ab)t^{-2}+c_{-1}t^{-1}\). Thus the represented stratum is the entire \(\mathbf A^5\), including three higher-root coordinates, and its evaluation is \(a+b\). Keeping only \(a,b\) would discard the actual bundle parameters and change the perverse shift from five to two.

The change between the ordered coordinate \(c_{-2}\) and the matrix coefficient is the polynomial addition of \(ab\), with polynomial inverse subtraction. Both remain inverse in characteristic two; no coefficient is divided out. Proposition G.3.1.1 gives the lisse perverse character object \((a+b)^*\mathcal L_\psi[5]\), geometric derived endomorphism complex \(E\) in degree zero, and zero entire stratum compact complex. The other three coordinates contribute ordinary affine parameters and do not change its nontrivial linear character.

Appendix H. Projective Hecke transport and a full-centre Whittaker reduction

Keep \(X=\mathbf P^1\), \(x=0\), the specified \(T\)-bundle and the coefficient conventions of Appendix G. We first construct actual projective modification maps and their action on meromorphic decorations. We then pass to a group with simply connected derived subgroup while retaining the original torus characters, cocharacters, slice labels and Weil maps. Whittaker patterns, §§5.1–5.3 supplies free comparison material. The constructions and coefficient comparisons used here are proved below or in the linked earlier programme lessons. The finite algebraic compactification and its boundary estimates remain further requirements.

H.1. Relative projective modifications, bounds and inversion

Theorem H.1.1.1. Let \(S\) be a parameter scheme, \(\mathcal E\) a \(G\)-bundle on \(X\times S\), and \(\lambda\) dominant in the original \(X_*(T)\). The modification functor \(\mathcal H_\lambda(\mathcal E)\) classifies a bundle \(\mathcal E'\) and an isomorphism \(\beta:\mathcal E|_{X-x}\simeq\mathcal E'|_{X-x}\), of relative type in the actual Schubert scheme \(Z_\lambda\). It is projective over \(S\) and compatible with arbitrary base change. At a sufficient finite jet level it is \[ \mathcal H_\lambda(\mathcal E)=\mathcal P_j(\mathcal E)\times^{G_j}Z_\lambda, \qquad G_j(R)=G(R[t]/t^j), \tag{H.1.1} \] where \(\mathcal P_j(\mathcal E)\) is the actual torsor of disc frames modulo \(t^j\). For a representation \(V\), its weight decomposition gives \[ c(V,\lambda)=\max_{\xi\in\operatorname{Wt}(V)}|\langle\xi,\lambda\rangle|, \qquad t^cV[[t]]\subset V_{\mathcal E'}|_{\mathcal D_x}\subset t^{-c}V[[t]]. \tag{H.1.2} \] The inverse modification has the same bounds. Exchanging the bundles gives dominant type \[ \lambda^*=-w_0\lambda. \tag{H.1.3} \] All assertions retain the full centre and parameter nilpotents.

Proof. Loop groups and the affine Grassmannian, Lemma 4.1, trivializes the formal bundle after an étale cover of an affine parameter base. Its proved successive lifting through the smooth jets gives the actual frame torsors. In a frame, Beauville–Laszlo gluing and the moduli interpretation, §§1–3 and 7–8 identifies the modification with its point in \(Z_\lambda\); changing that frame acts by \(G[[t]]\). Choose the faithful closed representation and the actual finite closed lattice stages of Loop groups and the affine Grassmannian, §§5–7 and Lemma 7.12. A finite stage containing \(Z_\lambda\) embeds into the ordinary Grassmannian of \(t^{-N}V[[t]]/t^NV[[t]]\). The action on this finite module factors through \(G_j\) for \(j\ge2N\), so its restriction to the closed Schubert scheme factors through that same group, with the defining ideals preserved.

Descend the finite module representation along \(\mathcal P_j\) to a vector bundle on \(S\). Its relative ordinary Grassmannian and the equivariant closed equations of \(Z_\lambda\) descend as well. This realizes (H.1.1) as an actual closed subscheme of that Grassmannian. Its Plücker immersion into the projective bundle of the corresponding exterior power proves projectivity over \(S\). This supplies the linearized ample construction; it does not presume that an arbitrary torsor twist of an arbitrary projective scheme is projective. These equations and frame torsors commute with every base change.

For \(V\), the \(t^\lambda\) weight decomposition gives the inclusions in (H.1.2). Multiplication on either side by a \(G[[t]]\)-element preserves the standard lattice and its scalar bounds, so they hold on the open orbit. Both inclusions are closed incidence conditions in the actual finite module of Loop groups and the affine Grassmannian, Lemma 2.1, and hence on its schematic orbit closure. They hold on every parameter-ring point of that scheme. If \(\Lambda=gV[[t]]\) has these inclusions, applying \(g^{-1}\) gives \(t^cV[[t]]\subset g^{-1}V[[t]]\subset t^{-c}V[[t]]\), proving the inverse bounds on local frame covers and then by descent.

The inverse of \(a t^\lambda b\) is \(b^{-1}t^{-\lambda}a^{-1}\). The dominant member of the orbit of \(-\lambda\) is \(-w_0\lambda\), retaining its original central coordinate. Inverting the punctured-curve isomorphism defines inverse morphisms of the actual bundle-modification functors; these exchange the two bundles and send the open types, and therefore their schematic closures, to the stated inverse types. This is inversion on the modification correspondence, not a supposed map sending a right coset \(gG[[t]]\) to another right coset \(g^{-1}G[[t]]\). Thus the maps and bounds retain nilpotents and the original lattice. ∎

H.2. Meromorphic decoration transport and both Cartesian squares

Proposition H.2.1.1. Fix the actual \(T\)-bundle \(\mathcal T_0\). A meromorphic decoration is a specified family of maps in actual algebraic \(G\)-representations, \[ \kappa^\chi:\mathcal L^\chi_{\mathcal T_0}\longrightarrow V^\chi_{\mathcal E}(*x), \tag{H.2.1} \] with specified \(G\)-equivariant tensor relations and highest-line conditions away from \(x\). If degeneracies away from \(x\) are allowed, impose sheaf injectivity and a finitely presented, parameter-flat quotient there. These quotients are coherent over Noetherian parameter schemes. Central-character isomorphisms and their inverses are included. Let \(\mathcal Y_x\) be this fibre category, with arrows preserving every map. Its definition here does not assume a finite algebraic presentation.

Let \(\mathcal Z_\lambda\) classify a decorated source, a bounded modification and its transported decorated target. Both squares in \[ \begin{array}{ccccc} \mathcal Y_x&\xleftarrow{\ l\ }&\mathcal Z_\lambda&\xrightarrow{\ r\ }&\mathcal Y_x\\ \downarrow&&\downarrow&&\downarrow\\ \operatorname{Bun}_G&\xleftarrow{\ h_l\ }&\mathcal H_\lambda&\xrightarrow{\ h_r\ }&\operatorname{Bun}_G \end{array} \tag{H.2.2} \] are Cartesian as fibre categories. The target-fibre interpretation uses inverse type \(\lambda^*\). Both \(l,r\) are representable by projective schemes: every base change by a parameter scheme is the actual projective scheme of Theorem H.1.1.1.

Proof. The associated punctured-curve isomorphism defines the actual transported map \[ (\kappa')^\chi=\beta_{V^\chi}\kappa^\chi. \tag{H.2.3} \] Allowing poles makes both associated bundles isomorphic. A finite pole bound for a chosen map increases by at most \(c(V^\chi,\lambda)\), by (H.1.2), so this remains a genuine meromorphic section in families. Tensor relations are preserved because \(\beta\) commutes with every equivariant tensor map, including central multiplication and inverse maps. Away from \(x\) it is an isomorphism of bundles, preserving the highest-line, injection and flat-quotient conditions. All these are equalities of maps on parameter rings.

Conversely \(\beta^{-1}\) recovers the unique source decoration from the target. Therefore a modification of a decorated source requires no extra target decoration choice or equation; its transported decoration is determined. This proves the source Cartesian square by the full groupoid-valued functor of points, including its arrows and stabilizers. The inverse construction proves the target square. Both commute with all base changes. Their pullbacks by a parameter scheme are exactly (H.1.1), with type \(\lambda\) or \(\lambda^*\), respectively. That theorem proves the stated representable projectivity. This statement does not by itself prove that \(\mathcal Y_x\) is an algebraic stack. ∎

H.3. Actual coefficient action and coherent composition

Theorem H.3.1.1. A coefficient object on \(\mathcal Y_x\) means a cartesian family of complexes \(F_S\) for every parameter scheme \(S\to\mathcal Y_x\), with specified coherent derived pullback comparisons. On existing algebraic charts these are the usual scheme complexes. An equivariant Satake complex \(A\) on a fixed finite Schubert support descends along (H.1.1) to its vertical complex \(A_{\lambda,S}\). For a normalized simple object it is \(I_\lambda(d_\lambda/2)\). Ordinary descent introduces no shift equal to the dimension of the frame group.

The left-frame action is the actual proper image \[ (\mathsf H_A F)_S=Rl_!(r^*F\otimes A_{\lambda,S}) =Rl_*(r^*F\otimes A_{\lambda,S}). \tag{H.3.1} \] It defines a cartesian family, commutes with base change, has its actual unit, and has the coherent comparisons \[ \mathsf H_A\mathsf H_B\simeq\mathsf H_{A*B}. \tag{H.3.2} \] The maps preserve coefficient transitions, units and Frobenius. Exchanging the two projections sends the normalized simple vertical object to the one labelled by \(\lambda^*\).

Proof. Trivialize the finite frame torsor on an étale cover of \(S\), as in Theorem H.1.1.1. On that cover the modification scheme is the product with \(Z_\lambda\), and its vertical complex is \(E\boxtimes A\). Changing the trivialization acts through \(G_j\). The actual equivariance of \(A\) gives the comparison on the overlap, with its group-law cocycle. The actual augmented Čech resolution in Hypercoverings, Theorem 3.3 glues these product complexes on the modification scheme. Explicitly, take the coherent complex on every level of the Čech nerve of the frame-trivializing cover, and its derived direct-image totalization on the modification scheme. Its bounded-below model has only finitely many terms contributing to each fixed total degree. Étale base change in Smooth base change and local acyclicity, Theorem 1.1, with its geometric foundations proved in Constructible complexes on algebraic varieties, §§AL–AM, pulls this totalization to the cover. The pulled-back nerve has its diagonal section; the extra degeneracy contracts its augmented complex onto the given complex there. The augmentation is consequently an isomorphism on that cover, and on all geometric stalks. This proves effective descent with the specified coherent cocycle. Apply it first to the finite coefficient reductions; their natural augmentations and uniform bounds give the completed and rational comparisons of Semi-infinite orbits and weight functors, Appendices M–N. At a larger jet level the same datum is pulled back along the jet homomorphism, so its cocycle and descended object are unchanged. Ordinary descent adds no torsor-dimension shift; normalization for a smooth atlas is a separate smooth-pullback operation.

The map \(l\) is proper on the finite support by Proposition H.2.1.1, so the actual \(Rl_!\) of Semi-infinite orbits and weight functors, Theorem O.4.2.1 is \(Rl_*\). Proper base change Theorem O.4.1.1 for the source Cartesian square identifies its pullback along \(S'\to S\) with (H.3.1) on \(S'\). The input target family and the vertical complex already have their coherent pullback comparisons. Composing them gives the required cartesian family. On existing finite algebraic charts the earlier coefficient functors supply the ordinary constructible complexes. This does not claim a completed enhancement for the full ind-stack.

For two kernels form the actual chain of decorated bundles \(\mathcal E_0\to\mathcal E_1\to\mathcal E_2\). Forgetting \(\mathcal E_1\) composes the actual punctured-curve isomorphisms and the decorations in (H.2.3). In disc frames this is the twisted convolution space and proper convolution map proved in Lesson 7 and used in The Mirković–Vilonen theorem: identifying the dual group, Theorem B.4.1.1. The composite finite support is bounded by the sum of the two dominant bounds. Both iterated actions use the same pullback of \(F\) from \(\mathcal E_2\) and the same ordered tensor of the vertical kernels \(A,B\). Proper base change for the intermediate projection, the projection formula Theorem O.4.3.1 and coherent composition Theorem O.4.2.1 identify them with the proper direct image of that common complex to \(\mathcal E_0\). Pushing first along the convolution map replaces the ordered vertical tensor by \(A*B\). This proves (H.3.2) as an actual comparison, not just a class identity.

For three and four kernels use the common full chain before forgetting any bundle. Every bracketing uses the same coherent direct-image composition and tensor associativity on that chain; their proved composition cocycles give the pentagon. The zero modification is the diagonal with its actual unit, so its action is \(F\) with the actual unit comparison. The inversion comparison on coefficient objects uses the actual finite frames of Convolution and rigidity, Lemma B.1.1.1 and Theorem B.2.1.1. On a sufficiently deep frame scheme its source map \(p\) and inverse-endpoint map \(f\) are smooth with connected fibres of the same dimension \(h_n=n\dim G\), in every geometric characteristic. That theorem constructs \(p^*I(Q)[h_n]=f^*Q[h_n]\) with the actual equivariance, involution and chain reversal. Applied to the raw simple kernel it gives \(I(I_\lambda)=I_{\lambda^*}\), with no additional raw Tate twist; the twist \(d_\lambda\) in the contragredient dual involves Verdier duality separately. Therefore applying the same half twist \(d_\lambda/2=d_{\lambda^*}/2\) gives exactly the stated normalized inverse kernel. These common-frame comparisons descend to our local disc-frame charts and agree under frame changes by that theorem's action cocycle. This supplies the actual coefficient comparison rather than inferring it just from inverse labels. Descent, base change, projection and composition are defined over the split finite field when the data are, so they preserve the specified Weil maps. ∎

Theorem H.1.1.1 and Proposition H.2.1.1 prove the full projective modification and both Cartesian squares. Theorem H.3.1.1 constructs the actual action on cartesian coefficient families and its common-chain composition. Editable SVG source.

H.4. An explicit smooth central torus cover

Theorem H.4.1.1. For a split connected reductive \(G\), let \(G_{\rm sc}\to G_{\rm der}\) be its actual simply connected central cover from that programme theorem, with torus \(T_{\rm sc}\), weight lattice \(P=X^*(T_{\rm sc})\), and coroot lattice \(P^*=X_*(T_{\rm sc})\). There is a split connected reductive group \[ \widetilde G=G_{\rm sc}\rtimes T,\qquad q(h,t)=\phi(h)t,\qquad \widetilde T=T_{\rm sc}\times T, \tag{H.4.1} \] whose kernel \(S\) is the smooth central torus \(T_{\rm sc}\). The displayed semidirect action is the descended conjugation action of \(T\) on \(G_{\rm sc}\). Its derived subgroup is exactly \(G_{\rm sc}\).

If \(X=X^*(T)\), the torus character map and its dual are \[ q^*:X\hookrightarrow P\oplus X,\quad x\mapsto(\phi_T^*x,x), \qquad q_*:P^*\oplus X_*(T)\to X_*(T),\quad (m,\ell)\mapsto\phi_{T*}m+\ell. \tag{H.4.2} \] The cokernel of \(q^*\) is the free lattice \(P\). A root is \((\alpha_{\rm sc},\alpha)\), and its coroot is \((\alpha^\vee_{\rm sc},0)\). In particular every original cocharacter has the canonical lift \((0,\lambda)\). The root subgroup maps and unipotent radical map \(\widetilde N\to N\) are isomorphisms on all parameter rings.

Proof. Pinnings and the classification of split reductive groups, Theorem 10.1 and its derived-group subsection proves the presentation of \(G\) is \((G_{\rm sc}\times C)/H\), with \(C\) a split torus and \(H\) the actual finite diagonalizable central subgroup, which may be nonreduced. Its torus is \((T_{\rm sc}\times C)/H\). On that finite faithfully flat cover of \(T\), let \((a,c)\) act on \(G_{\rm sc}\) by conjugation with \(a\). Changing the lift by an element of \(H\) changes \(a\) by a central element, so the actions agree as morphisms on the whole fibre product. Effective affine descent of this morphism gives an actual \(T\)-action; multiplication and its action identity descend from the same cover. It fixes \(T_{\rm sc}\), scales the root \(\alpha\) by the actual character \(\alpha(t)\), and satisfies \(\phi(\sigma_t(h))=t\phi(h)t^{-1}\). Thus (H.4.1) is a group homomorphism. The semidirect product is smooth and connected. Its kernel is determined on all test rings: if \(\phi(h)t=1\), then \(h\in T_{\rm sc}\), and \(t=\phi_T(h)^{-1}\). The first assertion follows from the same central quotient presentation, whose inverse image of \(T\) is \(T_{\rm sc}\times C\). Hence \[ S=\{(a,\phi_T(a)^{-1}):a\in T_{\rm sc}\}. \tag{H.4.3} \] For such a pair, \(\sigma_{\phi_T(a)^{-1}}\) is conjugation by \(a^{-1}\). Multiplication in the semidirect product shows that the pair commutes with \(G_{\rm sc}\); it also commutes with \(T\), since the latter fixes \(T_{\rm sc}\). Thus the kernel is central. The map is fppf surjective: the presentation cover \((h,c)\) of \(G\) lifts to \((h,\operatorname{image}(1,c))\). It is the actual torsor under the smooth split \(S\), and therefore is smooth.

The quotient of the semidirect product by \(G_{\rm sc}\) is \(T\). Since the programme theorem proves \(G_{\rm sc}\) perfect as an fppf sheaf, it is exactly the derived subgroup. Reductivity follows by an actual central quotient already covered by the programme theorem. The map \[ G_{\rm sc}\times T_{\rm sc}\times C\longrightarrow\widetilde G,\qquad (h,a,c)\longmapsto(ha^{-1},\operatorname{image}(a,c)) \] is a homomorphism: the descended torus action is \(\operatorname{Ad}(a)\), which gives the product \(h_1h_2(a_1a_2)^{-1}\) in its first coordinate. It is fppf surjective by the same torus cover. Its kernel is \(\{(a,a,c):(a,c)\in H\}\), an actual finite diagonalizable central subgroup. Thus \(\widetilde G\) is the central quotient of this split reductive product. Root data, Weyl chambers and the Bruhat decomposition, Theorem 7.1, proves that quotient reductive, including when its kernel is nonreduced. Its split torus is the displayed one. Conjugation gives the displayed root characters, and the embedded simply connected group gives their coroots. The character map in (H.4.2) is the graph of restriction; its quotient is \(P\) via \((p,x)\mapsto p-\phi_T^*x\). Its dual gives the cocharacter formula. The original root groups are unchanged under the finite central quotient in the programme construction; their ordered products therefore give \(\widetilde N\simeq N\) on all rings. This also proves the root-line and central-coordinate assertions, with no reduced substitute for \(H\). ∎

H.5. Schubert supports map by a universal homeomorphism

Theorem H.5.1.1. Fix dominant \(\lambda\in X_*(T)\), and put \(\widetilde\lambda=(0,\lambda)\). The induced map \[ u:Z_{\widetilde\lambda}(\widetilde G)\longrightarrow Z_\lambda(G) \tag{H.5.1} \] is proper, surjective and universally injective, hence a universal homeomorphism. This assertion does not identify the two schemes as schemes.

Proof. Extension of the structure group gives the actual map of Grassmannian functors, hence of their represented finite supports. The lift is dominant because its root pairings equal those of \(\lambda\). The map sends the open source orbit into the target orbit. The ideal of the target closed Schubert support therefore vanishes on the open source orbit and on its schematic closure. It factors through the displayed target scheme. Both supports are projective by Loop groups and the affine Grassmannian, Theorem 7.1, so the map is proper.

Over any algebraically closed extension \(\Omega\), the maps \(\widetilde G(\Omega[[t]])\to G(\Omega[[t]])\) and \(\widetilde G(\Omega((t)))\to G(\Omega((t)))\) are surjective. Indeed a fibre is an \(S\)-torsor; a split-torus torsor is a tuple of invertible modules, and over a field or a local power-series DVR each is free. The map on open orbit points is therefore surjective. Properness makes its geometric image closed; containing the dense open target orbit makes it all of the target Schubert support.

To prove uniqueness, take two source points \(g_1\widetilde G(O),g_2\widetilde G(O)\) with the same target point. Lift the integral element \(q(g_1)^{-1}q(g_2)\) to \(a\in\widetilde G(O)\). Then \(s=g_2a^{-1}g_1^{-1}\in S(F)\). Let \(A:\widetilde G\to T\) be the semidirect-product projection. On the whole reduced source Schubert scheme its torus-Grassmannian image is the reduced point \(t^\lambda\): this holds on the open orbit, and its closed incidence equations hold on the schematic closure. Thus the torus valuations of \(A(g_1),A(g_2)\) agree, and \(A(a)\) is integral. Write \(s=(b,\phi_T(b)^{-1})\), and let \(m\in P^*\) be the valuation of \(b\). Its projected valuation is \(-\phi_{T*}m=0\). The map \(\phi_{T*}\) identifies the coroot lattice with the original coroot lattice, so is injective. Therefore \(m=0\), whence \(b\in T_{\rm sc}(O)\) and \(s\in S(O)\). The two source right cosets are equal.

This uniqueness holds after every algebraically closed field extension. Each geometric fibre of a finite-type map consequently has exactly one geometric point. It is a zero-dimensional scheme: on an affine finite-type neighbourhood a unique maximal ideal is its nilradical by The Nullstellensatz and Jacobson rings, Theorem 2.2, and finitely many nilpotent algebra generators make its coordinate ring Artinian. Its residue extension is purely inseparable, since two separable embeddings would give two geometric points. These facts also show injectivity after every base change: over any point of the new base, pass to an algebraic closure of its residue field and use that unique geometric fibre. Thus the map is universally injective. A proper surjective map is universally closed, and the resulting universally closed bijection on points after any base change is a homeomorphism. This proves the assertion without assuming normality, separability or an isomorphism of supports. ∎

H.6. Actual coefficient equivalence on these supports

Lemma H.6.1.1. For a proper universal homeomorphism \(u:Y'\to Y\) between finite-type schemes, the actual comparisons are \[ u^*\simeq u^!,\qquad Ru_*=Ru_!,\qquad F\xrightarrow{\sim}Ru_*u^*F,\qquad u^*Ru_*F'\xrightarrow{\sim}F'. \tag{H.6.1} \] They are inverse equivalences of the constructible coefficient categories used here. They preserve tensor units, actual duality, the perverse heart and IC intermediate extensions. In (H.5.1), \[ u^*I_\lambda\simeq I_{\widetilde\lambda},\qquad u^*\bigl(I_\lambda(d_\lambda/2)\bigr) \simeq I_{\widetilde\lambda}(d_{\widetilde\lambda}/2). \tag{H.6.2} \] These are comparisons of actual objects and adjunction maps, preserving Frobenius when the map and data descend to the split finite field.

Proof. A geometric fibre is a finite Artinian scheme with a single point and algebraically closed residue field. Its finite étale algebras are just products of the unique étale lift of that residue field. The finite étale lifting and its algebra maps are proved in Semi-infinite orbits and weight functors, §§A.4.2 and A.5.5. Apply them successively through the nilpotent maximal ideal: for a square-zero step formal étaleness gives the unique lift and its unique maps; the finite nilpotent filtration finishes the argument. Thus its étale coefficient site has the same stalk functor as a point, with no positive cohomology. For a complex pulled from \(Y\) its entire fibre complex is the original geometric stalk, and the restriction adjunction unit is the identity. For a complex on \(Y'\), that same calculation identifies the fibre proper image with its unique stalk and identifies the counit with the identity.

The actual proper base-change theorem of Semi-infinite orbits and weight functors, Theorem O.4.1.1 now makes both the unit and the counit in (H.6.1) isomorphisms on every geometric stalk. Geometric stalks detect isomorphisms of the proved constructible complexes, so they are isomorphisms. Perform this first at the finite coefficient models and reductions of Semi-infinite orbits and weight functors, Appendices M–N; the uniform fibre bound zero retains coherent completion, rationalization and the algebraic scalar extension proved in The Mirković–Vilonen theorem: identifying the dual group, Theorem D.3.1.1. Hence the same actual comparisons hold for our coefficients, with their transition maps.

Properness gives \(u_!=u_*\), including their specified comparisons. An equivalence has the same inverse as its left and right adjoints, so the uniqueness of adjoints identifies \(u^*\) with \(u^!\), using these units and counits. Proper structural duality in Semi-infinite orbits and weight functors, Theorem P.3.3.1 and this identification give \(D_{Y'}u^*=u^*D_Y\). The tensor pullback preserves the unit and tensor comparisons by its actual construction. The map identifies the dimensions of closed supports: geometric fibres have dimension zero, and the ordinary transcendence-degree dimension formula, or chains under this universal homeomorphism, give the same dimension. Thus ordinary support bounds and their duals identify the middle perverse hearts. The functors commute with open restrictions and with closed supports by actual base change. Exactness therefore preserves the image of the canonical map from perverse \(j_!\) to perverse \(j_*\), which is the IC intermediate extension. The open raw IC unit has the same dimension shift and pulls back to the same constant unit, proving the first comparison in (H.6.2). Root pairings give \(d_{\widetilde\lambda}=d_\lambda\), so the identical chosen half twist gives the second.

Every unit, counit, proper base change, completion and duality comparison used is an actual natural map. If defined over the finite field, each commutes with its Weil pullback. The geometric fibre unit is fixed, so no additional Tate line or scalar has been introduced. ∎

H.7. Semi-infinite slices, character maps and Whittaker complexes

Theorem H.7.1.1. If \(X_{\lambda\nu}=Z_\lambda\cap S_\nu\) is nonempty, then \(\lambda-\nu\) belongs to the original coroot lattice. Let \(\theta_{\rm sc}\in P^*\) be its unique lift, and put \[ \widetilde\nu=(-\theta_{\rm sc},\lambda),\quad \widetilde\mu=(0,\mu). \tag{H.7.1} \] The map \(S_{\widetilde\nu}\to S_\nu\) is an isomorphism of the actual orbit functors on all parameter rings. The induced map \[ j:X_{\widetilde\lambda,\widetilde\nu}\hookrightarrow X_{\lambda\nu} \tag{H.7.2} \] is a closed immersion with nilpotent ideal and identical reduced finite slices. Under it the descended conductor characters satisfy \[ j^*\chi_\mu^\nu=\chi_{\widetilde\mu}^{\widetilde\nu},\qquad R\Gamma_c(X_{\lambda\nu},I_\lambda\otimes(\chi_\mu^\nu)^*\mathcal L_\psi) \simeq R\Gamma_c(X_{\widetilde\lambda,\widetilde\nu}, I_{\widetilde\lambda}\otimes (\chi_{\widetilde\mu}^{\widetilde\nu})^*\mathcal L_\psi). \tag{H.7.3} \] The comparison preserves the actual Weil operators, compact-image maps, the raw shifts \(h_\nu\), and the normalization \(D=(d_\lambda+h_\nu)/2\). The relevant global strata also correspond on all parameter bases. Use \(\widetilde{\mathcal T}_0\) induced by the sum of the positive source coroots, which projects to \(\mathcal T_0\); for any dominant lift \(\widetilde\mu\), one has \(\mathcal B_{\widetilde\mu}\simeq\mathcal B_\mu\) and \(\Psi_{\widetilde\mu}^{\mathrm n}\simeq\Psi_\mu^{\mathrm n}\), including the evaluation morphism and its unit.

Proof. A point of \(S_\nu\) has representative \(n t^\nu\). Lift \(n\) through \(\widetilde N\simeq N\), and use the initial lift \(\nu_0=(0,\nu)\). For a point of the source Schubert support lying over this point, lift the integral frame discrepancy as in Theorem H.5.1.1. The remaining discrepancy is a central \(S(F)\)-element of torus valuation \((m,-\phi_{T*}m)\). The semidirect abelianization label of every source point is \(\lambda\). Hence \(\nu-\phi_{T*}m=\lambda\), proving \(\lambda-\nu\) is a coroot and \(m=-\theta_{\rm sc}\). Its source semi-infinite label is exactly (H.7.1).

For arbitrary parameter rings the stabilizer of \(t^\nu\) in \(N((t))\) is \(\operatorname{Ad}(t^\nu)N[[t]]\). To see the scheme assertion, \(N\cap G[[t]]=N[[t]]\) follows from the closed subgroup equations and injectivity of \(R[[t]]\to R((t))\); conjugate by \(t^\nu\). The source root subgroup isomorphism identifies these stabilizers, since every root pairing with \(\widetilde\nu\) equals that with \(\nu\). Thus their actual quotient orbit functors are isomorphic, with the root coordinates and all nilpotents retained.

Both finite intersections are closed subfunctors of this common semi-infinite orbit. The map of Schubert supports sends the source intersection into the target, so it is the closed immersion (H.7.2). The point calculation and universal homeomorphism prove that its underlying sets coincide after all algebraically closed extensions. Its defining coherent ideal is therefore contained in the nilradical of the finite-type target slice, and is nilpotent by Noetherianity. This proves the exact nilpotent-immersion assertion instead of replacing that slice by its set of points.

For the global assertion, \(q\) identifies the source coroot-sum background with the original background. Every root line bundle therefore corresponds, including after twisting by the lifted label. The map \(\widetilde N\simeq N\) is equivariant for \(\widetilde T\to T\), so the two twisted unipotent group bundles are actually isomorphic. Their torsor categories with specified torus quotient are consequently equivalent on every parameter base, by the gluing description proved in Theorem G.1.1.1. The Laurent middle schemes, their dimensions and residue maps are literally the same. Thus the character objects have the same lisse line, perverse shift and half twist; their units and Weil maps correspond. Central data are fixed by the specified torus bundles throughout this comparison.

The simple-root quotient coordinates correspond under \(q\). Conjugation by \(\widetilde\mu\) scales them by the same powers as conjugation by \(\mu\), and the coefficient of \(t^{-1}\) has the same positive residue sign. Their descended functions therefore agree as actual morphisms, proving the first identity in (H.7.3). Dominance of \(\mu+\nu\), when required for descent, is equivalent to dominance of \(\widetilde\mu+\widetilde\nu\), by equality of all simple-root pairings.

Apply Lemma H.6.1.1 to the Schubert map and to the nilpotent closed immersion. Its IC comparison restricts to \(j^*I_\lambda=I_{\widetilde\lambda}\), and the character-line comparison pulls back along the equality of functions. The actual adjunction unit \(F\to Rj_*j^*F\), followed by coherent proper-support composition Theorem O.4.2.1, gives the compact comparison in (H.7.3). Root pairings identify \(d_\lambda,h_\nu\) with their lifted values. Hence the raw shifts and extra twist \(D\) are literally the same. The preceding actual units and residue maps preserve Frobenius, so this transfers concentration, vanishing and multiplicity calculations with their actual operators rather than only their dimensions. ∎

Theorems H.4.1.1, H.5.1.1 and H.7.1.1 and Lemma H.6.1.1 prove the group, support, coefficient and slice comparisons. The cover changes the central lift of a slice, as the \(PGL_2\) example shows. Editable SVG source.

H.8. Exercises

Exercise H.8.1 (medium). For \(GL_2\) and \(\lambda=(1,0)\), identify the full modification scheme over the framed stratum \(\mathcal B_0=\mathbf A^1\), its inverse dominant type, and \(c(V,\lambda)\) for the standard and symmetric-square representations. Compute the fibre of the normalized Hecke action on the constant coefficient family, retaining Frobenius.

Solution. On this Schubert support the actual closed incidence equations are \(tR[[t]]^2\subset\Lambda\subset R[[t]]^2\), with rank-one projective fibre quotient. They hold on the open orbit and its schematic closure. Their entire locus is the projective line of rank-one quotients of \(R^2\); local bases identify that locus with the orbit itself. Thus this is the projective line functor in Loop groups and the affine Grassmannian, §3. The supplied disc frame therefore gives the entire \(\mathbf A^1\times\mathbf P^1\), projective over \(\mathbf A^1\). The chart in Exercise G.7.1 is its open \(\mathbf A^1\times\mathbf A^1\); this chart alone is not proper. The inverse dominant type is \((0,-1)\). Standard weights pair with \(\lambda\) as \(1,0\), so \(c=1\). Symmetric-square weights pair as \(2,1,0\), so \(c=2\); these are the actual two pole bounds.

The normalized vertical kernel on \(\mathbf P^1\) is \(E[1](1/2)\). Its actual compact image is \[ R\Gamma(\mathbf P^1,E[1](1/2)) \simeq E(1/2)[1]\oplus E(-1/2)[-1]. \tag{H.8.1} \] Indeed localization for the affine line and the point at infinity gives the triangle \(R\Gamma_c(\mathbf A^1,E)\to R\Gamma(\mathbf P^1,E)\to E\). The unit \(E\to R\Gamma(\mathbf P^1,E)\) restricts to the identity at infinity, splitting this triangle. The actual affine trace identifies its first term with \(E(-1)[-2]\), by Semi-infinite orbits and weight functors, Theorem Q.1.2.1. These units, restrictions and traces preserve Weil maps. With \(a^2=q\) and the chosen \(E(1/2)\) having Frobenius \(a^{-1}\), the displayed summands have Frobenius \(a^{-1}\) in degree \(-1\) and \(a\) in degree \(1\). Thus a general coefficient-family fibre can have both degrees; the Whittaker concentration theorem requires its additional character and boundary arguments.

Exercise H.8.2 (advanced). For \(G=PGL_2\), write \(\lambda=n\varpi^\vee\), \(\nu=(n-2c)\varpi^\vee\), \(0\le c\le n\), and \(\mu=m\varpi^\vee\), \(m\ge0\). Identify the actual cover in Theorem H.4.1.1 and all lifted labels. Determine when \(\mu+\nu\) is dominant and calculate \(d_\lambda,h_\nu,D\). Explain the zero-slice lift when \(n=2\), and check characteristic two.

Solution. The adjoint torus character is \(\alpha\), with \(\langle\alpha,\varpi^\vee\rangle=1\) and \(\alpha^\vee=2\varpi^\vee\). The cover is \(SL_2\rtimes\mathbf G_m\), where \(z\) acts by conjugation with \(\operatorname{diag}(z,1)\). The actual group isomorphism to \(GL_2\) is \[ (h,z)\longmapsto h\operatorname{diag}(z,1),\qquad g\longmapsto \bigl(g\operatorname{diag}((\det g)^{-1},1),\det g\bigr). \] Both are morphisms on every parameter ring and are mutual inverses; multiplication is exactly the stated semidirect law. The map \(q\) becomes the ordinary \(GL_2\to PGL_2\), with full smooth scalar kernel.

In semidirect torus coordinates \(q_*(r,s)=2r+s\), and the change to the usual \(GL_2\) cocharacter pair is \((r,s)\mapsto(r+s,-r)\). Thus \[ \widetilde\lambda=(n,0),\quad \widetilde\nu=(n-c,c),\quad \widetilde\mu=(m,0),\quad \widetilde\eta=(m+n-c,c). \tag{H.8.2} \] Their images have the original labels \(\lambda,\nu,\mu,\mu+\nu\). The source root pairing of \(\widetilde\eta\) is \(m+n-2c\), so both source and target dominance mean \(m+n-2c\ge0\). The common values are \(d_\lambda=n\), \(h_\nu=n-2c\), \(D=n-c\). For \(n=2,c=1\), the target semi-infinite label is zero and its source label is \((1,1)\). The source support has determinant degree two, so replacing this by \((0,0)\) changes the slice.

The source and target root coordinates and conjugation powers agree. In particular their actual conductor functions take the same coefficient of \(t^{-1}\) with the same positive residue sign. Theorem H.7.1.1 and Lemma H.6.1.1 identify the whole compact complexes, with their actual units, raw shift \([n-2c]\), normalization \((n-c)\), and Weil operators. The group isomorphism, torus exponents and smooth scalar kernel use no division by two in the ground field. They therefore remain valid in characteristic two; the possibly inseparable Schubert map has precisely the coefficient equivalence proved in Lemma H.6.1.1.

Appendix I. Finite bounded decoration schemes and parameter-flat quotients

Here \(R\Gamma\) denotes derived Zariski sections of the indicated algebraic vector bundle. We construct bounded spaces of actual representation maps and polynomial tensor relations, retaining nonflat changes of parameter ring and central inverse maps. The free Whittaker patterns, §§2.1–2.2 describes the global decoration problem. The affine cohomology, projective coherent finiteness and universally exact two-term replacement needed below are fully proved in Constructible complexes on algebraic varieties, §§W.2, W.4 and U.1. Appendices J–K construct the full highest-line algebra, its affine flag cone, the projective fixed-bundle torus quotient and exact relative defect strata; the required global stack and boundary constructions remain further parts of the auxiliary compactification.

I.1. An actual finite two-term section scheme

Proposition I.1.1.1. Let \(R\) be Noetherian and \(\mathcal V\) a finite locally free sheaf on \(\mathbf P^1_R\). There are finite projective \(R\)-modules \(P^0,P^1\), an actual map \(d:P^0\to P^1\), and natural comparisons for every \(R\)-algebra \(A\), \[ R\Gamma(\mathbf P^1_A,\mathcal V_A)\simeq[P^0\otimes_R A\xrightarrow{d\otimes1}P^1\otimes_R A], \qquad \Gamma(\mathbf P^1_A,\mathcal V_A)=\ker(d\otimes1). \tag{I.1.1} \] The displayed complex has degrees zero and one. Consequently the entire section functor, including nonflat parameter changes, is the affine finitely presented scheme \[ \mathscr S(\mathcal V)= \operatorname{Spec}\left( \operatorname{Sym}_R((P^0)^\vee)/ \bigl(\operatorname{image}((P^1)^\vee\xrightarrow{d^\vee}(P^0)^\vee)\bigr) \right). \tag{I.1.2} \] This is a kernel scheme; it need not be a vector bundle.

Proof. Use the two standard affine charts with coordinates \(t,t^{-1}\). Their section modules and the intersection module give \(C=[C^0\xrightarrow{\delta}C^1]\). The actual augmented Čech comparison in the earlier Constructible complexes on algebraic varieties, §W.2 computes the derived sections. Each term is flat over \(R\): on a chart the bundle is a finite projective module over \(R[t]\), and on the overlap over \(R[t,t^{-1}]\); these rings are \(R\)-flat. The same affine calculation identifies the base-changed complex literally with \(C\otimes_R A\). Projective coherent finiteness in Constructible complexes on algebraic varieties, §W.4 makes \(H^0(C),H^1(C)\) finite over \(R\).

For clarity, apply the explicit finite replacement of Constructible complexes on algebraic varieties, §U.1 as follows. Take a finite free module \(P^1\) surjecting onto \(H^1(C)\), and lift its chosen basis to \(g:P^1\to C^1\). The map \((\delta,-g):C^0\oplus P^1\to C^1\) is surjective. Set \(P^0\) equal to its kernel and \(d\) equal to projection to \(P^1\). Since its target and middle term are flat, the kernel is flat and the exact sequence remains exact after every tensor operation, by the elementary exact-tensor proof in Constructible complexes on algebraic varieties, §U.1. The further sequence \[ 0\longrightarrow H^0(C)\longrightarrow P^0 \longrightarrow\ker(P^1\to H^1(C))\longrightarrow0 \tag{I.1.3} \] makes \(P^0\) finite; the right term is finite by Noetherianity. A finite flat module over a Noetherian ring is finite projective. Explicitly localize at a prime, lift a residue-field basis and use Nakayama to obtain a finite free surjection. Flatness makes its reduction left exact; the reduced map is an isomorphism, so Nakayama kills its finite kernel. Clearing denominators spreads these bases to open neighbourhoods.

The maps \(P^0\to C^0\) and \(g:P^1\to C^1\) give the actual complex comparison. Tensoring the universally exact kernel sequence with \(A\) identifies its degree-zero kernel with \(\ker(\delta\otimes1)\). Its degree-one cokernel is \(\operatorname{coker}(\delta\otimes1)\), since \(P^1\otimes A\) surjects onto that module and its kernel is the projection of \(P^0\otimes A\). Thus the complex comparison is a quasi-isomorphism for every \(A\), yielding (I.1.1) with the actual restriction maps and Čech augmentation.

Finite projectivity identifies the \(A\)-points of the affine space of \(P^0\) with \(P^0\otimes_R A\). The equations \(d^\vee((P^1)^\vee)=0\) say exactly \(d\otimes1=0\), proving (I.1.2) by the whole functor of points. Both the coordinate algebra and its ideal are finitely presented. The associated universal point is the actual global section on the universal parameter scheme. At no step was \(H^0(C)\otimes A\) incorrectly substituted for \(H^0(C\otimes A)\). ∎

I.2. Polynomial relations and closed pole-bound transitions

Proposition I.2.1.1. Fix a \(G\)-bundle \(\mathcal E\) and actual source line bundles \(\mathcal L_i\) on \(\mathbf P^1_R\). Fix finitely many actual algebraic \(G\)-representations \(V_i\), pole bounds \(N_i\ge0\), and finitely many polynomial relations in maps \[ \kappa_i:\mathcal L_i\longrightarrow (V_i)_{\mathcal E}(N_i x). \tag{I.2.1} \] Each relation is given by specified equivariant tensor maps and source-line identifications, so that after clearing its finite pole bound it is a section of an actual vector bundle \(\mathcal W_a\). Constant unit terms are allowed. The scheme of all such maps satisfying the relations is affine and finitely presented over \(R\), with its actual universal maps and arbitrary-base-change comparison. In particular the central-character inverse equations are retained as equations of maps.

Increasing any pole bound gives a closed immersion of the corresponding section and relation schemes. Their filtered union classifies the entire finite family of meromorphic maps with those relations, on every parameter algebra.

Proof. Apply Proposition I.1.1.1 to \(\mathcal V_i=\mathcal L_i^{-1}\otimes(V_i)_{\mathcal E}(N_i x)\). The finite product \(\mathscr S=\prod_i\mathscr S(\mathcal V_i)\) has the actual universal sections. Evaluating the specified polynomial tensor maps in these sections yields a global section of \(\mathcal W_a\) on \(\mathbf P^1_{\mathscr S}\). Yoneda and the actual section comparison give a morphism \(\Theta_a:\mathscr S\to\mathscr S(\mathcal W_a)\). The zero section of the target kernel scheme is closed, defined by all its linear coordinates. The common zero fibre of these finitely many morphisms is therefore an affine finitely presented closed subscheme. Its \(A\)-points are exactly the given equations of actual maps, for arbitrary \(A\); this uses the universal sections and (I.1.1), rather than equations merely on geometric points.

For a central character \(\chi\), include \(\kappa^\chi,\kappa^{-\chi}\) and the specified tensor-unit identification. The relation \(\kappa^\chi\kappa^{-\chi}=1\) is a polynomial relation of the stated form, after clearing the sum of the pole bounds. Off \(x\) it makes the two actual line maps inverse isomorphisms. Thus central units are imposed, not discarded by a lattice quotient or replaced by fibrewise scalar choices.

For \(N'\ge N\), the quotient of \(\mathcal V(N'x)\) by \(\mathcal V(Nx)\) is the jet bundle on the fixed Cartier divisor \((N'-N)x\). Its sections are a finite projective \(R\)-module: the divisor is finite locally free over \(R\), and its locally free bundle is consequently finite projective over \(R\). The exact sheaf sequence, including its left injection, remains exact after every base change. Restriction to those jets defines an actual morphism from the larger section scheme to their affine space. Its zero fibre is exactly the smaller section scheme on every algebra \(A\), and hence is a closed immersion. Intersecting with the polynomial relation locus gives the same assertion for the decoration schemes. Vanishing of the larger cleared relation is equivalent to vanishing of the smaller one: all further pole inclusions are injective on these locally free sheaves. The local Cartier equation is a non-zero-divisor even over a nilpotent parameter ring.

Finally \(\mathcal V(*x)=\varinjlim_N\mathcal V(Nx)\). On the two affine charts and their affine overlap, sections commute with this filtered union; its finite Čech equalizer does as well. A finite family of global meromorphic sections therefore has one common finite tuple of pole bounds. Its finitely many relations hold at that bounded stage, because the pole inclusions are injective. This proves the entire claimed filtered functor, with actual transition morphisms and all parameter nilpotents. ∎

I.3. The injection and flat-quotient open locus

Theorem I.3.1.1. For each of the finitely many maps of Proposition I.2.1.1, require that its meromorphic section is nonzero on every geometric curve fibre. This is an open condition on the bounded relation scheme. It is equivalent to sheaf injectivity with finitely presented, parameter-flat quotient away from \(x\), and it remains equivalent after every parameter base change. Over Noetherian parameters the quotients are coherent. These compactified injection conditions give a finitely presented open subscheme of the bounded relation scheme.

Proof. In the section kernel scheme (I.1.2), the zero section is closed. After any residue-field extension, its complement says exactly that the section is not identically zero. The finite intersection of these inverse-image complements is therefore the stated open locus. A nonzero section of a bundle on a geometric \(\mathbf P^1\) is nonzero on its generic point. Since its source is a line, the induced map is injective as a coherent sheaf on that integral curve, including on the complementary affine chart. Clearing poles adds only the fixed \(x\)-twist.

Here is a direct parameter-module proof that these fibre injections imply universal injectivity and a flat quotient. On either affine chart the source and target are finite projective \(R[t]\)-modules \(L,V\). Choose split embeddings into finite free modules, and write the source projector as a polynomial matrix \(e\) and the given map, followed by the target embedding, as a polynomial matrix \(F\). Then its kernel after tensoring by any \(R\)-module \(M\) consists of polynomial columns \(z\in M[t]^b\) with \[ ez=z,\qquad Fz=0. \tag{I.3.1} \] Let \(M_0\subset M\) be the finite \(R\)-submodule generated by the finitely many coefficients of \(z\). Both identities hold already in \(M_0[t]\), since its inclusion in \(M[t]\) is injective. Localize \(R\) at any maximal ideal \(\mathfrak m\). Modulo \(\mathfrak m\), the source image of \(e\) is \(L_{\kappa(\mathfrak m)}\), and \(F\) is injective there by the curve-fibre injection just proved. Tensoring that injective map over the residue field with its vector space \((M_0)_\mathfrak m/\mathfrak m(M_0)_\mathfrak m\) remains injective. Hence every coefficient of \(z\) vanishes in that quotient. Those coefficients generate \((M_0)_\mathfrak m\), so Nakayama gives \((M_0)_\mathfrak m=0\). This holds at every maximal ideal, whence \(M_0=0\) and \(z=0\).

Therefore the source-to-target map remains injective after tensoring by every \(R\)-module. The source and target themselves are \(R\)-flat. The exact tensor sequence, or the free-presentation tensor argument of the earlier Constructible complexes on algebraic varieties, §U.1, now shows that the cokernel is \(R\)-flat. This is a sheaf-local statement on the two charts, so it proves universal sheaf injectivity and finitely presented parameter-flat quotient on the curve. Inverting the fixed Cartier equation gives the assertion away from \(x\) for the original meromorphic maps. The same argument works over an arbitrary base-changed algebra: finiteness of the coefficient-generated module and Nakayama do not require that algebra Noetherian.

Conversely, if the meromorphic source map is injective away from \(x\) and its quotient is parameter-flat there, its exact sequence remains left exact on every curve fibre. A zero map on a geometric curve fibre could not be an injection from its rank-one source on that nonempty affine curve. Thus every section is geometrically nonzero, and the parameter point lies in the stated open locus. The cleared section is nonzero exactly when the meromorphic one is: the pole inclusions are universally injective. This proves the equivalence on all parameters. ∎

Propositions I.1.1.1–I.2.1.1 and Theorem I.3.1.1 retain the actual section scheme, polynomial relations, closed pole transitions and parameter-flat quotients. The axes are a coordinate schematic of the exact equation \(ax=0\), computed in Exercise I.4.1. Editable SVG source.

I.4. Exercises

Exercise I.4.1 (medium). Over \(R=k[a]\), define a rank-two bundle by the transition matrix \(\begin{pmatrix}t^{-2}&at^{-1}\\0&1\end{pmatrix}\). Compute its entire section scheme and derived coherent sections on every parameter algebra. Determine its injection/flat-quotient open locus and test \(a=\epsilon\) over the dual numbers.

Solution. Set \(R=k[a]\). On the two charts define a rank-two bundle with frames \(e_0,f_0\) and \(e_\infty,f_\infty\), and transitions \[ e_\infty=t^{-2}e_0,\qquad f_\infty=f_0+a t^{-1}e_0. \tag{I.4.1} \] Its transition determinant \(t^{-2}\) is a Laurent unit, so this is an actual bundle over every \(R\)-algebra. It is an extension \(0\to\mathcal O(-2)\to\mathcal V_a\to\mathcal O\to0\). The positive Čech difference of the two lifts of \(1\) is \(a t^{-1}e_0\), whose coefficient is \(a\) in \(H^1(\mathcal O(-2))=R[t,t^{-1}]/(R[t]+t^{-2}R[t^{-1}])=R\cdot[t^{-1}e_0]\).

For every \(A\), a global section has quotient coordinate a constant \(x\in A\). Comparing its two polynomial \(e\)-coordinates gives \(b_0(t)=t^{-2}b_\infty(t^{-1})+a x t^{-1}\). The three disjoint Laurent ranges force \(b_0=b_\infty=0\) and \(ax=0\). Therefore the actual section scheme is \[ \mathscr S(\mathcal V_a)=\operatorname{Spec}k[a,x]/(ax), \qquad R\Gamma(\mathcal V_a)\simeq[R\xrightarrow{a}R]. \tag{I.4.2} \] The Čech differential has components \(t^{-2}b_\infty-b_0+a t^{-1}c_\infty\) and \(c_\infty-c_0\). Polynomial subtraction eliminates the nonconstant quotient-coordinate tails, leaving the constant \(x\); subtraction of \(b_0\) and \(t^{-2}b_\infty\) eliminates the two outer \(e\)-ranges. The sole remaining differential is \(x\mapsto ax\) in the positive \(t^{-1}e_0\) coordinate. These eliminations use only additions and the unit signs, so give the actual two-term comparison on every parameter ring. The nonzero-section open locus is \(x\ne0\), which forces \(a=0\) as an equation over every parameter ring.

Over \(A=k[\epsilon]/\epsilon^2\), with \(a=\epsilon\), the ambient section scheme retains the sections \(x=b\epsilon\). They reduce to zero on the geometric fibre and consequently fail the injection/flat-quotient open condition. In contrast, using the generic \(H^0\) alone would omit all of these ambient nilpotent section parameters before imposing the correct open condition. This tests the actual finite equations, nonflat base change and the two different stages of the construction.

Exercise I.4.2 (advanced). The central target line is \(\mathcal O(-1)\). Give its map pole bound two and its inverse pole bound zero. Compute the entire scheme of inverse pairs on arbitrary parameter rings, the transition from pole bound one, and a nonreduced parameter point in characteristic two.

Solution. Let the central target line be \(\mathcal O(-1)\), with frame \(v_\infty=t^{-1}v_0\). With pole bounds two and zero, its map and inverse have the entire forms \[ \kappa^+=(b t^{-2}+c t^{-1})v_0,\qquad \kappa^-=(e+d t)v_0^\vee. \tag{I.4.3} \] Indeed the first global Laurent range is \(-2,-1\), and the dual line's global range is \(0,1\). Their actual product equals \(1\) exactly when \[ be=0,\qquad bd+ce=0,\qquad cd=1. \tag{I.4.4} \] Every solution on any \(k\)-algebra has \(c\) a unit, \(d=c^{-1}\), \(b=-c^2e\), and \(e^2=0\), and these formulas give every solution. Thus the precise central-inverse scheme is \[ \operatorname{Spec}k[c,c^{-1},e]/e^2. \tag{I.4.5} \] Over the dual numbers, take \(c=1,e=\epsilon\). The actual maps are \((t^{-1}-\epsilon t^{-2})v_0\) and \((t+\epsilon)v_0^\vee\), whose product is literally one. At pole bound one the extra coefficient \(b\) is zero, which forces \(e=0\); the closed transition embeds the reduced \(\mathbf G_m\) into this thickening. This calculation is valid in characteristic two and is a direct reason not to replace the torus meromorphic functor by its reduced integer labels.

Exercise I.4.2 solves every coefficient equation, retaining the actual tensor unit and inverse maps on all parameter rings. The lower-pole-bound scheme is the closed subscheme \(e=0\), including its scheme structure. Editable SVG source.

Appendix J. The full highest-line cone and geometric saturation

Let \(k\) be a field of arbitrary characteristic and \(G/k\) a pinned split reductive group with simply connected derived group. Theorem H.4.1.1 provides the required auxiliary group for every original root datum, preserving the original central labels and Weil comparisons. We use the actual root charts in Root data, Weyl chambers and the Bruhat decomposition, §§5 and 8, the smooth projective flag scheme and its charts in Automorphisms, forms and parabolic subgroups, Theorems 1.1 and 7.1, and the abelianization in Pinnings and the classification of split reductive groups, §10. The codimension-one extension theorem is proved in Discrete valuation rings, normal rings and Serre's criterion, Theorem 3.3. The cohomology and eventual-vanishing arguments are proved in Constructible complexes on algebraic varieties, §§W.2–W.4; descent of closed immersions is proved in Affine descent, Zariski Main and recognition of spaces, Corollary A3.2.

The free Geometric Eisenstein series, §§1.1–1.2 describes the full Plücker and defect constructions. The proofs below construct the entire algebra and all its finite equations, including central inverses and arbitrary parameter rings. Appendix K constructs the full fixed-bundle torus quotient and exact relative defect strata on arbitrary parameter rings.

J.1. A finite multisection algebra, without a multiplication-surjectivity assumption

Proposition J.1.1.1. Let \(Y\) be projective over a field and let \(A_1,\ldots,A_r\) be globally generated invertible sheaves. The entire multisection algebra \[ B_0=\bigoplus_{\mathbf n\in\mathbf N^r} H^0(Y,A_1^{n_1}\otimes\cdots\otimes A_r^{n_r}) \tag{J.1.1} \] is finite as a module over the image of \(S=\operatorname{Sym}_k(\bigoplus_i H^0(Y,A_i))\), with its natural multigrading. In particular it is a finitely presented algebra. This assertion does not assert that its degree-one spaces generate it as an algebra.

Proof. If \(r=0\), the assertion is projective coherent finiteness in degree zero. Otherwise put \(E=\bigoplus_i A_i\) and form the projective bundle \(p:W=\operatorname{Proj}_Y\operatorname{Sym}E\to Y\), with the quotient convention. On every trivializing affine chart the standard projective-space Čech calculation gives \(p_*\mathcal O_W(n)=\operatorname{Sym}^n E\) for \(n\ge0\). The identifications on overlaps are the actual changes of variables in the symmetric algebra. The global evaluation \(H^0(Y,E)\otimes\mathcal O_Y\twoheadrightarrow E\) embeds this relative Proj in \(Y\times\mathbf P(H^0(Y,E)^\vee)\); hence \(W\) is projective over \(k\). Thus \[ \bigoplus_{n\ge0}H^0(W,\mathcal O_W(n))=B_0 \tag{J.1.2} \] as algebras, where the right side is temporarily graded by total degree. The decomposition of \(\operatorname{Sym}^n(\bigoplus A_i)\) has one summand for each \(\sum n_i=n\); it uses monomials, with no division by factorials.

Global generation of \(E\) makes \(\mathcal O_W(1)\) globally generated by \(H^0(Y,E)\). It defines \(f:W\to P=\mathbf P(H^0(Y,E)^\vee)\), with \(f^*\mathcal O_P(1)=\mathcal O_W(1)\). This map is projective: its graph is closed in \(W\times P\), since \(P\) is separated, and projection from that product is projective. Projective coherent finiteness in W.4 makes \(F=f_*\mathcal O_W\) coherent. On an affine open of \(P\), that assertion is precisely finiteness of degree-zero cohomology for the projective inverse image. The ordinary projection formula here follows by trivializing \(\mathcal O_P(n)\) on affine opens, so \(H^0(W,\mathcal O_W(n))=H^0(P,F(n))\).

The graded module \(\bigoplus_{n\ge0}H^0(P,F(n))\) is finite over the homogeneous polynomial ring of \(P\). Indeed W.4 constructs a finite quotient \(Q=\bigoplus_j\mathcal O_P(-a_j)\twoheadrightarrow F\), with coherent kernel \(K\). For all sufficiently large \(n\), the proved eventual vanishing gives \(H^1(P,K(n))=0\). Hence \(H^0(P,Q(n))\to H^0(P,F(n))\) is surjective in those degrees. The nonnegative-degree tails of the finitely many shifted polynomial modules of \(Q\) are finite polynomial modules: every monomial above their starting degree is a multiple of one of the finitely many monomials in that degree. Add bases of the finitely many lower-degree spaces \(H^0(P,F(n))\), which are finite by W.4. Their images and those tail generators generate the entire module. The action is exactly the map from \(S\) in the statement.

Decomposing these finitely many total-degree generators into their finitely many multihomogeneous components gives multihomogeneous generators too. Finite module generation implies finite algebra generation. The polynomial-ring Noetherian proof in W.4 makes the kernel of a polynomial presentation finitely generated. This proves both claims, in every characteristic. ∎

J.2. Fundamental flag line bundles and an ample product

Proposition J.2.1.1. Write \(T\subset B\), \(N=\operatorname{rad}_u B\), and \(Y=G/B\). Let \(C=X^*(G/G_{\rm der})\). Restriction to the maximal torus of the simply connected derived group gives an exact sequence of free lattices \(0\to C\to X^*(T)\to P\to0\), where \(P\) is its weight lattice. To see exactness, \(T\cap G_{\rm der}\) is its split maximal torus, the quotient is the torus of the proved abelianization, and dualizing that split-torus quotient gives the character sequence. Choose lifts \(\omega_i\in X^*(T)\) of the fundamental weights and a basis of \(C\). Then \[ X^*(T)=\bigoplus_i\mathbf Z\omega_i\oplus C, \qquad X^*(T)^+=\bigoplus_i\mathbf N\omega_i\oplus C. \tag{J.2.1} \] No quotient of the original central lattice is taken.

For a character \(\lambda\), put \(A_\lambda=G\times^B k_{-\lambda}\). Its tensor identifications are induced by multiplication of characters. For dominant \(\lambda\), this bundle is globally generated. For \(\rho=\sum_i\omega_i\), a positive power of \(A_\rho\) is very ample if there are roots. For a torus \(Y\) is a point.

Proof. The earlier flag theorem makes \(Y\) smooth, projective and geometrically integral. Its finitely many Weyl translates of the opposite root chart cover every Bruhat cell, by the actual closed-root-chart construction in Theorem 7.1. These are affine polynomial charts. Their polynomial rings are normal by the content and factorization argument, so the flag scheme is normal over every field extension. The opposite big cell is \(U^-B/B\simeq\mathbf A^{|\Phi^+|}\). Its root-product lift to \(G\) gives an actual frame \(\sigma_\lambda\) of \(A_\lambda\). The codimension-one components of its complement are the opposite cells indexed by the simple reflections: the Bruhat dimensions are \(|\Phi^+|-\ell(w)\), and the only elements of length one are those reflections.

At the generic point of the component indexed by \(i\), use the open minimal-parabolic chart \(U_i^-\times P_i/B\simeq U_i^-\times\mathbf P^1\). Here \(U_i^-\) has the negative roots other than \(-\alpha_i\). The cocharacter open-cell product \(U_i^-\times P_i\to G\), proved in the earlier relative flag theorem, gives this chart after its right \(B\)-quotient. The ordered root-product coordinates put the omitted \(-\alpha_i\) last. Thus on its affine \(z\)-chart the frame is exactly the frame from the rank-one matrix \(\left(\begin{smallmatrix}1&0\\z&1\end{smallmatrix}\right)\). The rank-one construction identifies \(P_i/B\) with \(\mathbf P^1\), and the character \(-\lambda\) gives \(\mathcal O(m_i)\), where \(m_i=\langle\lambda,\alpha_i^\vee\rangle\). In the second coordinate \(w=z^{-1}\), its first frame is \(w^{m_i}\) times the second frame. Consequently \[ \operatorname{ord}_{D_i}(\sigma_\lambda) =\langle\lambda,\alpha_i^\vee\rangle. \tag{J.2.2} \] There are no other possible poles: the section was a frame on the big cell. If \(\lambda\) is dominant, all these integers are nonnegative. Normality of the smooth flag scheme and the actual codimension-one intersection theorem extend the section over every affine trivialization. It is a nonzero global section. Translation makes the evaluation of all global sections surjective at every geometric point; finite coherent cohomology makes that evaluation a finite map of coherent sheaves, and Nakayama kills its cokernel. Field base change of \(H^0\) here is the flat tensor of its finite affine-cover equalizer, so the argument also proves global generation over the original split field.

For \(\rho\), every order in (J.2.2) is one. The nonvanishing locus of \(\sigma_\rho\) is precisely the affine big cell. Finitely many geometric translates of this section cover \(Y\) by such affine nonvanishing loci. They and their chart coordinates are defined over a finite extension of \(k\). On each chart take finitely many algebra generators. A chart function extends after multiplication by a sufficiently large power of its defining section: on a finite affine cover it is a localization fraction, and finitely many denominators can be cleared. These local extensions agree, since \(Y\) is integral and they agree on their dense nonvanishing locus. Raise all the finitely many exponents to one \(M\). The sections \(s_j^M\) and the extended coordinate functions multiplied by \(s_j^{M-m}\) define a map to projective space. On \(s_j^M\ne0\), their coordinate ratios generate the entire coordinate ring of that affine chart, so that restricted map is a closed immersion. The source is projective, so its image is closed; these charts together with the complement of the image prove that the whole map is a closed immersion. Thus \(A_\rho^M\) is very ample. The complete linear system of that same power is a closed immersion as well, by the same chart argument with the additional sections. Flat field base change identifies its section space. Descent of a closed immersion through the finite field extension is the affine ideal descent of the earlier faithfully flat descent lesson. Hence a positive power is very ample over \(k\) too.

Finally a central character extends to \(G\); its homogeneous line bundle is trivial as an ordinary line bundle, with that specified \(G\)-linearization. Also \(H^0(Y,\mathcal O_Y)=k\): projective finiteness makes it a finite-dimensional domain, hence a field; evaluation at the split base point embeds that field in \(k\) and is the identity on \(k\). These assertions include the empty-root case. ∎

J.3. The entire cone, its units and its actual open flag torsor

Theorem J.3.1.1. Set \(W_\lambda=H^0(Y,A_\lambda)^\vee\) for dominant \(\lambda\). Dual evaluation at the split base point gives a specified \(B\)-stable line of character \(\lambda\) in this module. These are geometric representations of the original group, distinct from the dual-group coefficient modules denoted by \(V_\lambda\) earlier. This fixes the geometric Weyl-module convention; the modules need not be simple in positive characteristic. Multiplication of sections gives actual maps \(\Delta_{\lambda,\mu}:W_{\lambda+\mu}\to W_\lambda\otimes W_\mu\). All associativity and unit diagrams are the duals of ordinary multiplication diagrams.

The full graded algebra and its affine cone are \[ B_G=\bigoplus_{\lambda\in X^*(T)^+}H^0(Y,A_\lambda), \qquad \mathcal C_G=\operatorname{Spec}B_G. \tag{J.3.1} \] This algebra is finitely presented. Its central homogeneous generators \(u^c\), \(c\in C\), retain the actual equations \(u^cu^{-c}=1\). On every algebra, the open subscheme where each fundamental-degree vector is unimodular is exactly \(G/N\), with its full stabilizers and parameter nilpotents.

Proof. The chosen central character trivializations give \(B_G=B_0\otimes_k k[C]\), with \(A_i=A_{\omega_i}\) in (J.1.1). The preceding two proofs give finite presentation, and \(k[C]\) is the Laurent polynomial algebra on the actual chosen central basis. Its inverse equations are identities of functions.

For the open subscheme form the frame torsor \[ Q=\operatorname{Spec}_Y\left(\bigoplus_{\lambda\in X^*(T)}A_\lambda\right). \tag{J.3.2} \] Locally it is the spectrum of the Laurent polynomial algebra on the basis (J.2.1). Its points are nonzero dual frames of those line bundles, with their tensor identifications. The map \(G\to Q\) sends the fixed frames at the base point to their translates. After a faithfully flat lift of a point of \(Y\), arbitrary such frames differ by a unique point of \(T\), because the chosen characters are a lattice basis. The stabilizer of all the frames is precisely \(N\) as a group scheme: in \(B\), every basis character must equal one, which is exactly the kernel of \(B\to T\). Thus \(Q=G/N\) on the entire fppf functor, rather than only on field points.

There are no negative fundamental degrees in \(\Gamma(Q,\mathcal O_Q)\). Indeed, if \(\langle\lambda,\alpha_i^\vee\rangle<0\), restrict a section of \(A_\lambda\) to every translate of \(P_i/B\simeq\mathbf P^1\). Its degree there is negative, so the elementary two-chart calculation gives no section. These curves pass through every geometric point, whence the original section is zero. Direct sums commute with the finite affine-cover equalizer. Therefore \(\Gamma(Q,\mathcal O_Q)=B_G\), and evaluation defines \(Q\to\mathcal C_G\).

Take sections \(s_i\in H^0(Y,A_i)\) and put \(F=\prod_i s_i\). If their common nonvanishing locus \(U\) is nonempty, it is affine. In fact \(F\) is a section of \(A_\rho\); a very ample power embeds \(Y\) in projective space, and \(U\) is the intersection with the affine complement of the hyperplane defined by \(F^M\). On \(U\), the \(s_i\) trivialize their line bundles. Clearing a power of \(F\), as in the preceding proof, extends every section on \(U\) of any \(A_\lambda\). Adding enough multiples of \(\rho\) makes all its fundamental coordinates nonnegative. Hence the actual graded localization satisfies \[ B_G[F^{-1}]=\Gamma(U,\mathcal O_U)[s_1^{\pm1},\ldots,s_r^{\pm1}] \otimes_k k[C]=\Gamma(Q|_U,\mathcal O_Q). \tag{J.3.3} \] The symbols on the right indicate the trivialized homogeneous frame coordinates; their coefficients are the ordinary degree-zero functions. Equality is by extension and restriction of sections, so it is an equality of algebras. The localization inverts each \(s_i\), since its inverse is \((\prod_{j\ne i}s_j)/F\).

Choose finite bases of the fundamental section spaces. Their simultaneous nonvanishing principal opens cover exactly the locus where each fundamental-degree functional is unimodular. On each such open (J.3.3) identifies the evaluation map with an isomorphism. These isomorphisms agree by their restriction maps and glue. For a torus the product is empty, \(Y\) is a point and the cone itself is \(T\). This proves the asserted open immersion and its full functor of points. No normalization, reduction, or removal of central directions was used. ∎

J.4. Finite equations for the full highest-line datum

Theorem J.4.1.1. For a \(G\)-bundle \(\mathcal E\) and a \(T\)-bundle \(\mathcal T\) on \(\mathbf P^1_R\), with \(R\) Noetherian, consider every dominant-weight map \(\kappa^\lambda:\mathcal L_{\mathcal T}^\lambda\to(W_\lambda)_{\mathcal E}\), with the unit map, every central inverse map, and all equations \[ \kappa^\lambda\otimes\kappa^\mu =\Delta_{\lambda,\mu}\kappa^{\lambda+\mu}. \tag{J.4.1} \] The entire regular datum is represented over this family by an affine finitely presented relation scheme. Requiring injection with finitely presented parameter-flat quotient is a finitely presented open condition. The same assertions hold for bounded meromorphic data away from a fixed point; their closed pole transitions give the entire meromorphic datum. These assertions hold on every parameter algebra, including nonflat and nilpotent ones.

Proof. A map \(\kappa^\lambda\) is exactly a functional \((B_G)_\lambda{}_{\mathcal E}\otimes\mathcal L_{\mathcal T}^\lambda\to\mathcal O\). The equations (J.4.1), including the unit, are precisely multiplicativity of these functionals. Thus the entire collection is an algebra map from the actual sheaf algebra \(\bigoplus_\lambda (B_G)_\lambda{}_{\mathcal E}\otimes\mathcal L_{\mathcal T}^\lambda\) to \(\mathcal O\).

Choose finitely many homogeneous algebra generators in (J.3.1). Enlarge their spaces to the whole finite-dimensional homogeneous spaces in their degrees, and include all fundamental degrees and a basis of central characters with its negatives. Work first with a polynomial algebra in the noncentral generator spaces tensored with the Laurent algebra \(k[C]\). This gives a finite \(G\)-equivariant presentation. Its relation ideal is finitely generated; converting the central Laurent generators to pairs of variables adds the stated tensor-unit equations. Decompose ideal generators into homogeneous components and take their finite-dimensional \(G\)-stable spans. These spans are finite: an actual coaction on a polynomial is a finite sum; its coefficient span is a finite-dimensional stable submodule by coassociativity. Fixed homogeneous components here are finite-dimensional, since each noncentral generator has a positive total fundamental degree and the Laurent central monomial is fixed by its total central degree. Thus there are finitely many actual equivariant polynomial relations. The central relations are included as the actual tensor-unit equations.

Evaluating those finitely many spaces and relations in the given bundles is exactly Proposition I.2.1.1. Its universal sections recover the entire algebra map, and hence every \(\kappa^\lambda\), uniquely. This is an equivalence of functors on all algebras, not an identification of reduced points. For meromorphic maps the same argument takes place in \(\mathcal O(*x)\); a finite generator family has one finite tuple of pole bounds, and the finite relation list can be cleared there. Proposition I.2.1.1 supplies the actual closed transitions. No uniform bound for infinitely many independent generators is needed, since those generators have now been proved finite.

It remains to show that the full injection condition is finite. On a geometric curve fibre, if the fundamental maps are not identically zero, their evaluation functionals on its function field are nonzero. For \(\lambda=\sum_i n_i\omega_i+c\), select a fundamental section whose evaluation is nonzero for each \(n_i>0\). Its product, multiplied by the central unit \(u^c\), has nonzero evaluation. Hence the entire \(\lambda\)-degree functional is nonzero. This uses only multiplication and the field property, with no surjectivity assertion for any Cartan multiplication. Conversely the full condition includes the fundamental maps. Theorem I.3.1.1 identifies their finite geometric-nonzero open locus with universal sheaf injection and finitely presented parameter-flat quotient. The same argument then applies to every recovered weight. Over Noetherian bases these quotients are coherent.

Finally, at a point of the curve, the full datum is a genuine flag reduction with its torus framing exactly when all the fundamental vectors are unimodular: Theorem J.3.1.1 identifies that actual open with \(G/N\). For regular data, the locus of parameters where this holds everywhere on the proper curve is open, since the finite fundamental zero schemes are closed and their proper projections are closed. This distinguishes the genuine reduction open from the compactified, generically nonzero open. ∎

Propositions J.1.1.1–J.2.1.1 and Theorems J.3.1.1–J.4.1.1 construct the entire algebra and actual parameter functor. The localization is an equality of coordinate algebras for the torus frame torsor. Editable SVG source.

J.5. Canonical all-root saturation on geometric curve points

Theorem J.5.1.1. Let \(K\) be algebraically closed, let \(X/K\) be a smooth projective connected curve, and let \((\mathcal E,\mathcal T,\kappa)\) be regular full highest-line data with all maps generically nonzero. There is a unique effective coroot-valued divisor \[ D=\sum_{x\in X}\sum_i m_{i,x}\alpha_i^\vee[x], \qquad m_{i,x}\ge0, \tag{J.5.1} \] such that replacing \(\mathcal T\) by \(\mathcal T(D)\) makes every highest-line map a subbundle map. Its fundamental line at \(i\) is twisted by \(\sum_xm_{i,x}[x]\); all central lines and their actual inverse maps are unchanged. The maps yield a unique saturated \(B\)-reduction of \(\mathcal E\).

Proof. On the function field, the fundamental maps are nonzero, so Theorem J.3.1.1 gives a point of the actual \(G/N\)-bundle and hence of its \(G/B\)-bundle. The latter is projective. At every local DVR of the smooth curve its valuative extension is unique. A morphism from that local ring into this finitely presented flag bundle spreads to an open neighbourhood by clearing its finitely many coordinate denominators; uniqueness on the common generic point and separatedness glue these extensions. Thus the generic flag extends to one \(B\)-reduction on all of \(X\).

At a point \(x\), trivialize the source torus line bundles and the torus lines of this reduction over its DVR, using frames for the basis (J.2.1). Their tensor identifications provide compatible frames for all weights. Dual evaluation, which is surjective by Proposition J.2.1.1, makes the highest line of the reduction a primitive submodule of \((W_\lambda)_{\mathcal E}\). Write its primitive vector as \(v_\lambda\). On the fraction field the original map has the form \(a_\lambda v_\lambda\), with \(a_\lambda\ne0\). The Cartan maps on these specified highest vectors give \(\Delta_{\lambda,\mu}v_{\lambda+\mu}=v_\lambda\otimes v_\mu\), because they are dual to evaluating a product at the same flag. Therefore (J.4.1) gives \(a_{\lambda+\mu}=a_\lambda a_\mu\), and the central inverse equation gives \(a_ca_{-c}=1\).

Primitivity and regularity force \(a_\lambda\) into the DVR: a coordinate of \(v_\lambda\) is a unit and its product with \(a_\lambda\) is regular. Thus the central \(a_c\) are units and \[ v_x(a_\lambda)=\sum_i n_i m_{i,x}, \qquad m_{i,x}=v_x(a_{\omega_i})\ge0 \quad(\lambda=\sum_i n_i\omega_i+c). \tag{J.5.2} \] Only finitely many of these fundamental orders are nonzero: each nonzero regular vector section on a proper curve has a finite zero locus. Its order is the minimum of its coordinate valuations, so the numbers are intrinsic and independent of every chosen frame.

The source line of weight \(\lambda\) in \(\mathcal T(D)\) is the old line twisted by the divisor with multiplicities \(\langle\lambda,D\rangle\). Locally its generator is \(t^{-\sum n_im_{i,x}}\) times the old generator. The resulting map is consequently a unit times the primitive \(v_\lambda\). All tensor and inverse equations remain equalities of maps. These saturated maps lie in the actual \(G/N\) open and recover the same extended flag. Conversely a subbundle map must have zero residual order at each DVR, so any divisor with this property has exactly the multiplicities in (J.5.2). This proves uniqueness.

This is a geometric-point saturation theorem. It does not assert that saturation commutes with arbitrary nilpotent parameter changes. The actual finite parameter schemes of Theorem J.4.1.1 retain those changes; relative fixed-defect strata need their own scheme construction. ∎

Theorem J.5.1.1 determines every geometric-point defect and its saturated flag. The rank-two example is realized by a fixed flag in the trivial bundle, with fundamental sources \(\mathcal O(-2[-1])\) and \(\mathcal O(-[-1]-3[1])\), central unit maps, and the canonical divisor sections. The points are schematic positions on \(\mathbf P^1\); all labels and orders are exact. Editable SVG source.

J.6. Exercises

Exercise J.6.1 (advanced). For \(SL_2\), compute the whole affine flag cone and its maps in characteristic two, and saturate a pair of binary forms defining \(\mathcal O(-d)\to\mathcal O^2\) on \(\mathbf P^1\).

Solution. The flag scheme is \(\mathbf P^1\), the fundamental line is \(\mathcal O(1)\), and its entire section ring is \(k[x,y]\). Thus \(\mathcal C_{SL_2}=\mathbf A^2\), with open \(SL_2/N=\mathbf A^2\setminus\{0\}\). On every algebra, a primitive column \((a,b)\) extends to a determinant-one matrix by choosing \(c,d\) with \(ad-bc=1\); its stabilizer is the actual upper unitriangular group. The degree-\(n\) map has coordinates \(a^{n-j}b^j\), \(0\le j\le n\), in the basis dual to the monomial basis of \(H^0(\mathcal O(n))\). These are divided-symmetric coordinates, not a divided factorial formula. In degree two its Cartan map into the tensor square is the symmetric matrix \[ \begin{pmatrix}a^2&ab\\ab&b^2\end{pmatrix}, \tag{J.6.1} \] including in characteristic two. There is no factor of two in the mixed coordinate. The primitive dual-number column \((1,\epsilon)\), \(\epsilon^2=0\), therefore gives mixed coordinate \(\epsilon\), which must be retained.

For a nonzero pair of degree-\(d\) binary forms \((f,g)\), take their homogeneous greatest common factor \(h\) of degree \(e\), including the case that one form is zero. The residual pair has no common zero on \(\mathbf P^1\). Hence the unique defect is \(\operatorname{div}(h)\alpha^\vee\), and the saturated source is \(\mathcal O(-d+e)\). The full compactified parameter scheme is \(H^0(\mathcal O(d))^2\setminus\{0\}\); its scalar quotient is \(\mathbf P^{2d+1}\). The genuine-reduction open is the locus with \(e=0\). This computes the complete rank-one quotient, not merely the highest-degree maps.

Exercise J.6.2 (medium). Compute the full cone for \(GL_2\), preserving determinant inverses, and identify its defect coroot.

Solution. Put \(\omega=e_1\), \(\delta=e_1+e_2\). Every dominant weight is uniquely \(n\omega+b\delta\), with \(n\ge0\), \(b\in\mathbf Z\). The fundamental line is again \(\mathcal O(1)\), and the determinant character gives the canonically trivial underlying line with its actual linearization. Therefore \[ B_{GL_2}=k[x,y,z,z^{-1}],\qquad \mathcal C_{GL_2}=\mathbf A^2\times\mathbf G_m. \tag{J.6.2} \] The coordinates of the entire weight map are \(z^b x^{n-j}y^j\). The open consists of primitive columns and a unit \(z\). Choose a second column whose determinant with the first is \(z\); this is a \(GL_2\)-matrix on every algebra, with the exact unitriangular stabilizer. In particular \((x,y,z)=(1,\epsilon,1+\epsilon)\) is a genuine dual-number point, with inverse \(z^{-1}=1-\epsilon\), also in characteristic two. The coroot is \(\alpha^\vee=(1,-1)\); it pairs to one with \(\omega\) and to zero with \(\delta\). A zero of order \(e\) twists the first fundamental source by \(e[x]\), leaves the determinant line unchanged, and has defect \(e(1,-1)[x]\). This retains the full determinant character and its inverse, rather than an adjoint or reduced substitute.

Appendix K. The full fixed-bundle compactification and relative defect strata

Let \(R\) be Noetherian, \(X=\mathbf P^1_R\), and \(G\) be split with simply connected derived group, as in Appendix J. Fix the actual \(G\)- and \(T\)-bundles. We construct the full regular highest-line parameter scheme, its entire auxiliary torus quotient and the relative fixed-defect strata on every parameter algebra. The geometric modules remain \(W_\lambda\), distinct from the dual-group coefficient modules \(V_\lambda\).

The cohomology and denominator arguments are proved in Constructible complexes on algebraic varieties, §§W.2–W.4, with their finite two-term application in Proposition I.1.1.1. Finite algebra descent and the proper quasi-finite theorem are proved in Affine descent, Zariski Main and recognition of spaces, Lemmas A1.2–A1.3, Corollaries A3.2 and D7.3. The full highest-line algebra, its flag open and all-weight equations are proved in Appendix J. The free Geometric Eisenstein series, §1.2 describes the corresponding compactification and saturation problem; each construction needed here is proved below.

K.1. Detecting an entire section at finitely many reduced points

Proposition K.1.1.1. Let \(\mathcal V_1,\ldots,\mathcal V_s\) be finitely many vector bundles on \(X\). After a finite faithfully flat extension of the ground field, there is a divisor \(D\) consisting of finitely many distinct constant sections in the affine \(t\)-chart such that restriction gives closed immersions \[ \mathscr S(\mathcal V_j)\longrightarrow \mathbb A\bigl(\Gamma(D,\mathcal V_j|_D)\bigr) \quad(1\le j\le s). \tag{K.1.1} \] These are closed immersions on arbitrary parameter algebras, not merely injections on reduced points.

Proof. Eventual global generation applied to \(\mathcal V_j^\vee\) gives a finite free surjection \(\mathcal O(-a_j)^{b_j}\twoheadrightarrow\mathcal V_j^\vee\). It splits locally because its target is locally free. Dualizing gives a universally injective map \(\mathcal V_j\hookrightarrow\mathcal O(a_j)^{b_j}\). Choose \(N>\max_j a_j\), with \(N>0\). If the field has fewer than \(N\) affine points, a finite field extension supplies them; an infinite field already does. Take \(N\) distinct constant affine points for \(D\). Its divisor line is \(\mathcal O(N)\). On every algebra \(A\), a section of \(\mathcal V_j\) restricting to zero on \(D_A\) lies in \(H^0(X_A,\mathcal V_j(-D))\), which injects into \(H^0(X_A,\mathcal O(a_j-N))^{b_j}=0\), by the two-chart monomial calculation. Thus the restriction is universally injective as a map of section functors.

Make its scheme assertion explicit. Proposition I.1.1.1 gives the section scheme from \([P^0\xrightarrow dP^1]\). Projection of its explicit kernel construction to the section module on the affine \(t\)-chart, followed by evaluation at the \(N\) points, gives an actual linear map \(j:P^0\to J=\Gamma(D,\mathcal V|_D)\). This module is finite projective. On every residue field the map \((d,j):P^0\to P^1\oplus J\) is injective: a vector in its kernel is the global section whose restrictions were just proved to detect it. Therefore the finite cokernel of the dual map \((P^1)^\vee\oplus J^\vee\to(P^0)^\vee\) has zero residue fibres. Nakayama kills every maximal localization, so that dual map is surjective. Consequently \[ \operatorname{Sym}J^\vee\twoheadrightarrow \operatorname{Sym}(P^0)^\vee/(d^\vee(P^1)^\vee). \tag{K.1.2} \] This is exactly the coordinate map of (K.1.1), proving closedness and all its base changes. The divisor is reduced and finite étale over the parameter base. Replacing it by one thick jet would not justify the next finite-morphism argument: restriction of scalars along a nonreduced finite scheme need not preserve finiteness. ∎

K.2. The central-isomorphism torsors

Proposition K.2.1.1. For each basis character \(c\) of the actual central lattice \(C\), let \(\mathcal M_c=\mathcal L_{\mathcal T}^{-c}\otimes (W_c)_{\mathcal E}\). The two central highest-line maps, with their tensor-unit equation, are exactly a nowhere-vanishing section of \(\mathcal M_c\) and its actual inverse. There is an open and closed part \(S_0\subset\operatorname{Spec}R\) on which every \(\mathcal M_c\) has degree zero on geometric fibres. The complete regular datum is empty off \(S_0\). On \(S_0\), each \(\mathcal M_c\) is the pullback of an invertible parameter module, and its isomorphism scheme is the actual frame \(\mathbf G_m\)-torsor of that module. Their product is a torsor under the central coordinate torus.

Proof. A line bundle on a geometric projective line is \(\mathcal O(d)\). Here is the elementary input: a rational section gives a divisor, every finite point divisor is the divisor of its monic polynomial plus its degree at infinity, and the polynomial quotient moves that divisor to \(d[\infty]\). The two-chart computation then gives \(h^0-h^1=d+1\), with \(h^0=\max(d+1,0)\) and \(h^1=\max(-d-1,0)\). The finite projective complex of Proposition I.1.1.1 computes these fibres. Its Euler characteristic is the difference of its two finite projective ranks, which is locally constant. Thus the degree-zero conditions define the claimed open and closed part.

On that part, the differential \(P^0\to P^1\) is surjective on every residue field, since all the \(H^1\) vanish. Its finite cokernel is zero by Nakayama. Since \(P^1\) is projective, its kernel \(P\) is finite projective, of rank one. The actual arbitrary-base-change comparison gives \(H^0(X_A,\mathcal M_{c,A})=P\otimes_R A\). Evaluation \(\mathcal O_X\otimes_RP\to\mathcal M_c\) is an isomorphism: on every geometric curve fibre it evaluates the constant section of the trivial line, so its finite coherent cokernel vanishes by Nakayama at every curve point. A surjection between two invertible modules is an isomorphism on their local frames. A nowhere-vanishing section is consequently a frame of \(P\) on every algebra; its dual is its uniquely specified inverse. Off the degree-zero part, the central inverse equation would give an isomorphism of lines of different degree on a geometric fibre, which is impossible. This proves every asserted functor and retains the entire central tensor unit. ∎

K.3. A finite morphism to fundamental section parameters

Theorem K.3.1.1. Fix actual central isomorphisms, after the faithfully flat frame torsor of Proposition K.2.1.1 if necessary. Let \(\mathscr D\) be the affine scheme of all regular highest-line maps and relations, before imposing geometric generic nonzero, and let \[ \mathscr A=\prod_i\mathscr S\bigl( \mathcal L_{\mathcal T}^{-\omega_i}\otimes (W_{\omega_i})_{\mathcal E}\bigr). \tag{K.3.1} \] The morphism \(\mathscr D\to\mathscr A\) forgetting the extra homogeneous generators is finite. Its inverse image of the simultaneous geometrically-nonzero locus \(\mathscr A^+\) is precisely the full compactified injection/parameter-flat-quotient scheme \(\mathscr D^+\).

Proof. Proposition J.1.1.1 makes \(B_0\) a finite module over the image of the polynomial algebra of the fundamental section spaces. With the full central Laurent algebra this says that the cone map to the fundamental vector spaces and central units is finite. Descent through the given bundles gives the corresponding finite morphism of cone bundles on \(X\); fixing the actual central isomorphisms is a base change of this map, not their omission.

Take the entire finite homogeneous generator list from Theorem J.4.1.1 and apply Proposition K.1.1.1 simultaneously to their associated map bundles. Over \(\mathscr A\), evaluate the fixed fundamental sections at its divisor of distinct constant points. The product of the pullbacks of the finite cone morphism at those points is a finite scheme \(Z\to\mathscr A\): each point is a copy of the parameter base, and products of finite algebras are finite tensor algebras. The extra homogeneous coordinates of \(Z\) place it as a closed subscheme of the product of their finite evaluation spaces. Restriction of all extra sections embeds their section schemes as closed subschemes of those evaluation spaces, by (K.1.2). The full polynomial relations cut out another closed subscheme. Hence the induced map \(\mathscr D\to Z\) is a closed immersion: both coordinate maps are restrictions from that same evaluation affine space, and the coordinate map for the source is already surjective. A closed subscheme of a finite affine scheme is finite. This proves the assertion after the finite field extension used to choose the points; finite module/algebra descent proves it over the original base as well.

Finally Theorem J.4.1.1 proves on every parameter algebra that the fundamental generic-nonzero conditions imply those for every dominant map, with all central inverses included. Theorem I.3.1.1 identifies them with universal injection and finitely presented parameter-flat quotient. Thus \(\mathscr D^+\) is exactly the indicated inverse image, and its map to \(\mathscr A^+\) remains finite. ∎

K.4. The full torus quotient is projective

Theorem K.4.1.1. Let \(\overline{\mathscr D}=\mathscr D^+/T\), where \(T\) acts by multiplying every map by its weight character, including the central unit maps. This quotient is represented by a scheme projective over \(S_0\), and by the empty scheme over its complement. The map from genuine reductions, whose fundamental sections have no zeros on the curve, is an open immersion into this scheme. Every assertion is on the full parameter functor, including nilpotents.

Proof. The character basis (J.2.1) splits the acting torus into the \(r\) independent fundamental scalars and the central coordinate torus. Quotienting the latter consumes exactly the frame torsor of Proposition K.2.1.1. Thus first choose those frames faithfully flatly, as in Theorem K.3.1.1, and form the remaining quotient. No central character of the original group has been discarded; its line isomorphism and inverse equation were imposed before this descent.

Write each fundamental section scheme as \(\operatorname{Spec}\operatorname{Sym}(P_i^0)^\vee/(d_i^\vee(P_i^1)^\vee)\). Its geometrically-nonzero open consists exactly of the kernel vectors whose coordinates generate the unit ideal on the parameter base. Such a vector is a frame of a rank-one locally direct summand of \(P_i^0\). The quotient by its actual scalar \(\mathbf G_m\)-action is therefore \[ P_i=\operatorname{Proj}_R \bigl(\operatorname{Sym}(P_i^0)^\vee/ (d_i^\vee(P_i^1)^\vee)\bigr). \tag{K.4.1} \] This identification follows on all tests by choosing a local frame of the tautological line, with the linear equations saying \(d_i=0\). Each \(P_i\) is a closed subscheme of the relative projective space of the finite projective module \(P_i^0\). Thus \(P=\prod_iP_i\) is projective, and \(\mathscr A^+\to P\) is the actual \((\mathbf G_m)^r\)-frame torsor.

The finite map \(\mathscr D^+\to\mathscr A^+\) is equivariant. Its finite algebra and multiplication descend through this torsor, by effective affine module/algebra descent, to a finite scheme \(\overline{\mathscr D}\to P\). The two inverse pullbacks identify it with the quotient on every fppf test, so no unproved geometric quotient is being assumed. The initial central frame descent gives the same finite scheme over the original \(P\): changing central frames acts equivariantly on the higher generators and leaves the fundamental parameter spaces unchanged. Finiteness and its descent datum are preserved. In rank zero, the product \(P\) is the base and the cone is the central torus itself, so the quotient is just \(S_0\).

For completeness a finite scheme over this projective \(P\) is projective over \(R\). A finite map is separated and universally closed: its diagonal is the multiplication quotient of finite algebras, and integrality plus lying-over makes the image of each closed ideal locus closed after every base change. It is therefore proper. Pull back the Segre very ample line of \(P\). Its finitely many homogeneous coordinate nonvanishing opens have affine finite inverse images and cover \(\overline{\mathscr D}\). Take finitely many algebra generators on each such inverse image. Powers of the defining section extend each generator globally: on a finite affine cover, clear its localization denominators; multiply by a common further power to kill the finitely many differences on overlaps. This is the actual denominator-clearing proof of W.4, valid even if that section is a zero-divisor. Raise all these powers to one degree and include the powers of the original covering sections. Their coordinate ratios generate each affine inverse-image algebra. The resulting morphism to a finite projective space is a closed immersion on the corresponding target charts. It is proper, so its image is closed; those charts and the complement of its image prove that it is a closed immersion globally. Thus the pulled-back line has a very ample power and the source is projective over \(R\).

The genuine-reduction condition is that every fundamental map is a subbundle map at every curve point. Its complement is the proper projection of their finite closed zero schemes on \(X\). It is therefore open before the quotient, and its invariant open descends through the same torsors. Theorem J.3.1.1 identifies it with the actual flag reduction and torus-framing datum. This proves the stated open immersion. ∎

Propositions K.1.1.1–K.2.1.1 and Theorems K.3.1.1–K.4.1.1 construct the finite map and the actual quotient. Distinct reduced sampling points are essential for the finite product-of-values step. Editable SVG source.

K.5. Relative fixed-defect strata retain their own scheme equations

Theorem K.5.1.1. For each tuple \(\mathbf d=(d_i)\) of nonnegative integers there is a locally closed subfunctor \(\mathscr D^+_{\mathbf d}\) of the full compactified injection scheme, defined on arbitrary parameter algebras. Its points have effective relative Cartier divisors \(D_i\), finite flat of degree \(d_i\), as the exact fundamental zero schemes; every homogeneous generator of weight \(\lambda=\sum_i n_i\omega_i+c\) is divisible by the canonical divisor section for \(\sum_i n_iD_i\). Dividing gives an actual saturated reduction for the twisted torus bundle \(\mathcal T(\sum_i\alpha_i^\vee D_i)\), with its central lines unchanged. Multiplication by these sections gives the inverse construction. The strata partition geometric points and preserve their specified nilpotent subfunctors; a general nilpotent family need not factor through one stratum.

Proof. For a fundamental map, let \(Z_i\) be the scheme cut out by all its local vector coordinates. Its fibres are finite, since its section is geometrically generically nonzero. It is closed in the proper curve, so it is proper and quasi-finite over the parameter scheme; Corollary D7.3 of Affine descent, Zariski Main and recognition of spaces makes it finite; that corollary is the proper quasi-finite consequence of its finite-completion proof. Thus its coordinate pushforward \(Q_i\) is a finite presented parameter module, and formation of that finite algebra commutes with every base change.

Choose a finite presentation of \(Q_i\) on each parameter affine chart. The condition that \(Q_i\) become locally free of rank \(d_i\) is an actual locally closed scheme condition: \(\operatorname{Fitt}_{d_i}(Q_i)\) must be the unit ideal and \(\operatorname{Fitt}_{d_i-1}(Q_i)\) must be zero. Here \(\operatorname{Fitt}_{-1}=0\). To check the assertion, on an open where a presentation minor of size \(n-d_i\) is invertible, row and column operations isolate that identity block. The remaining presentation has \(d_i\) rows. The minors defining the next Fitting ideal are exactly the entries of the remaining matrix. Their vanishing makes the quotient free of rank \(d_i\). Conversely a free quotient has precisely these two ideals. Both ideals are generated by the stated finite minors and commute with arbitrary scalar extension. Their open and closed equations therefore represent the asserted flattening locus on every algebra. The preceding equivalence with the module functor also proves independence of the chosen presentation and gluing on chart overlaps, by Yoneda; no separate Fitting-invariance theorem is being assumed.

On this locus, \(Z_i\) is an effective relative Cartier divisor. Its ideal \(I_i\) is parameter-flat, since both \(\mathcal O_X\) and \(\mathcal O_{Z_i}\) are. On every geometric curve fibre the coordinate ideal is \((t^{m_i})\) in its local DVR. Lift that one generator at a curve point. Nakayama gives a local surjection from \(\mathcal O_X\) to \(I_i\); the parameter-flatness of \(I_i\) makes its kernel reduce injectively, and its reduced map is the DVR isomorphism onto \((t^{m_i})\). The finite kernel vanishes by Nakayama. Thus \(I_i\) is invertible locally. Its inclusion in \(\mathcal O_X\) is multiplication by a non-zero-divisor. Parameter-flatness of its quotient preserves that injection after every base change. This is the relative Cartier assertion, with the exact zero ideal rather than just its support.

These divisors have their usual projective parameter spaces on \(\mathbf P^1\), including nonreduced tests. One can verify this directly without an unproved Hilbert scheme. Away from a constant point missing a fibre divisor, the divisor lies in \(\mathbf A^1\). Multiplication by \(t\) on its finite free degree-\(d\) algebra has a monic characteristic polynomial \(q\). Cayley–Hamilton gives a surjection \(A[t]/(q)\to\mathcal O_D\). On every residue field its defining ideal is generated by a monic degree-\(d\) polynomial, which is the same characteristic polynomial; hence that map is a fibre isomorphism. A surjection between finite locally free modules of rank \(d\) is an isomorphism. The coefficients of \(q\), homogenized and taken up to a unit, give the point of \(\mathbf P^d\). Such charts cover faithfully flatly: a finite extension of the ground field with more than \(d\) projective points supplies a missing constant point over every fibre. The local polynomials agree by their exact ideal, so glue. Conversely, a primitive homogeneous degree-\(d\) polynomial has these same charts where its value at the omitted point is a unit; its monic affine equation gives a finite free degree-\(d\) Cartier divisor there. This proves the divisor functor and its universal divisor on all rings. Degree zero means the empty divisor.

It remains essential to impose the higher-generator equations, rather than infer them from geometric points. On the finite flat zero-divisor locus just constructed, form \(D_\lambda=\sum_i n_iD_i\) for every member of the entire finite generator list. Restriction of its map to \(D_\lambda\) must vanish. This is a closed scheme condition: the divisor is finite locally free and the restriction bundle has finite projective sections, so its actual universal restriction vector has finitely many linear coordinate equations. Its vanishing is exactly divisibility by the canonical section of \(D_\lambda\), since the Cartier sheaf sequence is universally exact. Impose these finitely many closed conditions. Their result is \(\mathscr D^+_{\mathbf d}\), locally closed in the original full scheme.

Division now yields all the regular saturated generator maps. The fundamental ones have coordinate ideal the unit ideal: locally divide all coordinates by the generator of their exact common ideal \(I_i\). Every polynomial relation is homogeneous in the full weight lattice. Multiplying its divided expression by the canonical section for its total nonnegative fundamental divisor gives the original zero relation. That section is a non-zero-divisor after every base change, so the divided relation is zero as an actual map. Central relations have divisor zero and are the unchanged inverse equations. Thus these divided generators recover the entire regular highest-line algebra map, and all fundamental vectors are unimodular at every curve point. Theorem J.3.1.1 gives an actual saturated flag and torus-framing datum.

Conversely, any such saturated datum for \(\mathcal T(\sum\alpha_i^\vee D_i)\), multiplied by the canonical sections, has fundamental coordinate ideals exactly \(I_i\), finite flat zero schemes of the given degrees, and satisfies every higher divisibility equation. Multiplication is universally injective, so division is its unique inverse, including all nilpotents and isomorphisms of bundles. This establishes the equivalence of the stated full subfunctors. Scaling by any weight character preserves the exact zero ideals and every divisibility equation. These strata are therefore invariant under the entire acting torus. Their open and closed ideals descend on the frame-torsor charts used in Theorem K.4.1.1, giving actual locally closed subschemes of the projective quotient as well. Theorem J.5.1.1 shows that every geometric point lies in the unique degree tuple supplied by its canonical saturation. A nilpotent family may have a nonflat zero scheme or fail a higher divisibility equation; it is retained in the ambient scheme and is not silently reduced into one of these strata. ∎

K.6. Exercises

Exercise K.6.1 (advanced). For the finite scheme \(Z=\operatorname{Spec}k[u]/u^2\), compare its restriction-of-scalars functors along two reduced points and along \(\operatorname{Spec}k[\epsilon]/\epsilon^2\). Explain why reduced sampling was needed in Theorem K.3.1.1, including characteristic two.

Solution. Along \(k\times k\), an \(A\)-point is two elements \(u_1,u_2\in A\) with \(u_i^2=0\). Its entire coordinate algebra is \(k[u_1,u_2]/(u_1^2,u_2^2)\), a finite four-dimensional algebra with basis \(1,u_1,u_2,u_1u_2\). Along the dual numbers, the image of \(u\) is \(a+b\epsilon\), and its square is \(a^2+2ab\epsilon\). Thus the entire scheme is \[ \operatorname{Spec}k[a,b]/(a^2,2ab). \tag{K.6.1} \] Its reduced scheme is the affine \(b\)-line, in every characteristic. In characteristic two the mixed equation disappears, and its algebra is \(k[a,b]/a^2\), still infinite-dimensional. Thus restriction of scalars of a finite morphism along a finite nonreduced scheme need not be finite. Theorem K.3.1.1 instead uses a product of copies of the base, together with actual closed-immersion restrictions for every whole section; both ingredients were necessary.

Exercise K.6.2 (medium). In the \(SL_2\) compactification take the family \(\mathcal O(-1)\to\mathcal O^2\) with homogeneous coordinates \((T,aS)\) over \(k[a]\), where \(t=T/S\). Compute its exact zero scheme and degree-zero and degree-one defect strata. Test \(a=\epsilon\) over the dual numbers and explain why saturation cannot be inferred from its geometric fibre.

Solution. The map is generically nonzero on every geometric curve fibre, so is a point of the ambient injection/parameter-flat-quotient scheme. At infinity its first homogeneous coordinate is a unit in the source frame. On the finite chart its coordinate ideal is exactly \((t,a)\). Therefore its entire zero scheme and finite pushforward are \[ Z=\operatorname{Spec}k[a,t]/(t,a),\qquad Q=k[a]/(a). \tag{K.6.2} \] The Fitting ideals are \(\operatorname{Fitt}_0(Q)=(a)\) and \(\operatorname{Fitt}_1(Q)=k[a]\). Degree zero is the open \(a\ne0\); degree one is the exact closed subscheme \(a=0\). There are no other strata for this family. On \(a=0\), the defect is \(\alpha^\vee[0]\), the source twists to \(\mathcal O\), and division by \(t\) gives the primitive saturated pair \((1,0)\).

For \(A=k[\epsilon]/\epsilon^2\) and \(a=\epsilon\), the zero algebra is \(A/(\epsilon)\). It is not flat over \(A\): tensoring the injection \((\epsilon)\hookrightarrow A\) with \(A/(\epsilon)\) sends its nonzero generator to zero. This point factors through neither \(a\ne0\) nor \(a=0\), though its geometric fibre lies on the degree-one boundary. The attempted division \((t,\epsilon)/t\) has second coordinate \(\epsilon/t\), which is not regular at zero. The actual nilpotent family remains in the full compactification; it is not assigned the saturated map of its reduced fibre. The calculation is valid also in characteristic two. For \(GL_2\) with determinant map one the same family retains that central unit and has the coroot defect \((1,-1)[0]\).

Theorem K.5.1.1 imposes every higher divisibility equation as a scheme condition. Exercise K.6.2 computes the entire zero algebra and its two Fitting loci; the nilpotent point remains in the ambient projective compactification. Editable SVG source.

Appendix L. Finite jet presentations and intrinsic duality of genuine strata

Let \(G\) be the actual split group, \(T\subset B\), and \(N\) its positive unipotent subgroup. Work on \(X=\mathbf P^1_\kappa\), choose \(y\in X(\kappa)\), and fix a \(T\)-bundle \(\mathcal T\). Its root line bundles and twisted unipotent group are \[ \mathcal L_\alpha=\mathcal T\times^T \kappa_\alpha \simeq\mathcal O(d_\alpha),\qquad U=\mathcal T\times^T N,\qquad n=|\Phi^+|. \tag{L.1.1} \] The ground field is \(\kappa\); the jet level \(k\) below is an integer. The integers \(d_\alpha\) add on roots whose sums are roots. An integral basis of the original \(X^*(T)\), with all its central characters, specifies the bundle. A \(B\)-bundle with its \(T\)-quotient identified with \(\mathcal T\) is exactly a \(U\)-torsor, by twisting the fixed split \(T\subset B\) frame. The finite-jet construction below includes arbitrary parameter algebras and their nilpotents.

Ordered root coordinates are proved in Root data, Weyl chambers and the Bruhat decomposition, §§5–8. The exact additive descent argument is Affine descent, Zariski Main and recognition of spaces, Lemma A1.1; the two-chart cohomology is Constructible complexes on algebraic varieties, §§W.1–W.3. We use the actual scheme smooth normalization, biduality, dual exchanges and affine-line unit of Lesson 5, Theorems P.2.4.1, P.3.2.1, P.3.3.1 and Lemma Q.1.2.1. The site nerve comparison is proved in Hypercoverings, Theorems 3.3–4.1. The free Whittaker patterns, §§3.1–3.2 and 6.1 describes the finite-jet and self-extension problem; all constructions used here are proved below.

L.1. The entire framed torsor functor is an affine tower

Proposition L.1.1.1. Let \(k\geq1\) be an integer strictly greater than every \(d_\alpha\), and put \(D_k=k y\). The functor of \(U\)-torsors on \(X_A\), together with a specified trivialization on \(D_{k,A}\), has no automorphisms and is represented by a smooth affine scheme \(M_k\). It is a tower of torsors for the vector spaces \[ H^1(X,\mathcal L_\alpha(-k y)),\qquad \dim M_k=m_k=\sum_{\alpha>0}(k-d_\alpha-1). \tag{L.1.2} \] Each tower torsor is trivial over its affine predecessor. Thus \(M_k\simeq\mathbf A^{m_k}\), with a choice of such trivializations. Its representing universal torsor and the functor comparison hold on every \(\kappa\)-algebra, not just on geometric points.

Proof. First make the kernel of a level frame precise. Ordered root products identify the underlying \(X\)-scheme of \(U\) with the product of its root line bundles. Replace each root coordinate line by \(\mathcal L_\alpha(-k y)\). In each root multiplication polynomial substitute \(\sigma^k x_\alpha\) for the old coordinate, where \(\sigma\) is the local equation of \(y\), and divide the resulting output coordinate by \(\sigma^k\). A nonlinear monomial of total degree \(r\geq2\) has the remaining factor \(\sigma^{k(r-1)}\). Hence all these new multiplication and inverse formulas are regular and glue by the root line tensor identifications. They define a smooth affine group scheme \(U^{(k)}\) over \(X\), with ordered root lines \(\mathcal L_\alpha(-k y)\), and a homomorphism \(U^{(k)}\to U\).

On the flat site of \(X_A\) this homomorphism is injective and its image consists exactly of the sections equal to the unit on \(D_{k,A}\). Indeed \(\sigma^k\) stays a nonzerodivisor after every such flat pullback, so a coordinate vanishing modulo \(\sigma^k\) divides uniquely by it. This is not an assertion that the group-scheme homomorphism remains injective on a test scheme supported at \(y\); that different assertion would be false. The small flat site, where torsors and their trivializing covers are defined, gives the stated kernel on every parameter algebra \(A\).

A framed \(U\)-torsor can be trivialized locally on this site by frames extending its prescribed frame on \(D_k\). To see existence, first take a trivializing flat cover, then choose affine subcovers with root line frames. Every ordered root coordinate of the difference on \(D_k\) lifts to the affine covering ring. Multiplying by that lifted \(U\)-element corrects the trivialization. The corrected transition functions belong to the preceding kernel. Conversely extending a \(U^{(k)}\)-torsor to \(U\) gives its canonical \(D_k\)-frame. These operations and their isomorphisms are inverse.

Filter \(U^{(k)}\) by root height, successively retaining roots of height at most \(h\). The last retained height is central; its kernel is the vector-group bundle \(\bigoplus_{\operatorname{ht}(\alpha)=h}\mathcal L_\alpha(-k y)\). The ordered root projection is surjective on sections over any affine chart: lift the earlier coordinates and set the new ones to zero. No Chevalley coefficient is inverted.

Here are the torsor calculations used in the induction. A vector-bundle torsor over an affine base is trivial. Its faithfully flat additive transition cocycle is a boundary in the augmented Amitsur complex of its module. Exactness follows by tensoring with the faithfully flat covering algebra, applying the multiplication retraction to the first factor as a contracting homotopy, and descending vanishing by faithful flatness. This is also the actual proof of Lemma A1.1 in Affine descent, Zariski Main and recognition of spaces. Central root induction therefore trivializes every torsor for each truncated group on an affine chart.

For the two standard affine charts of \(\mathbf P^1_A\), a line of degree \(e<0\) has no sections and has the two-chart cohomology \[ H^0(\mathbf P^1_A,\mathcal O(e))=0,\qquad H^1(\mathbf P^1_A,\mathcal O(e)) =A^{\,\max(-e-1,0)}. \tag{L.1.3} \] Explicitly, use the overlap coordinate \(t\) and the frame whose other-chart sections have coefficients in \(t^e A[t^{-1}]\). The quotient of \(A[t,t^{-1}]\) by \(A[t]+t^e A[t^{-1}]\) has the monomial basis \(t^{e+1},\ldots,t^{-1}\); its intersection is zero. Both chart rings and their intersection are affine, so the affine-module contraction and the two-chart derived comparison in GL-PERV W.1–W.3 prove that these calculations compute the actual groups. They commute with every algebra change. No higher Čech obstruction is present.

Suppose a truncated torsor over \(X_A\) has been given. It is trivial on the two charts by the affine argument. Lift its overlap transition through the next ordered root projection and glue with the inverse transition in the other direction. This constructs an actual lift to the next group; on a two-chart cover there is no additional triple equation. Two such lifts with their specified quotient identification differ by a torsor for the central vector bundle. Their isomorphism classes form its \(H^1\), by subtracting their central transitions; their automorphisms are its \(H^0\). Formula (L.1.3), with \(e=d_\alpha-k<0\), makes those automorphisms zero and gives the displayed finite vector space of lifts.

Start with the trivial group. At each induction stage the previous torsor functor is represented by an affine scheme carrying its universal torsor. Apply the preceding transition lift over that scheme. It supplies a universal lift, and adding the finite monomial representatives in (L.1.3) represents all its lifts by the indicated vector-space torsor. No automorphism of the quotient is lost: induction has already made its automorphism group zero. The same calculation kills the automorphisms at this stage. The affine vector torsor is trivial by the affine argument. Thus the resulting representing scheme is affine space, with its entire universal family and all its arrows. Root height terminates after finitely many steps. Adding the dimensions gives (L.1.2). This proves the proposition, including arbitrary-base functoriality. ∎

L.2. The genuine bundle stack and every level change

Theorem L.2.1.1. Let \(H_k=\Gamma(D_k,U|_{D_k})\). It is a smooth split unipotent group of dimension \(kn\), with underlying scheme \(\mathbf A^{kn}\). Changing the specified frame gives an actual action on \(M_k\), and \[ \operatorname{Bun}_U(X)=[M_k/H_k],\qquad d:=\dim\operatorname{Bun}_U =m_k-kn=-\sum_{\alpha>0}(d_\alpha+1). \tag{L.2.1} \] The atlas is an \(H_k\)-torsor over the stack, not an assertion that its action on \(M_k\) has trivial stabilizers. For \(k\geq1\), forgetting the last jet gives \(M_{k+1}\to M_k\), a torsor for a vector group of dimension \(n\). All the quotient presentations identify the same torsor groupoid. The construction extends locally over every Noetherian family of \(T\)-bundles on \(\mathbf P^1\), with arbitrary changes of its parameter algebra.

Proof. Near \(y\), choose the full actual \(T\)-frame. Ordered root coordinates in \(A[t]/t^k\) represent \(H_k\), with precisely \(k\) coefficients per root. Their height filtration has additive kernels of these finite coefficient spaces, which proves the group assertions in every characteristic.

Every \(U\)-torsor over \(X_A\) has a frame on \(D_{k,A}\). The latter is affine, its restricted root lines have the chosen frames, and the central induction in Proposition L.1.1.1 trivializes its torsors. The scheme of choices is the \(H_k\)-torsor of frames. Changing a frame in the universal family of \(M_k\) gives a new framed family and hence, by its representing property, an action morphism \(H_k\times M_k\to M_k\). The same representing property makes the identity and action-composition equations actual equations of morphisms.

More generally an object of \([M_k/H_k](S)\) is an \(H_k\)-torsor over \(S\) and an equivariant map to \(M_k\). Its pulled-back universal \(U\)-torsor descends with its maps, by flat torsor descent, and forgetting its frame gives a \(U\)-torsor on \(X_S\). Conversely the frame torsor of any \(U\)-torsor has its natural equivariant map to \(M_k\). After its flat cover both constructions are the same framed family, so descent proves that they are inverse on objects and on all isomorphisms. Thus the claimed equality is an equality of stacks, including their automorphism groups.

The kernel of \(H_{k+1}\to H_k\) consists of root coordinates divisible by \(t^k\) modulo \(t^{k+1}\). Products of two such coordinates vanish, since \(2k\geq k+1\). It is consequently the vector group \(\Gamma(y,\operatorname{Lie}(U)\otimes\mathcal O(-k y)|_y)\) of dimension \(n\). A frame on \(D_k\) lifts to one on \(D_{k+1}\): first trivialize the torsor on this affine thickening, then lift the correcting ordered root coordinates. Its choices are exactly that kernel. Since the original \(k\)-framed torsor has no automorphisms, they represent \(M_{k+1}\to M_k\) as its vector-group torsor. Over the affine \(M_k\) it is trivial. Forgetting a frame and applying torsor descent commutes with each level change, giving the same quotient stack.

For a Noetherian parameter ring \(R\), each line of a \(T\)-bundle has locally constant fibre degree, by the Euler complex in I.1. On an open-and-closed constant-degree part, a line \(\mathcal L\) of degree \(e\) is \(\mathcal O(e)\otimes p^*P\) for a parameter line \(P\): apply K.2's degree-zero evaluation proof to \(\mathcal L(-e)\). Apply this to an integral basis of \(X^*(T)\), retaining its entire lattice. Locally on \(\operatorname{Spec}R\) all the finitely many parameter lines have frames. The \(T\)-bundle is then the pullback of the bundle associated with the corresponding actual cocharacter of degrees. The preceding root-coordinate construction works over that base, with free coefficient spaces in (L.1.3). Globally they are the corresponding finite locally free modules and their affine towers. The constructions commute on overlaps by their representing torsor functors and therefore glue. A uniform \(k\) exists on each quasi-compact constant-degree part.

The smooth atlas has relative dimension \(kn\), and its source has dimension \(m_k\); subtraction gives (L.2.1). This is independent of \(k\), as is also checked by the \(n\)-dimensional level increment. If there are no roots, every construction is the point stack and the sums are zero. Central characters were never discarded or replaced by a finite central quotient. ∎

L.3. Actual cohomology and rank-one self-extensions

Theorem L.3.1.1. Over an algebraically closed field with \(\ell\) invertible, let \(E/\mathbf Q_\ell\) be finite. The augmentation unit gives \[ E\xrightarrow{\ \sim\ } R\Gamma(\operatorname{Bun}_U(\mathbf P^1),E). \tag{L.3.1} \] The same assertion holds integrally for the constant complete coefficient ring and at every finite reduction. For a rank-one lisse line \(\mathcal L\) on this stack and any integer shift \(s\) and invertible Tate twist \(r\), \[ R\operatorname{Hom}(\mathcal L[s](r),\mathcal L[s](r))=E. \tag{L.3.2} \] Both comparisons retain actual composition, the identity and, when all data descend to a finite field, Frobenius. In particular every positive derived self-extension group is zero.

Proof. The atlas nerve is \(M_k\times H_k^p\) in degree \(p\). Its underlying scheme is affine space of dimension \(m_k+pkn\). Every face pulls its constant coefficient identity to the same identity. Iterating the actual ordinary affine-line unit of Lesson5 Lemma Q.1.2.1 identifies the cohomology of each nerve level with the coefficient ring in degree zero, integrally and at every reduction; all positive groups vanish.

For completeness the nerve computes the stack cohomology used here. On the smooth site of the quotient, the atlas sheaf is an epimorphism: its pullback to a smooth test scheme is the smooth \(H_k\)-torsor of frames, which has étale local sections. The augmented free nerve complex is exact. Locally choose a lift to the atlas and insert it as first vertex; deleting that vertex gives the identity and all other face terms cancel in pairs. Apply an injective coefficient resolution and take the first-quadrant double complex of sections over the nerve. The horizontal augmentation is then a quasi-isomorphism; each total degree involves finitely many terms. This is precisely the finite-cone and comparison proof of Hypercoverings, Theorems3.3–4.1, applied to this site. The vertical calculation is the actual scheme cohomology just proved. Hence its totalization computes the required cohomology without replacing the quotient by its set of points.

It remains the augmented complex with one coefficient-ring term in every nonnegative degree. Its degree-\(p\) differential is multiplication by \(\sum_{i=0}^{p+1}(-1)^i\): zero for even \(p\), the identity for odd \(p\). Its cohomology is the ring in degree zero and zero otherwise; its generator is the augmentation of \(1\). This proves the actual unit comparison. Applying it at each finite coefficient level gives the same transition identities; the constant inverse tower has surjective maps and no degree-one-limit term. Completion and the actual rationalization preserve the displayed unit. Over a finite field that unit is defined over the field, so Frobenius acts on its coefficient generator as the identity.

The evaluation \(\mathcal L^\vee\otimes\mathcal L\to E\) is an actual isomorphism of lisse lines, with its unit and equivariance. Rank-one internal Hom and the cancelling shifts and twists identify the derived self-mapping complex with the preceding constant cohomology. Its composition is multiplication of constant endomorphisms, and its generator is the identity. This proves (L.3.2), including the specified operators.

If a shifted rank-one line is perverse, its perverse Yoneda self-\(\operatorname{Ext}^1\) is its degree-one derived self-mapping group: a short exact sequence gives its connecting map, and the cone of such a map gives the inverse construction by the heart truncation sequence. Thus it is zero. Induction on length splits every finite perverse extension whose factors are this one line. This last assertion concerns that stratum and those factors; it supplies no claim yet about extensions across different compactified strata. ∎

L.4. Intrinsic normalized duality on the finite quotient

Theorem L.4.1.1. Use the bounded cartesian constructible enhancement of the actual atlas nerve. Perverse normalization and structural duality on \(\operatorname{Bun}_U\) are independent of the sufficiently large jet level. If \(q_k:M_k\to\operatorname{Bun}_U\), put \(q_k^\dagger=q_k^*[kn](kn/2)\), using the fixed half-Tate normalization. The intrinsic duality is determined by \[ q_k^\dagger\mathbb D_{\operatorname{Bun}_U}K =\mathbb D_{M_k}q_k^\dagger K. \tag{L.4.1} \] It has the actual evaluation and biduality maps and retains finite coefficient comparisons and Frobenius. For a descended evaluation \(e:\operatorname{Bun}_U\to\mathbf A^1\), set \(\Psi=e^*\mathcal L_\psi[d](d/2)\), where \(d\) is (L.2.1). Then \(\Psi\) is simple perverse and \[ \mathbb D\Psi=e^*\mathcal L_{\psi^{-1}}[d](d/2). \tag{L.4.2} \] The sign of the additive character changes; it must not be erased when invoking duality.

Proof. Describe the enhancement and descent comparisons explicitly. A cartesian object is its constructible complexes on the nerve, their pullback identifications, and the coherent face-composition identifications. The corresponding derived mapping complex is the totalized nerve of scheme mapping complexes. Uniform boundedness is tested on the atlas; perverse bounds are tested after the smooth normalization. This construction uses the actual scheme coefficient categories of Lesson5 M–P, rather than an equivalence with bounded complexes in a prescribed abelian heart.

Every face map between successive nerve levels is smooth of relative dimension \(kn\). The actual scheme dual exchange of Lesson5 Theorem P.3.3.1 and smooth normalization P.2.4.1 give, for such a map \(f\) of dimension \(h\), \[ \mathbb D f^*[h](h/2) =f^*[h](h/2)\mathbb D . \tag{L.4.3} \] Indeed \(\mathbb D f^*=f^!\mathbb D\), with \(f^!=f^*[2h](h)\), and dualizing the added shift and twist subtracts \(h,h/2\). Apply (L.4.3) to the normalized complexes in every nerve degree. Dualize each descent isomorphism and invert it to restore its direction. The coherent composition equations remain the same equations: the scheme comparisons were constructed from the same trace counits and evaluation and have coherent transitivity. The degeneracy identifications are the inverse consequences of these face equations and the identity section, so they are retained as well. Thus the dual complexes descend in this enhancement and satisfy (L.4.1).

Equivalently its unnormalized atlas complex is \(\mathbb D_{M_k}(q_k^*K)[-2kn](-kn)\). This explains why ordinary scheme duality of the unnormalized pullback would give the wrong shift. The scheme double-evaluation maps commute with the same descent identifications and therefore descend to the actual intrinsic double evaluation. They are isomorphisms on the conservative atlas by P.3.2.1. Tensor evaluation and its graded signs, as well as their finite reductions and Weil maps, descend in the same way.

To check independence, use the common frame atlas with both a \(D_k\)-frame and a \(D_l\)-frame. It is the fibre product \(M_k\times_{\operatorname{Bun}_U}M_l\), a smooth \(H_l\)-torsor over \(M_k\) and a smooth \(H_k\)-torsor over \(M_l\). It is a scheme: the second-frame torsor of the universal family on the affine \(M_k\) is trivial by the central affine argument. Both comparisons with its scheme duality are exactly (L.4.3), and their normalized shifts are \(kn+ln\) in either order. Scheme transitivity therefore identifies the two descended dualities, evaluations and their nerve coherences. The same common smooth atlas identifies the perverse bounds. No assumption about a globally free action was used.

The normalized pullback of \(\Psi\) is \((e q_k)^*\mathcal L_\psi[m_k](m_k/2)\). It is a rank-one lisse perverse line on the connected smooth affine space \(M_k\). Such a line is simple: at a proper smooth closed support of positive codimension its ordinary and exceptional restrictions have strictly negative and positive perverse degrees, respectively; the scheme smooth orientation and closed localization exclude a subobject or quotient there. On a dense smooth open, rank one excludes a proper nonzero lisse subobject. Stratifying any proposed support finishes the argument, as in G.3. Since normalized atlas pullback is exact and conservative for these perverse bounds, the descended object is simple.

Scheme smooth duality sends that normalized line to \((e q_k)^*\mathcal L_{\psi^{-1}}[m_k](m_k/2)\). Its evaluation uses the actual multiplication pairing \(\mathcal L_{\psi^{-1}}\otimes\mathcal L_\psi=E\) of E.1.1.1, not an identification of the two character lines. Descent gives (L.4.2) with the same pairing, half twist, positive trace orientation and Frobenius. This proves the theorem. It constructs intrinsic stratum duality; ordinary and exceptional restrictions from a compactification still require their own geometric bounds. ∎

Proposition L.1.1.1 and Theorems L.2.1.1–L.4.1.1 construct the framed schemes, their exact quotient stacks and the intrinsic duality. The negative dimension of \(B\mathbf G_a\) records its smooth automorphisms. Editable SVG source.

L.5. Exercises

Exercise L.5.1 (medium). Take \(SL_2\) with the trivial \(T\)-bundle. Compute the entire level-one presentation, its stack dimension, constant cohomology and normalized duality. Explain the negative shift.

Solution. There is one root line \(\mathcal O\). At \(k=1\) its framed line is \(\mathcal O(-1)\), with \(H^0=H^1=0\) by (L.1.3). Thus \(M_1=\operatorname{Spec}\kappa\), \(H_1=\mathbf G_a\), and the entire torsor stack is \(B\mathbf G_a\), of dimension \(0-1=-1\). This is the groupoid of trivial additive bundles and their additive automorphisms on every affine parameter ring, rather than a point scheme. Its nerve terms are \(\mathbf A^p\); the alternating constant cochain complex gives \(R\Gamma(B\mathbf G_a,E)=E\), with its identity operator. The zero-evaluation normalized perverse line is \(E[-1](-1/2)\). Pulling it to the point and applying \(q^\dagger=[1](1/2)\) gives \(E\) in degree zero. Its structural dualizing line is \(E[-2](-1)\); tensor duality therefore takes \(E[-1](-1/2)\) to itself with the actual pairing. The negative shift records the smooth additive automorphisms; replacing the stack by its point set would lose it.

Exercise L.5.2 (advanced). For \(GL_3\), take the full torus bundle with coordinate line degrees \((-2,0,0)\). Compute the level-one and level-two dimensions. Give a smaller quotient presentation of its genuine unipotent bundle stack and its scheme-theoretic stabilizer when the first extension class is a dual-number nilpotent.

Solution. The three positive root lines have degrees \(-2,0,-2\). Hence \(m_1=2+0+2=4\), \(\dim H_1=3\), \(m_2=3+1+3=7\), and \(\dim H_2=6\); both stack dimensions are one. \(H_1\) is the actual upper unitriangular group, including its central multiplication term, not its abelianization. Forgetting the last jet is a three-dimensional vector torsor.

For a smaller presentation, trivialize on the two standard charts and successively normalize the upper entries \(x,y,z\). The \(x,z\) root lines \(\mathcal O(-2)\) each retain exactly their \(t^{-1}\) class; the \(y\) line \(\mathcal O\) has no \(H^1\). Every transition reduces to \[ n(a,b)= \begin{pmatrix} 1&a/t&b/t\\0&1&0\\0&0&1 \end{pmatrix}. \tag{L.5.1} \] The remaining equal global and local \(y\)-changes are a constant \(c\). Conjugating by \(u_{23}(c)\) takes \(n(a,b)\) to \(n(a,b-ac)\), by actual matrix multiplication. Conversely a gauge arrow between two normalized transitions has zero \(x\)-change, since \(\mathcal O(-2)\) has no global section; its \(y\)-change is exactly that constant. Its \(z\)-equation then gives \(b'=b-ac\) and no further arrow, since the \(z\) line also has no global section. Thus the entire groupoid is \[ [\mathbf A^2_{a,b}/\mathbf G_{a,c}], \qquad c:(a,b)\longmapsto(a,b-ac). \tag{L.5.2} \] It has trivial stabilizer over \(a\) invertible and additive stabilizer over \(a=0\). More precisely over any parameter algebra its stabilizer is the group scheme \(\operatorname{Spec}A[c]/(ac)\). For \(A=\kappa[\epsilon]/(\epsilon^2)\), \(a=\epsilon\), this is \(A[c]/(\epsilon c)\); it is not flat over \(A\). Indeed the nonzero polynomial class of \(c\) modulo \(\epsilon\) tensors with the generator of \((\epsilon)\), but its image under multiplication into \(A[c]/(\epsilon c)\) is zero. Flatness would preserve that injection. The actual quotient stack and the smooth finite-jet atlas retain this nilpotent stabilizer; neither replacing \(a\) by its reduced fibre nor declaring the frame-group action free is valid. All matrix computations, including their minus sign, remain valid in characteristic two.

Exercise L.5.2 computes the entire action \(c:(a,b)\mapsto(a,b-ac)\), its scheme-theoretic stabilizers and the two jet levels. The smooth frame atlas retains the nonflat stabilizer \(A[c]/(\epsilon c)\); its action is not assumed free. Editable SVG source.

Appendix M. Compactified global jet schemes and the affine genuine open

Write \(X=\mathbf P^1_\kappa\), with homogeneous coordinates \(S,T\), \(y=[1:0]\), and \(u=T/S\) near \(y\). A jet level \(k\) is a positive integer and \(D_k=k y\). We work with the regular compactification for split \(G\) with simply connected derived group, as in Appendix J, retaining the full original central torus. The following construction uses the diagonal sequence and finite Grassmannians to prove the needed framed bundle schemes directly.

The exact cohomology inputs are Constructible complexes on algebraic varieties, §§W.1–W.4 and Appendix I. The finite Grassmannian is constructed in Roots and reductive groups of rank one, Lemma 5.A. We use Affine descent, Zariski Main and recognition of spaces, Lemmas A1.1–A1.3 and Corollary A3.2. The subgroup-detecting line is proved in Tori, maximal tori and their conjugacy, §2. Its full schematic orbit and actual fppf quotient torsor are Group schemes over a field, Theorem 7.13 and Quotients and torsors, Theorem 11.1b. These earlier programme proofs include nonreduced test schemes. The free Whittaker patterns, §3.3 supplies the geometric setting; the bounded scheme charts and affine-open argument are proved here.

M.1. The universal two-term presentation on the projective line

Proposition M.1.1.1. Let \(R\) be Noetherian and \(\mathcal E\) a vector bundle on \(X_R\). Assume \(H^1(X_{\bar s},\mathcal E_{\bar s}(-1))=0\) on every geometric parameter fibre. There are finite locally free parameter modules \[ P=p_*\mathcal E,\qquad Q=p_*\mathcal E(-1) \] whose formation commutes with every parameter algebra change, and a canonical sequence \[ 0\longrightarrow\mathcal O(-1)\otimes Q \xrightarrow{\,T i-S j\,}\mathcal O\otimes P \longrightarrow\mathcal E\longrightarrow0, \quad i=S\cdot:Q\to P,\quad j=T\cdot:Q\to P. \tag{M.1.1} \] If \(\mathcal E\) has rank \(r\) and degree \(e\), then \(e\geq0\), \(\operatorname{rk}Q=e\) and \(\operatorname{rk}P=e+r\). The sequence is universally exact.

Proof. Multiplication by \(S\) gives \(0\to\mathcal E(-1)\to\mathcal E\to\mathcal E|_y\to0\). Its first arrow is injective even after a parameter change, because \(S\) is universally regular on the curve and \(\mathcal E\) is locally free. Fibre cohomology and the two-chart vanishing in degrees at least two show \(H^1(\mathcal E_{\bar s})=0\) as well. For each of the two bundles use I.1's actual finite projective complex \([P^0\to P^1]\). Its finite differential cokernel has zero geometric fibres and is therefore zero by Nakayama. The differential splits, since \(P^1\) is projective. Its kernel is finite projective, and its tensor by any \(R\)-algebra is the kernel of that same split differential. This proves the asserted parameter and cohomology comparisons.

On \(X_R\times_R X_R\) the diagonal has equation \(S_1T_2-T_1S_2\), and the sequence \[ 0\to\mathcal O(-1,-1)\to\mathcal O\to\mathcal O_\Delta\to0 \tag{M.1.2} \] is universally exact. On each product chart the equation is either a monic coordinate difference or its equivalent homogeneous frame expression; it cuts out exactly the diagonal and is a nonzerodivisor on every coefficient ring. Tensor by the first pullback of \(\mathcal E\) and take cohomology along the first factor. The preceding zero \(H^1\) terms and universally split finite complexes give precisely (M.1.1). The sign follows from the displayed diagonal equation: its two multiplication maps are \(T i-S j\). Its last map is evaluation. In particular \(\mathcal E\) is generated by \(P\), and the sequence remains exact after every parameter change.

On a connected parameter part write \(q=\operatorname{rk}Q\) and \(p=\operatorname{rk}P\). Taking ranks in the sequence gives \(p-q=r\). Taking determinants gives \(\det\mathcal E=\mathcal O(q)\otimes p^*(\det P\otimes(\det Q)^{-1})\). Thus its geometric fibre degree is \(q\). This proves the rank and degree formulas without needing a splitting theorem or a separate vector-bundle Riemann–Roch assertion. ∎

M.2. An explicit separated scheme of framed vector bundles

Theorem M.2.1.1. Fix integers \(r\geq1,e\geq0,k\geq1\), and let \(J=(\kappa[u]/u^k)^r\), viewed as a \(kr\)-dimensional vector space with endomorphism \(u\). The groupoid of rank-\(r\), degree-\(e\) vector bundles \(\mathcal E\) on \(X_A\), with a specified isomorphism \(\mathcal E|_{D_{k,A}}\simeq J_A\), satisfying

1. \(H^1(\mathcal E_{\bar s}(-1))=0\);
2. \(H^0(\mathcal E_{\bar s}(-k))=0\),

on every geometric parameter fibre, is represented by a separated finite-type scheme \(Z_{r,e,k}\), on every parameter algebra \(A\). Empty Grassmannian ranks give the empty scheme. It has no framed automorphisms.

Proof. In the product of subbundle Grassmannians \(\operatorname{Gr}(e+r,J)\times\operatorname{Gr}(e,J)\), let \(P,Q\subset J\) be the universal subbundles. Impose the closed incidence equations \[ Q\subset P,\qquad uQ\subset P. \tag{M.2.1} \] They are the zero maps \(Q\to J/P\) and \(Q\xrightarrow uJ\to J/P\), so their scheme equations are finite matrix entries and impose the assertions on all rings. Let \(i:Q\to P\) be inclusion and \(j:Q\to P\) be \(u\) followed by inclusion into \(P\). Form the universal map \(T i-S j\) on the projective line.

Require it to have rank \(e\) everywhere on every curve fibre. This is open in the parameter scheme: its failure locus is the closed locus of its \(e\)-minors in the proper curve, whose projection is closed. For \(e=0\) there is no failure locus. On this open the map is a subbundle map. Indeed at any point an \(e\)-minor is a unit; row and column operations split off its invertible block, and leave a locally free quotient of rank \(r\). Denote that quotient by \(\mathcal E\). It is a vector bundle and the sequence is universally split on those local frames.

The natural map \(P\otimes_\kappa\kappa[u]/u^k\to J\) kills the relation \(u i(q)-j(q)\), so gives \(\mathcal E|_{D_k}\to J\). Require this to be an isomorphism. This is a further open condition, either by its determinant in finite local frames or by the proper finite projection of its failure locus on \(D_k\). It specifies the entire jet frame, including its nonreduced coefficients. These conditions define \(Z_{r,e,k}\) as an open in a closed incidence scheme in the projective Grassmannian product; hence it is separated and of finite type.

The sequence now computes \(H^0(\mathcal E)=P\), \(H^1(\mathcal E)=0\), \(H^0(\mathcal E(-1))=Q\), and \(H^1(\mathcal E(-1))=0\), universally: \(H^0(\mathcal O(-1))=H^1(\mathcal O(-1))=0\), and \(H^1(\mathcal O(-2))\) is the single two-chart monomial class. The natural section corresponding to \(q\in Q\) is the class of \(i(q)\) on \(S\ne0\), and the class of \(j(q)\) in the \(T\)-frame on \(T\ne0\); the relation \(T i-S j\) identifies them. This also checks the indicated \(Q\) comparison and its sign. Multiplication by \(S,T\) gives the original \(i,j\). Their jet restrictions are exactly the original inclusions \(Q,P\subset J\). In particular the section restriction \(P\to J\) is injective with locally free cokernel, so \(H^0(\mathcal E(-k))=0\) on every base change.

Conversely a framed bundle as in the theorem has the sequence of Proposition M.1.1.1. Its section restrictions \(P\to J_A\) and \(Q\to J_A\), using the fixed \(S\)-frames on the jet, are injective with locally free cokernels. To justify the scheme statement, work first over a Noetherian model. On geometric fibres the first restriction is injective by condition2; the second is injective too, since multiplication by \(S\) embeds its sections into the first. The dual maps have finite cokernels with zero fibres, so Nakayama makes them surjective. Their locally free targets make these surjections split locally. Thus the restrictions define the actual Grassmannian subbundles. Their ranks are \(e+r,e\) by Proposition M.1.1.1. Multiplication by \(T\) at the jet is exactly \(u\), giving (M.2.1). The canonical sequence and the original jet isomorphism give exactly the open conditions above, and reconstruct the bundle and its frame.

These constructions are inverse on morphisms. A framed automorphism induces the identity on the injective section subbundles \(P,Q\subset J_A\), and hence the identity on their evaluation quotient. Thus no automorphisms are omitted. For an arbitrary \(A\), descend the finite presentation of the bundle, its frame and all these maps to a finitely generated coefficient ring. The two fibre vanishing conditions define opens there by the finite complexes of I.1: zero cokernel for \(H^1\), and split injectivity for the actual finite restriction map. Their preimages contain all of \(\operatorname{Spec}A\). Restrict to them and apply the preceding universally exact construction. It gives the same comparison after tensor by \(A\). This proves the whole functor assertion on arbitrary parameter rings. ∎

M.3. Framed actual \(G\)-bundles are separated schemes

Theorem M.3.1.1. Choose a faithful closed representation \(\rho:G\hookrightarrow GL(V)\). There is a finite \(GL(V)\)-module \(W\) and a line \(\ell\subset W\) whose full schematic stabilizer is \(G\). Let \(\chi\in X^*(G)\) be its line character. Fix degrees \(\delta_V,\delta_\chi\), and a twist \(a\) with \(e=\delta_V+a\dim V\geq0\). Impose the two conditions of Theorem M.2.1.1 on \(E_V(a)\) and on \(E_\chi(-\delta_\chi)\). Actual \(G\)-bundles of these degrees with a specified full \(G\)-frame on \(D_k\) are then represented by a separated finite-type scheme. All assertions concern the entire torsor and its isomorphisms on arbitrary parameter algebras.

Proof. Here the faithful representation and detecting line are concrete finite data. A finite-dimensional Hopf subcomodule of \(\kappa[G]\) containing algebra generators gives the faithful closed representation: evaluation at the identity expresses each generator as a matrix coefficient, so the coordinate algebra map from the general linear group is surjective. For the closed \(G\subset GL(V)\), choose a finite regular \(GL(V)\)-submodule containing generators of its ideal and intersect it with that ideal. Its stabilizer is \(G\) on every test algebra, because preserving that subspace preserves the ideal it generates, and evaluating a preserved right translate at the identity gives precisely membership in \(G\). The determinant line of that subspace has the same schematic stabilizer. This is the line construction proved in AG-RG-01 §2.

The full orbit theorem gives a locally closed immersion \[ O=GL(V)/G\hookrightarrow\mathbf P_{\rm lines}(W), \tag{M.3.1} \] with its original scheme structure, and \(GL(V)\to O\) is its actual fppf \(G\)-torsor. We use Group schemes over a field, Theorem7.13, and Quotients and torsors, Theorem11.1b, whose scheme and torsor proofs include nonreduced test schemes. Thus a reduction of the frame torsor of a vector bundle \(E_V\) to \(G\) is exactly a line subbundle \(F\subset E_W\) whose corresponding projective section factors through the associated orbit \(O\). Pull back the displayed \(G\)-torsor to reconstruct the reduction; extension to \(GL(V)\) is the inverse. Its associated character line is \(F\), with its natural inclusion in \(E_W\).

Over the product \(Z_{\dim V,e,k}\times Z_{1,0,k}\) take the first universal bundle and untwist it by \(\mathcal O(-a)\), and take the second and twist it by \(\mathcal O(\delta_\chi)\). These are \(E_V,F\) with their entire \(D_k\)-frames. The finite section scheme of I.1 represents maps \(F\to E_W\) as a finite-type affine scheme over this Noetherian base. Associated \(W\)-bundles exist by affine frame descent, and are vector bundles; no statement that \(W\) is a symmetric power in the geometric characteristic is required. Impose the closed jet equation that this map is the inclusion of the fixed line \(\ell\) under the two specified frames.

Require the map to be a subbundle everywhere. Its rank-one failure locus is closed on the proper curve, giving an open parameter condition by projection. On that open it defines a projective line section. Require that section to lie in the closure of the orbit and avoid its boundary. The first condition is closed: the invariant projective orbit closure has finitely many homogeneous defining equations. Their finite-dimensional degree pieces are \(GL(V)\)-stable; evaluating their associated bundle maps on the line gives the actual finite polynomial zero conditions of I.2. They enforce factorization as a scheme morphism, including nilpotents. The second condition is open: pull back the closed projective boundary and remove its proper closed projection to the parameter base. These two conditions are exactly factorization through the locally closed orbit, on every parameter algebra.

The resulting locally closed subscheme of the finite affine map scheme is separated and of finite type. It recovers the full \(G\)-reduction by (M.3.1). Its jet equation says that the specified \(GL(V)\)-frame actually belongs to the reduction over \(D_k\); the specified frame of \(F\) is precisely its induced \(\ell\)-frame. Thus it gives the specified \(G\)-frame, with no extra line-frame choice. Conversely a \(G\)-framed bundle gives these two framed bundles, their orbit line and the jet equation. The constructions are inverse, including their isomorphisms. A framed automorphism is already the identity on \(E_V\) by Theorem M.2.1.1, and \(\rho\) is a closed embedding, so it is the identity on the \(G\)-torsor. This proves the scheme assertion. ∎

M.4. Uniform bounds from the complete geometric defect

Proposition M.4.1.1. Fix the actual \(T\)-bundle \(\mathcal T\) and nonnegative integers \(b_i\), one per simple coroot. Let regular compactified data have geometric fundamental zero-divisor lengths \(m_i\leq b_i\). For any fixed finite \(G\)-module \(V\) with weight multiset \(\Xi\), all the line degrees in a \(B\)-filtration of \(E_V\) lie between \[ L_V=\min_{\xi\in\Xi} \left(\deg\mathcal L_\xi+ \sum_i\min(0,b_i\langle\xi,\alpha_i^\vee\rangle)\right), \qquad U_V=\max_{\xi\in\Xi} \left(\deg\mathcal L_\xi+ \sum_i\max(0,b_i\langle\xi,\alpha_i^\vee\rangle)\right). \tag{M.4.1} \] Consequently \(a\geq\max(0,-L_V)\) and \(k>a+U_V\) ensure the two framed-bundle conditions on \(E_V(a)\). The full determinant and detecting-line degrees are fixed by the entire central maps. The condition \(m_i\leq b_i\) is an actual open condition in the compactified parameter scheme, and does not require a nilpotent family to factor through one fixed-defect stratum.

Proof. On a geometric curve, J.5 supplies the unique saturated \(B\)-reduction with torus bundle \(\mathcal T(D)\), where \(D=\sum_i\alpha_i^\vee D_i\), \(\deg D_i=m_i\). This is a geometric-fibre argument, exactly where J.5 applies. For a \(T\)-weight \(\xi\), its line degree is \(\deg\mathcal L_\xi+\sum_i m_i\langle\xi,\alpha_i^\vee\rangle\). Taking the minimum and maximum on the finite integer box \(0\leq m_i\leq b_i\) gives (M.4.1).

The required \(B\)-filtration of a representation is explicit in every characteristic. Its split \(T\)-coaction decomposes it into its weight spaces. In the polynomial coaction of a positive root \(u_\alpha(z)\), the coefficient of \(z^j\) sends a weight \(\xi\) into \(\xi+j\alpha\), by conjugating with \(T\); the \(j=0\) coefficient is the identity. Order the finite weights by a linear functional positive on every positive root. Successive spans of the highest remaining weight spaces are \(B\)-stable, and refine each equal-weight block by any basis flag, since \(N\) has no degree-zero off-identity action there. This gives successive line quotients with the stated weights, and hence the associated filtration of the actual bundle.

For the twist \(E_V(a-1)\), each quotient line has degree at least \(-1\), so its \(H^1\) vanishes by the two-chart calculation. The long exact sequences give \(H^1(E_V(a-1))=0\). For \(E_V(a-k)\), every quotient line has negative degree, so \(H^0=0\) by the same calculation and exact sequences. These are precisely the two conditions of Theorem M.2.1.1 for \(E_V(a)\). The sums of the representation weights give \(\det V\in X^*(G)\), which pairs to zero with all coroots. The detecting line character also pairs to zero with all coroots. Thus their degrees are respectively \(\deg\mathcal L_{\det V}\) and \(\deg\mathcal L_\chi\), and are fixed by the actual central inverse-map equations. No adjoint torus or rational lattice is substituted.

Finally each fundamental zero scheme is finite over the parameter scheme, as proved in K.5. Its finite algebra has a finite module presentation. The condition that its geometric fibre dimension is at most \(b_i\) is the open locus where the corresponding presentation matrix has rank at least the number of generators minus \(b_i\). It is the union of its indicated minor opens; if that number is nonpositive it is the whole base. The same minor description commutes with arbitrary parameter algebra change. Intersect the finitely many such opens. This describes the bound on the entire ambient family even if its finite zero algebra is nonflat, as in K.6.2. The preceding geometric fibre bounds imply the framed chart conditions, which themselves are opens proved by finite complexes. Nothing here divides a nilpotent family by a geometric defect divisor. ∎

M.5. The full compactified \(N\)-jet chart is a separated scheme

Theorem M.5.1.1. In the regular compactification for fixed \(\mathcal T\), let \(\overline{\mathcal B}_{b,y}^{\mathcal T}\) be the open substack with all fundamental zero lengths at most \(b_i\) and all fundamental maps primitive at \(y\). For a sufficiently large positive jet level \(k\) it has a presentation \[ \overline{\mathcal B}_{b,y}^{\mathcal T} =[\,\overline M_{b,y,k}^{\mathcal T}/N_k\,], \qquad N_k=N(\kappa[u]/u^k), \tag{M.5.1} \] where \(\overline M_{b,y,k}^{\mathcal T}\) is a separated finite-type scheme. Its entire genuine open is exactly the framed scheme \(M_k\) of L.1. All parameter nilpotents and all stabilizers belong to this presentation. These opens cover the full compactification after passage to an algebraic closure; the construction and the open immersion descend over the field of definition.

Proof. Choose \(V,\ell\subset W\) as in Theorem M.3.1.1. Use the finite degree bounds of Proposition M.4.1.1 to choose \(a,k\) for \(E_V\), and increase \(k\) if needed for the fixed-degree detecting line and for L.1's negative-root frame condition. Every geometric object in the stated bound then lies in the actual \(G\)-framed scheme of Theorem M.3.1.1. The fixed \(T\)-bundle has a chosen frame over the jet. Its entire character basis fixes every source line frame there, including the central characters and their inverses.

Over that Noetherian \(G\)-framed scheme, the universal \(G\)-bundle and fixed \(\mathcal T\) define the finite highest-line section scheme of I.1–I.2 and J.4. Its finite generators and relations represent the complete datum, not just the fundamental maps. The geometric generic-nonzero open gives universal sheaf injection and finitely presented parameter-flat quotients by I.3; impose that open. The length bounds are the opens of Proposition M.4.1.1. Impose the finitely many closed jet equations \[ \kappa_{\lambda}|_{D_k}=v_\lambda \tag{M.5.2} \] for a finite homogeneous generator list of the full highest-line algebra, including the central inverse generators, in the specified \(G\)- and \(T\)-frames. Here \(v_\lambda\) is the standard highest vector. The polynomial relations recover (M.5.2) for every weight. The finite restriction maps are actual maps on jet modules, so these are closed scheme equations by I.2.

This constructs \(\overline M_{b,y,k}^{\mathcal T}\) as an open of a closed subscheme in a finite-type affine scheme over the separated \(G\)-framed base. Thus it is a separated finite-type scheme. The chosen jet equations guarantee primitivity at \(y\). Conversely any primitive-at-\(y\) compactified datum has the full \(G/N\) open along the jet by J.3–J.4, giving its genuine \(N\)-reduction there with the chosen \(T\)-frame. The compatible \(G\)-frames are the torsor of \(N_k\)-frames of this reduction. It has sections locally on the parameter base; in fact on an affine base the ordered central root argument trivializes its unipotent jet torsor. Choosing such a section gives exactly the object of the constructed scheme and equations (M.5.2).

Changing that frame gives the \(N_k\)-action on the scheme. The schematic stabilizer of all standard highest vectors is exactly \(N\), by J.3's actual torus frame construction of \(G/N\). The same is therefore true on every jet algebra. The constructions from an \(N_k\)-torsor with its equivariant map to \(\overline M\), and from a compactified datum to its compatible frame torsor, are inverse by flat torsor descent, including every isomorphism. This proves (M.5.1) and does not declare the action globally free. Its atlas is smooth of relative dimension \(k|\Phi^+|\).

On the genuine locus there is an actual \(N^{\mathcal T}\)-torsor on the entire curve. The compatible jet frame is precisely its \(D_k\)-frame, and L.1 represents that whole groupoid by \(M_k\). Conversely every such framed genuine torsor lies in the constructed chart: its defect is zero and the uniform bounds included zero. Thus its open pullback is the entire \(M_k\), rather than a selected affine subopen.

At a geometric point only finitely many fundamental zeros occur, so their lengths have finite bounds \(b_i\), and some point \(y\) of \(\mathbf P^1\) lies outside their supports over the algebraic closure. The corresponding opens cover. Each has the smooth scheme atlas above. Representability on scheme tests follows directly: pull back the primitive and length opens, then their compatible jet frame torsor, which is an affine \(N_k\)-torsor over that open test scheme by affine descent. Thus these atlases are representable, and their quotient presentations construct the required locally finite-type algebraic stack. Overlaps represent exactly the same torsor-and-map groupoid, so glue with its existing descent. A finite extension makes any needed chosen \(y\) rational, and the constructions preserve their field-of-definition maps and descent. This proves the stated full chart result. ∎

Propositions M.1.1.1 and M.4.1.1, and Theorems M.2.1.1, M.3.1.1 and M.5.1.1 identify the section subbundles, their exact two-term relation, the schematic orbit reduction and the full compactified jet equations. Geometric defect provides uniform bounds; all scheme equations retain arbitrary parameter nilpotents. Editable SVG source.

M.6. The genuine open immersion is affine

Theorem M.6.1.1. For the full regular compactification with fixed \(\mathcal T\), the genuine-reduction open immersion \[ j:\operatorname{Bun}_{N^{\mathcal T}}(X) \hookrightarrow\overline{\operatorname{Bun}}_N^{\mathcal T}(X) \tag{M.6.1} \] is representable and affine. It retains the full datum on arbitrary parameter schemes.

Proof. Representability as an open immersion is already explicit in the finite highest-line equations: all fundamental maps must be subbundles at every point of the curve. Their failure is the proper projection of the finitely many closed zero loci. Its complement is open on every parameter scheme, and J.3 identifies its entire datum with the genuine reduction, including its inverse central maps. This is an open subfunctor, not a reduced-point condition.

Pull back \(j\) to an atlas \(\overline M_{b,y,k}^{\mathcal T}\) of Theorem M.5.1.1. Its source is the full affine scheme \(M_k\) of L.1; its target is separated. An open immersion from an affine scheme \(U\) into a separated scheme \(Z\) is affine. Indeed for any affine open \(V\subset Z\), the intersection \(U\cap V\) is the inverse image of the closed diagonal in the affine product \(U\times V\), and hence is affine. This is the affine-test definition of an affine morphism. Therefore the pulled-back open immersion is affine.

For any scheme test \(S\to\overline{\operatorname{Bun}}_N^{\mathcal T}\), first restrict to the primitive-at-\(y\), bounded-length open cover just constructed. The compatible jet-frame torsor is a faithfully flat affine cover of this test open. After that cover the pullback of \(j\) is a base change of \(M_k\hookrightarrow\overline M\), and is affine. Corollary A3.2 of Affine descent, Zariski Main and recognition of spaces descends affineness of this existing open scheme morphism. Affineness is local on target opens, so it holds on all of \(S\). If the point choices were made over a field extension, the same fpqc descent applies before the final conclusion over the original field. This proves representable affineness on every test.

Every bound and atlas was defined on the entire section and jet schemes. A nonflat zero algebra can cross fixed-defect strata while remaining in those bound opens. Its family and every nilpotent automorphism are therefore retained in the argument. The theorem proves the affine geometric input; the perverse direct-image and compactified boundary estimates still require their coefficient and stratification arguments. ∎

Theorem M.6.1.1 proves affineness on every test scheme through the closed-diagonal intersection and affine descent. Exercise M.7.2 shows exactly why separatedness is required. The quotient presentation retains full stabilizer schemes. Editable SVG source.

M.7. Exercises

Exercise M.7.1 (medium). For a degree-\(d\) line bundle, twist by \(\mathcal O(-d)\). Compute its entire level-\(k\) framed parameter scheme and a dual-number frame. Compare with the unframed torus component.

Solution. The twisted bundle has degree zero, so \(e=0,r=1\). The two-term presentation has \(Q=0\) and \(P\) a line in \(J=A[u]/u^k\). Its jet evaluation is an isomorphism precisely when a local generator of \(P\) is a unit of \(J\), equivalently when its constant coefficient is a unit of \(A\). Normalize that coefficient to one by the unique parameter scalar. The entire scheme is \[ Z_{1,0,k}=\mathbf A^{k-1},\qquad f(u)=1+c_1u+\cdots+c_{k-1}u^{k-1}. \tag{M.7.1} \] Its coordinates survive on every algebra. For \(A=\kappa[\epsilon]/\epsilon^2\) and \(k\geq2\), \(f(u)=1+\epsilon u\) is a nonconstant jet frame with inverse \(1-\epsilon u\). This is valid also in characteristic two. In general the inverse is the finite geometric series in \(f-1\), whose \(k\)-th power vanishes because it is divisible by \(u^k\). Forgetting the jet frame gives the degree-\(d\) component \(B\mathbf G_m\) of the actual torus-bundle stack: its full frame group is \((A[u]/u^k)^\times\), not merely its constant or reduced units. This example retains the entire central torus and explains the necessary nonreduced jet equations.

Exercise M.7.2 (advanced). Explain why separatedness cannot be omitted from the affine-open argument. Glue two copies of \(\mathbf A^2_\kappa\) along their common punctured plane \(W=\mathbf A^2\setminus\{0\}\), by the identity. Test the open immersion of the first affine copy.

Solution. The glued object is a scheme by ordinary open gluing. Pulling that open immersion back to the second affine copy gives \(W\). This is not affine. Indeed its two-chart cover \(D(x)\cup D(y)\) gives \[ \Gamma(W,\mathcal O) =\kappa[x,y]_x\cap\kappa[x,y]_y =\kappa[x,y] \tag{M.7.2} \] inside \(\kappa[x^{\pm1},y^{\pm1}]\). A Laurent polynomial in the first ring has no negative \(y\)-exponent, and one in the second has no negative \(x\)-exponent; their intersection therefore has neither. If \(W\) were affine, its canonical map to the spectrum of this section ring would be an isomorphism. That map is the usual inclusion into \(\mathbf A^2\), whose image omits the origin, a contradiction. Thus the open immersion from an affine scheme is not affine in this example. The glued scheme is not separated: otherwise its two affine opens would have affine intersection by the closed-diagonal argument just contradicted. In Theorem M.6.1.1 the explicit separated compactified jet scheme is what makes that argument valid.

Appendix N. Actual rational affine images and strict compactified boundary bounds

Work geometrically over an algebraically closed field \(\kappa\), with \(\ell\ne\operatorname{char}\kappa\), and a finite coefficient field \(E/\mathbf Q_\ell\). Write \(O\) for its valuation ring, \(\pi\) for a uniformizer, and \(F=O/\pi\). In the arithmetic setting every functor and comparison retains actual Weil descent. Geometric stalk and support tests use geometric cohomology.

The finite affine support theorem and its strict-local proof are Cohomological dimension and the Künneth formula, Theorem 6.2 and §§2–6. The exact integral truncation construction is Affine morphisms, Artin vanishing and perverse cohomology, Lemma 4.2. Its rational coefficient hypotheses are supplied by Lesson 5, Theorems N.4.3.1, O.2.1.1, O.2.2.2, P.2.4.1, P.3.2.1 and P.3.3.1, and Lemma Q.1.1.1, with the finite stratal perfect models of its §M.5. Those proofs construct the actual global models, nonproper image finiteness, reduction and rationalization, exceptional comparisons, smooth orientation and structural duality. Appendices K–M supply the full defect subfunctors and finite-jet quotient charts used below. The free Whittaker patterns, §6.2 provides the geometric application; the coefficient and boundary arguments are proved here.

N.1. An actual upper integral model

Proposition N.1.1.1. Let \(Y/\kappa\) be separated of finite type. Every \(K\in{}^pD^{\le0}(Y,E)\) has an integral constructible derived model \(L\in{}^pD^{\le0}(Y,O)\), and its actual derived reduction \(L_1=L\otimes_O^LF\) is upper perverse over \(F\). Integral self-duality is not an assumption.

Proof. A global integral derived model \(L_0\) exists by Lesson5 Theorem N.4.3.1; it is a constructible complex, not an asserted global locally free lattice. All the required closed/open operations preserve integral constructibility: open and closed restriction and extension by zero are proved in M.6 and N.4.1.1, ordinary image in O.2.1.1, and supported exceptional restriction in O.2.2.2. Their rational comparisons are the actual coefficient maps. Smooth orientation and its coefficient comparisons are P.2.4.1. Thus every operation and rational comparison required in the finite gluing construction exists here.

For clarity perform that construction. On a dense smooth open union \(a:V\hookrightarrow Y\) where the cohomology of \(L_0\) is lisse, cut \(a^*L_0\) in ordinary degree \(-s\) on each component of dimension \(s\). Call its upper truncation \(V_0\). With \(c:C\hookrightarrow Y\) the complementary closed subset, form \[ a_!V_0\longrightarrow L_0\longrightarrow L'_0, \qquad C_0={}^p\tau_C^{\le0}c^!L'_0,\qquad c_*C_0\longrightarrow L'_0\longrightarrow B. \tag{N.1.1} \] The closed truncation is constructed by induction on dimension. The octahedron gives triangles \(L\to L_0\to B\) and \(a_!V_0\to L\to c_*C_0\). Hence \(a^*L=V_0\), \(c^*L=C_0\), whereas \(a^*B\) is above its shifted cut and \(c^!B\) is the positive closed truncation. The stalk support test makes \(L\) upper. The open/closed lower tests make \(B\) lower of perverse degree at least one.

Here is the orthogonality which identifies a genuine truncation. For an upper \(A\) and a lower \(B'\) of degree at least one, apply \(\operatorname{Hom}(-,B')\) to \(a_!a^*A\to A\to c_*c^*A\). The outer morphism groups are respectively the open shifted ordinary group and the closed inductive group, both zero. Thus the middle group is zero. Repeating the construction on the smaller closed subset terminates. Finite constructibility of each operation allows its adapted stratification to be refined during this induction. The upper condition is exactly the ordinary stalk support condition, so the resulting upper part is independent of that refinement; its orthogonal lower part and its truncation are then determined as well. This is the proved integral gluing argument, with all its operation hypotheses now supplied.

Flat rationalization commutes with \(a^*,a_!,c^*,c^!\), including the actual exceptional comparison O.2.2.2. It therefore carries the constructed triangle to an upper/lower rational truncation triangle. Its uniqueness gives \(L[1/\pi]={}^p\tau^{\le0}K=K\).

Finally the derived coefficient triangle is \[ L\xrightarrow{\pi}L\longrightarrow L_1\longrightarrow L[1]. \tag{N.1.2} \] The upper part contains \(L[1]\) and is extension closed, so \(L_1\) is upper when viewed as an integral complex. Its stalk support test is the same over \(F\); hence it is upper over that finite field too. The adjacent-degree Tor term in derived reduction is retained. This proves the assertion. ∎

N.2. The rational affine support bound and affine-open exactness

Theorem N.2.1.1. For an affine finite-type morphism \(f:X\to Y\) of separated finite-type \(\kappa\)-schemes, \[ Rf_*({}^pD^{\le0}(X,E))\subset{}^pD^{\le0}(Y,E), \qquad Rf_!({}^pD^{\ge0}(X,E))\subset{}^pD^{\ge0}(Y,E). \tag{N.2.1} \] For an affine open immersion \(a\), both \(a_!\) and \(Ra_*\) are t-exact, and hence exact on the perverse hearts. These are actual coefficient functors and maps.

Proof. First take finite-field coefficients \(F\). The finite relative Artin theorem gives, for an ordinary constructible sheaf \(H\), \[ \dim\operatorname{Supp} R^qf_*H \le \dim\operatorname{Supp}H-q. \tag{N.2.2} \] This is Theorem6.2 of the stated earlier étale lesson, not a geometric-fibre formula. Its proof calculates a stalk on \(X\times_Y\operatorname{Spec}\mathcal O^{\mathrm{sh}}_{Y,\bar y}\). If \(\operatorname{trdeg}_\kappa\kappa(y)=b\) and \(\dim\operatorname{Supp}H\le d\), its affine finite models over the embedded separably closed field have support dimension at most \(d-b\). The affine dimension theorem and continuity make their cohomology zero for \(q>d-b\). This is exactly the asserted support drop. No nonproper direct-image stalk is replaced by cohomology of the geometric fibre.

For an upper finite complex \(C\), its ordinary layer \(\mathcal H^bC[-b]\) has support dimension at most \(-b\). Equation (N.2.2) bounds the degree-\(n=q+b\) cohomology of the image layer by \(-b-q=-n\). The upper support condition is extension closed: the middle cohomology support in a triangle lies in the union of the two outer supports. The finite ordinary truncation tower therefore makes \(Rf_*C\) upper.

Now take a rational upper \(K\) and the upper model \(L\) of Proposition N.1.1.1. Actual \(M=Rf_*L\) is integral constructible by Lesson5 Theorem O.2.1.1, and its actual first reduction is \(Rf_*L_1\). The preceding finite argument makes that reduction upper. At a geometric point over a point of closure dimension \(b\), the long exact sequence of the coefficient triangle gives \[ H^n(M_{\bar y})/\pi \hookrightarrow H^n((M_1)_{\bar y})=0 \qquad(n>-b). \tag{N.2.3} \] The integral stalk complex here has finite \(O\)-cohomology by the actual stratal perfect models of Lesson5 M.5. Thus Nakayama forces \(H^n(M_{\bar y})=0\) in those degrees. Explicitly if finite generators express \(M'=\pi M'\), their relation matrix \(I-\pi A\) has unit determinant, and its adjugate kills every generator. This checks every support point, not only closed points. It follows that \(M\) is upper. Its rationalization is the actual \(Rf_*K\) by O.2.1.1, proving the first inclusion in (N.2.1).

Rational field duality reverses the perverse inequalities; its actual exchange is \(D_YRf_!=Rf_*D_X\), proved in Lesson5 P.3.3.1. Applying the first inclusion to \(D_XK\) proves the second. All these comparisons include their units and counits, so no objectwise replacement functor was used.

For an open immersion, restriction \(a^*\) is t-exact by the same support/costalk tests. Its left adjoint \(a_!\) preserves the upper part, and its right adjoint \(Ra_*\) preserves the lower part: for example orthogonality computes \(\operatorname{Hom}(A,Ra_*B)=\operatorname{Hom}(a^*A,B)=0\) for upper \(A\) and positive lower \(B\). Affineness supplies the other two inequalities in (N.2.1). Thus both functors are t-exact. The perverse long exact sequence of the image of a short exact sequence has only degree zero, proving exactness on hearts. No absence of boundary stalks follows from exactness alone. ∎

N.3. Exact extension on the actual compactified jet quotients

Theorem N.3.1.1. The representable affine genuine open \(j\) of M.6 has t-exact actual rational \(j_!,Rj_*\) on every bounded compactified chart and compatibly on their overlaps. The perverse and duality normalizations are defined on their whole quotient presentations, including all parameter nilpotents and stabilizers. In particular both extensions of the genuine rank-one character sheaf \(\Psi=e^*\mathcal L_\psi[d_0](d_0/2)\) of L.4 are perverse.

Proof. Write a bounded chart as \(\mathcal Y=[Z/N_k]\), with \(Z=\overline M_{b,y,k}\) from M.5, and set \(r=k|\Phi^+|\). The full genuine pullback \(U=M_k\) is the affine open in \(Z\) of M.6. The smooth atlas \(q:Z\to\mathcal Y\) has relative dimension \(r\), with its actual nerve \(Z\times N_k^p\). Use the normalization \[ q^\dagger=q^*[r](r/2). \tag{N.3.1} \] The half Tate line means the same chosen arithmetic square root as in §1 and Appendix L; geometrically it is an invertible scalar line. Neither it nor a normalization changes the unipotent stabilizer schemes.

Here are the required descent facts at this exact scope. A sheaf object on the quotient is the enhanced Cartesian system on that atlas nerve, with its action/unit/cocycle and their higher coherences. Smooth normalized pullback is t-exact on schemes. For its upper inequality, a smooth relative dimension \(r'\) pullback raises every support dimension by \(r'\), while shift \([r']\) lowers its degree by \(r'\), giving exactly the same support bound. Its lower inequality follows by actual smooth orientation and duality, Lesson5 P.2.4.1 and P.3.3.1. The two projections of the frame relation are smooth of the same relative dimension; hence the descent isomorphism is preserved by perverse truncation. The truncated objects and their maps satisfy the cocycle by uniqueness of the truncation triangle. Orthogonality and the triangles can be checked on the conservative atlas. This constructs the perverse structure on the Cartesian category.

Intrinsic duality is constructed by the same actual smooth comparison used in L.4, now on the finite-type scheme \(Z\) and its nerve rather than only a genuine affine tower. The equality \(q^\dagger D_\mathcal Y=D_Zq^\dagger\) is the smooth orientation formula with its evaluation and trace maps. The face and degeneracy coherences are the transitivity and adjunction coherences of P.2.4.1/P.3.3.1. Applying them to the actual nerve gives descent for duality and its bidual evaluation. Every scheme level has actual field duality by P.3.2.1. Thus the duality comparison, not just a dual object, descends.

Extension by zero commutes with restriction on every atlas square. Actual ordinary image for the open commutes with the smooth squares of the nerve by Lesson5 Lemma Q.1.1.1; its smooth base-change comparison preserves the restriction unit. Consequently the scheme functors \(a_!,Ra_*\), for \(a:U\hookrightarrow Z\), give Cartesian enhanced systems on the target nerve. Their adjunctions descend: maps into and out of them are the levelwise adjunction maps, compatible with all nerve maps. These are therefore the actual quotient \(j_!,Rj_*\), and \[ q^\dagger j_!K=a_!q_U^\dagger K,\qquad q^\dagger Rj_*K=Ra_*q_U^\dagger K. \tag{N.3.2} \] Both are perverse by Theorem N.2.1.1; atlas conservativity proves their t-exactness on the quotient.

For different levels, point choices or degree bounds, the common double-frame atlas is an actual scheme: over either chart the second compatible frame functor is its unipotent jet torsor, represented by an affine scheme by affine descent. Its two projections are smooth. The same smooth comparisons and adjunction maps identify the two constructions there; their unit/cocycle gives their agreement on the stack overlap. The bounded opens cover M.5's whole regular compactification. Thus the statements are compatible on all these opens. All constructions used the original group actions on entire schemes; a nilpotent family crossing degree strata remains in the ambient chart. Finally L.4 proves that the displayed genuine \(\Psi\) is perverse with its actual dimension and character; applying the established t-exact functors proves the last assertion. ∎

Proposition N.1.1.1 and Theorems N.2.1.1–N.3.1.1 keep the derived reduction, strict-local image stalks, actual finite integral cohomology and normalized frame-atlas comparisons. Editable SVG source.

N.4. Strict ordinary and exceptional boundary inequalities

Theorem N.4.1.1. Let \(j:\mathcal U\hookrightarrow\mathcal Y\) be the genuine open of a bounded chart, let \(L\) be perverse on \(\mathcal U\), and set \[ P=j_{!*}L=\operatorname{im}_{\operatorname{Perv}(\mathcal Y)} (j_!L\longrightarrow Rj_*L). \tag{N.4.1} \] For the closed complement \(i:\mathcal C\hookrightarrow\mathcal Y\), \[ i^*P\in{}^pD^{\le-1}(\mathcal C,E), \qquad i^!P\in{}^pD^{\ge1}(\mathcal C,E). \tag{N.4.2} \] In particular on a smooth boundary stratum \(s:\mathcal S\hookrightarrow\mathcal Y\) of an adapted stratification, of stack dimension \(d_\mathcal S\), the actual ordinary cohomology bounds are \[ \mathcal H^n(s^*P)=0\quad(n\ge-d_\mathcal S),\qquad \mathcal H^n(s^!P)=0\quad(n\le-d_\mathcal S). \tag{N.4.3} \] These are strict bounds. The theorem does not assert that either restriction is zero in every degree.

Proof. Exactness in Theorem N.3.1.1 makes (N.4.1) an image between perverse objects. It restricts to \(L\). It has no nonzero quotient supported in \(\mathcal C\): such a quotient would receive a nonzero map from \(j_!L\), whereas \(\operatorname{Hom}(j_!L,i_*A)=\operatorname{Hom}(L,j^*i_*A)=0\). It has no nonzero subobject supported there, since any such map would compose into \(Rj_*L\), and \(\operatorname{Hom}(i_*A,Rj_*L)=0\) by the other open adjunction.

Closed restriction \(i^*\) preserves the upper perverse part, and \(i^!\) preserves the lower part, by the closed/open gluing tests. Hence \(i^*P\) is upper and \(i^!P\) lower. If \({}^pH^0i^*P\ne0\), the canonical projection to this heart object and the adjunction give a nonzero map \(P\to i_*{}^pH^0i^*P\). Its image is a nonzero supported quotient of \(P\), a contradiction. Dually a nonzero \({}^pH^0i^!P\) maps into \(P\) by its inclusion and the adjunction, producing a forbidden supported subobject. Thus their degree-zero perverse cohomologies vanish, proving (N.4.2).

One can check the numerical bounds without guessing a stack shift. On an atlas of relative dimension \(r\), the preimage \(S\) of \(\mathcal S\) has dimension \(d_\mathcal S+r\), and its two restrictions of \(q^\dagger P\) are the smooth-normalized pullbacks of the displayed restrictions. On an adapted smooth scheme stratum the upper strict bound permits only ordinary degrees \(m\le-(d_\mathcal S+r)-1\); the lower bound permits only \(m\ge1-(d_\mathcal S+r)\). The ordinary cohomology degree of the unnormalized stratum object is \(n=m+r\), by (N.3.1). These are exactly \(n\le-d_\mathcal S-1\) and \(n\ge1-d_\mathcal S\). Smooth conservative pullback detects the vanishings. This proves (N.4.3), including negative-dimensional strata.

The construction and adjunction maps are compatible with duality. In particular duality interchanges these ordinary and exceptional inequalities and sends \(j_{!*}L\) to \(j_{!*}D_\mathcal UL\). It follows either by taking the dual of the image in the field perverse heart or by its no-supported-subobject/quotient characterization. The actual evaluation maps from Theorem N.3.1.1 retain their Weil descent. ∎

N.5. The actual dimensions of complete defect strata

Proposition N.5.1.1. For the actual fixed \(T\)-bundle on \(\mathbf P^1\), put \(d_\alpha=\deg\mathcal L_\alpha\) and \[ d_0=-\sum_{\alpha>0}(d_\alpha+1). \] Let the exact fundamental defect degrees be \(m_i\ge0\), and put \(m=\sum_i m_i\). The full fixed-degree defect stratum, with moving divisors, is smooth of dimension \(d_0-m\). Its fibre at a specified entire tuple of divisors is smooth of dimension \(d_0-2m\). Central degrees are unchanged. These are the actual locally closed subfunctors of K.5; arbitrary ambient families are not forced into them.

Proof. Theorem K.5.1.1 proves the full divisor-and-saturated-datum description on every test algebra. The divisor base is \(\prod_i\mathbf P^{m_i}\), with its actual universal Cartier divisors, so is smooth of dimension \(m\). Over it the stratum is exactly the genuine torsor stack for \(\mathcal T'=\mathcal T(\sum_i\alpha_i^\vee D_i)\). The degree of its root line is \[ d'_\alpha=d_\alpha+\sum_i \langle\alpha,\alpha_i^\vee\rangle m_i. \tag{N.5.1} \] For every simple coroot, \(\sum_{\alpha>0}\langle\alpha,\alpha_i^\vee\rangle=2\). Indeed the simple reflection permutes the positive roots other than \(\alpha_i\), so it sends their sum \(2\rho\) to \(2\rho-2\alpha_i\); comparing with its reflection formula gives that equality.

Appendix L.1–L.2 supplies the relative smooth torsor atlas on this entire divisor base. To see the dimension and smoothness directly, cover the base by actual character line frames and choose one uniform sufficiently large jet level. The root frame lines have constant degrees \(d'_\alpha-k<0\). Their two-chart \(H^1\) modules are finite projective of ranks \(k-d'_\alpha-1\), with arbitrary-base comparison. Central height lifting gives successive torsors under their vector bundles. Thus the framed scheme is smooth over the divisor base, of relative dimension \(\sum_{\alpha>0}(k-d'_\alpha-1)\), and the smooth jet frame group has dimension \(k|\Phi^+|\). Subtracting yields the relative stack dimension \[ -\sum_{\alpha>0}(d'_\alpha+1)=d_0-2m. \tag{N.5.2} \] The frame torsor arguments and their overlaps are exactly those of L.2; they require no free action. Adding the smooth divisor-base dimension \(m\) gives \(d_0-m\), and fixing its full point gives (N.5.2). Divisors which meet the current chosen jet point use a different point chart, available over a finite field extension and descended as in M.5. Smoothness and the dimension agree on the common frame atlases.

Every central character pairs to zero with the coroots, so all its divisor contributions vanish. This is an equality in the original character lattice. K.5's higher divisibility equations and exact finite flat zero ideals define these particular subfunctors with their nilpotents. Ambient nonflat zero families or failures of a higher divisibility equation remain outside this stratum and inside M.5's compactification. Smooth degree strata partition geometric points; before applying (N.4.3) on a whole such stratum, one must establish that it is adapted to the complex in question. The full Whittaker equivariance argument supplying that assertion is still a separate step. ∎

Theorem N.4.1.1 proves the strict inequalities on adapted strata. Proposition N.5.1.1 and Exercise N.6.2 compute complete defect dimensions with the divisor parameter and actual stabilizers. The moving degree-one SL2 stratum has dimension zero; its fixed divisor fibre has dimension minus one. Editable SVG source.

N.6. Exercises

Exercise N.6.1 (medium). Show that affine-open exactness does not itself prove cleanness. Take \(a:\mathbf G_m\hookrightarrow\mathbf A^1\), with geometric rational coefficients. Compute the boundary stalk of \(Ra_*E[1]\), compare it with \(a_!E[1]\), and repeat with a nontrivial finite Kummer rank-one local system.

Solution. Write \(R\) for the strict henselian local DVR of \(\mathbf A^1\) at zero and \(K=\operatorname{Frac}R\). It is algebraic over \(\kappa(t)\), so the field-extension bound in the earlier étale lesson, Lemma2.3, gives \(\operatorname{cd}_\ell(K)\le1\). Every unit of \(R\) has an \(\ell^n\)-th root by Hensel's lemma: its residue has a root in \(\kappa\), and the derivative is a unit. Thus valuation identifies \(K^\times/(K^\times)^{\ell^n}\) with \(\mathbf Z/\ell^n\). The Kummer sequence gives \(H^1(K,\mathbf Z/\ell^n)=\mathbf Z/\ell^n(-1)\); degree zero is the constant module, and higher degrees vanish by that field bound. Finite coefficient comparison, followed by the surjective inverse systems in these two degrees and actual rationalization of Lesson5 O.2.1.1, gives \(E\) in degree zero and \(E(-1)\) in degree one. This is the strict-local higher-image stalk, not a fibre formula. After shift \([1]\), the boundary of \(Ra_*E[1]\) is \[ H^{-1}=E,\qquad H^0=E(-1). \tag{N.6.1} \] The boundary stalk of \(a_!E[1]\) is zero by extension by zero. Thus the canonical map cannot be an isomorphism, although both objects are perverse by Theorem N.2.1.1. Their intermediate extension is \(E_{\mathbf A^1}[1]\), whose boundary stalk is \(E\) in degree \(-1\) and costalk is \(E(-1)\) in degree \(1\). These satisfy precisely the strict inequalities (N.4.3) for the dimension-zero stratum.

For a nontrivial Kummer character choose an integer \(h>1\) prime to both \(\operatorname{char}\kappa\) and \(\ell\), the cover \(p:z\mapsto z^h\), and a coefficient field containing the chosen nontrivial character. On the strict puncture the cover is the field extension obtained by the Eisenstein equation \(z^h-t=0\). Its normalization is a strict henselian DVR with uniformizer \(z\) and the same residue field, so the preceding calculation again gives just \(E,E(-1)\). Every deck transformation acts trivially on these groups: it fixes the constant module, and replaces \(z\) by \(\zeta z\), which has the same Kummer valuation class because the unit \(\zeta\) has every required \(\ell^n\)-th root. Exact finite pushforward identifies \(R\Gamma(K,p_*E)\) with the cohomology of that extension. The nontrivial character idempotent \(h^{-1}\sum_g\chi(g)^{-1}g\) annihilates both trivial deck modules. Its summand is precisely the chosen rank-one local system \(L\), so its actual boundary complex is zero. This proves the local-system calculation without assuming a general analytic comparison. The cone of \(a_!L[1]\to Ra_*L[1]\) is the closed pushforward of that zero boundary restriction. Hence it is clean. In the constant case the cone is nonzero. The contrast proves exactly the need for a character boundary argument beyond affineness.

Exercise N.6.2 (advanced). Let \(G=SL_2\), with its full torus line \(\mathcal L=\mathcal O(-1)\). Compute all complete degree-\(m\) defect strata, including their parameter line, stabilizers and dimensions. Compare a moving degree-one defect with a fixed point defect.

Solution. The positive root line is \(\mathcal L^2=\mathcal O(-2)\). With no defect, \(H^1(\mathcal O(-2))=\kappa\) and \(H^0(\mathcal O(-2))=0\), so the genuine stack is \(\mathbf A^1\), of dimension \(d_0=1\). For a degree-\(m\) defect divisor the torus line is \(\mathcal O(-1)(D)\). The full divisor base is \(\mathbf P^m\). Its universal divisor line is \[ \mathcal O(D)=\mathcal O_{\mathbf P^1}(m) \boxtimes\mathcal O_{\mathbf P^m}(1), \tag{N.6.2} \] because the tautological homogeneous polynomial is a section of exactly that line. The new positive root line is therefore \(\mathcal O_{\mathbf P^1}(2m-2)\boxtimes\mathcal O_{\mathbf P^m}(2)\). For \(m\ge1\) its \(H^1\) is zero, and its \(H^0\) vector bundle on the parameter base is \[ V_m=H^0(\mathbf P^1,\mathcal O(2m-2)) \otimes\mathcal O_{\mathbf P^m}(2), \qquad\operatorname{rk}V_m=2m-1. \tag{N.6.3} \] The root group is additive. Thus the full moving defect stratum is \(B_{\mathbf P^m}V_m\), with its entire vector-group stabilizers, of dimension \(m-(2m-1)=1-m\). At a specified divisor, choosing a basis of its parameter line identifies its fibre with \(B\mathbf G_a^{2m-1}\), of dimension \(1-2m\). These formulas hold on nilpotent parameter algebras because the universal divisor sequence and its two-chart cohomology are universally exact. For \(m=1\) the moving stack has dimension zero, whereas the fixed-point fibre is \(B\mathbf G_a\), of dimension minus one. Omitting the divisor parameter or treating that stabilizer as a point would give the wrong boundary dimension. The original torus line and its character are retained throughout.

Appendix O. Meromorphic unipotent actions and boundary characters

Throughout this argument \(X=\mathbf P^1_\kappa\), initially over an algebraically closed field. Keep the original split group \(G\), its actual torus \(T\), every central character map and the entire compactified datum of Appendices J–M. For compactifications retain their hypothesis that the derived group is simply connected. The full compactification transfer to arbitrary \(G\) is a separate requirement; it is not supplied by the slice comparison alone. Put \(N=U\), write \(h_\alpha\) for the positive root height, and fix \(y\) outside the defects and outside the zeros of the nonzero maps \(\omega_i:\mathcal L_{\alpha_i}\to\Omega_X\). A coordinate \(u\) vanishes at \(y\). All moduli assertions below concern arbitrary parameter algebras. For character sheaves assume \(\operatorname{char}\kappa=p>0\), \(\ell\ne p\), and finite \(E/\mathbf Q_\ell\) containing the values of the specified nontrivial additive character \(\psi\). Half-Tate normalizations are those of Appendix L. In an arithmetic form keep the actual Weil descent; geometric semisimplicity will not assert semisimplicity of arbitrary Frobenius operators.

The algebraic completion inputs are proved in Completion, Theorems 3.1–3.2. Affine additive descent and the root-height universal lifting are proved in Appendix L; the full framed actual G-schemes and detecting-line reconstruction are in Appendix M. The coefficient operations, smooth comparison and localization are proved in Lesson 5, Appendices P–Q, and compactified perverse descent is Appendix N. The free Whittaker patterns, §6.2 supplies the geometric context; the all-base gluing, group action and boundary proofs below supply the assertions used here.

O.1. Anisotropic root frames and finite groups

Proposition O.1.1.1. On a bounded compactified chart there are integers \(m,k\) with the following properties. Define \[ H_m(A)=\delta_mN(A[[u]]),\quad K_{m,k}(A)=\delta_m\ker\bigl(N(A[[u]])\to N(A[u]/u^k)\bigr), \quad a_\alpha=k-mh_\alpha, \tag{O.1.1} \] where \(\delta_m\) multiplies the root coordinate for \(\alpha\) by \(u^{-mh_\alpha}\). Then \(K_{m,k}\) is normal in \(H_m\), lies in \(N(A[[u]])\), and \[ H=H_m/K_{m,k}\simeq N(\kappa[u]/u^k) \tag{O.1.2} \] is a smooth split unipotent group of dimension \(kn\), where \(n=|\Phi^+|\). Partial \(K_{m,k}\)-frames give a separated finite-type compactified scheme \(\widehat{\overline M}\). Its genuine open \(\widehat M\) is smooth affine of dimension \[ \dim\widehat M=\sum_{\alpha>0}(a_\alpha-d_\alpha-1). \tag{O.1.3} \] The regular frame group \(B=N(\kappa[[u]])/K_{m,k}\) is a closed smooth subgroup of \(H\), of dimension \(\sum_\alpha a_\alpha\), and the full compactified chart is \([\widehat{\overline M}/B]\).

Proof. The root-coordinate multiplication and inverse polynomials are homogeneous for the root lattice. A nonlinear monomial contributing to root \(\gamma\) has roots summing to \(\gamma\), hence heights summing to \(h_\gamma\). This proves that \(\delta_m\) is an automorphism over \(\kappa((u))\), directly from the defining polynomials, in every characteristic. It uses no fractional cocharacter of \(T\). Transporting the ordinary jet kernel proves normality and (O.1.2). Choose \(m\) large enough that \(d'_\alpha+1\ge-mh_\alpha\) for every root degree occurring in the bounded complete defects. Choose \(k\) still larger, so \[ s=\min_\alpha a_\alpha\ge1,\qquad 2s\ge\max_\alpha a_\alpha,\qquad a_\alpha>d'_\alpha \tag{O.1.4} \] and \(s\) exceeds the framed-bundle bounds of M.3–M.5. We may also require any finite jet depth needed in the next proof. These are finitely many inequalities; increasing \(k\) satisfies them.

Ordered coordinates identify \(K_{m,k}\) with the root coefficients divisible by \(u^{a_\alpha}\). Inside \(H\), \(B\) is given by vanishing of coefficients of degrees below \(mh_\alpha\) after applying \(\delta_m^{-1}\); this is a closed subgroup with precisely \(a_\alpha\) remaining coefficients per root. Let \(J_s=N(u^s\kappa[[u]])\). In \(J_s/K_{m,k}\) every nonlinear coordinate monomial vanishes, since its order is at least \(2s\ge a_\gamma\). Thus this quotient is the actual vector group of root coefficients of degrees \(s,\ldots,a_\alpha-1\), of rank \(\sum_\alpha(a_\alpha-s)\).

By M.5 the full level-\(s\) chart \(\overline M_s\) is a separated finite-type scheme. Above its universal reduction near \(y\), partial \(K_{m,k}\)-frames lifting its specified level-\(s\) frame form a torsor for \(J_s/K_{m,k}\). Such lifts exist fppf locally: a split unipotent torsor on an affine formal neighborhood is trivial by the central root-height induction and the vanishing of quasi-coherent \(H^1\) on affines proved in L.1; finite root coordinates lift through successive truncations. Affine descent represents this vector-group torsor by an affine morphism over \(\overline M_s\). It is separated and finite type. A partial frame includes the full level-\(s\) \(G\)-frame, so M.3 kills all automorphisms, not only their geometric points. Forgetting the partial frame is a full \(B\)-torsor, giving the asserted quotient, with every stabilizer retained.

For the genuine open repeat L.1 with root lines \(\mathcal L_\alpha(-a_\alpha y)\). Multiplication extends across \(y\): a monomial with \(r\) factors has excess order \[ \sum_{j=1}^r a_{\alpha_j}-a_\gamma=(r-1)k\ge0. \tag{O.1.5} \] On the small flat site its torsors are exactly the \(N\)-torsors with partial \(K_{m,k}\)-frame. The inclusion is an inclusion on flat tests; no inclusion on arbitrary tests supported at \(y\) is being claimed. At each central height the two-chart calculation is \(H^0(\mathcal L_\alpha(-a_\alpha y))=0\), with \(H^1\) free of rank \(a_\alpha-d_\alpha-1\). The universal lifting and affine vector-torsor induction in L.1 applies unchanged, including arbitrary base change. It supplies the affine tower, its universal torsor, and zero automorphisms. This proves (O.1.3). ∎

O.2. Algebraic gluing and the actual meromorphic action

Theorem O.2.1.1. There is a natural left action of the finite group \(H\) on the whole scheme \(\widehat{\overline M}\). It preserves the genuine open and the entire complete defect subfunctors. In formal frames its convention is \[ \phi\longmapsto n\phi,\qquad g\longmapsto ng, \tag{O.2.1} \] where \(\phi\) maps the framed bundle to the standard bundle and \(g\) is its transition from the outside trivialization to that frame. The action keeps every central map and every higher divisibility equation.

Proof. We first prove the bounded lattice gluing used here. Let \(A_0\) be Noetherian, \(R=A_0[u]\), \(\widehat R=A_0[[u]]\), and let \(E\) be a finite projective \(R\)-module. Given a formal frame \(\phi\) and a meromorphic matrix \(v\) with determinant one, choose \(b\) bounding the poles of \(v\) and \(v^{-1}\). The desired new lattice is \[ \Lambda=\phi^{-1}v^{-1}\widehat R^r,\qquad u^b\widehat E\subset\Lambda\subset u^{-b}\widehat E. \tag{O.2.2} \] The quotient \(Q=u^{-b}E/u^bE\) is finite over \(A_0\) and has the identical formal quotient. Let \(P=\Lambda/u^b\widehat E\subset Q\), and define \(E'\) as its preimage in \(u^{-b}E\). Noetherianity makes \(E'\) finitely presented. The proved completion exactness and flatness in AG-CA, Completion, Theorems 3.1–3.2, give \[ E'\otimes_R\widehat R=\Lambda,\qquad E'[1/u]=E[1/u]. \tag{O.2.3} \] The map \(R\to\widehat R\times R[1/u]\) is faithfully flat. Flatness is the completion theorem and localization. A prime containing \(u\) is covered by completion: completion of its finite quotient \(R/\mathfrak p\), on which \(u=0\), is that quotient itself. A prime not containing \(u\) is covered by localization. These nonzero residue fibres prove faithfulness. Both pullbacks of \(E'\) are finite locally free; flatness descends by the faithful-flat Tor test. For completeness, a finitely presented flat module over the local ring \(R_{\mathfrak p}\) is free: lift a residue basis to a surjection from a finite free module; its finite kernel has zero residue by flatness, hence is zero by Nakayama. Thus \(E'\) is a vector bundle. The preimage construction also proves uniqueness of the algebraic bundle and of maps with these two specified restrictions.

We need universal base change, not merely Noetherian existence. Near a prime containing \(u\), choose bases of \(E'\) and \(u^{-b}E\). The determinant of their inclusion has, in completion, the form \(u^{br}\) times a unit, by (O.2.2) and \(\det v=1\). The equality of the quotients modulo \(u^{br}\) implies algebraic divisibility by \(u^{br}\). Its remaining factor is a unit near that prime, since its reduction modulo \(u\) is a unit there. Thus, after shrinking, the determinant is \(u^{br}\) times a unit. For any \(A_0\)-module \(M\), multiplication by \(u\) is injective on \(M[u]\) and on its localizations. The adjugate identity therefore makes the tensor of the inclusion injective. Away from \(u=0\) it is already an isomorphism. Source and target are \(A_0\)-flat, so their cokernel \(C\) is \(A_0\)-flat by the Tor exact sequence. It is finite over \(A_0\), since \(u^{2b}C=0\). Consequently \[ 0\longrightarrow E'\longrightarrow u^{-b}E\longrightarrow C\longrightarrow0 \tag{O.2.4} \] remains exact after every \(A_0\to A\). The same is true of its finite-jet preimage description. This constructs the lattice over arbitrary \(A\) and proves the required base-change comparison without assuming \(A[u]\to A[[u]]\) flat for every \(A\).

Apply this to \(v=\rho(n)\), for the faithful representation in M.3. Its determinant is one: the character \(\det\rho|_N\) is trivial on each root \(\mathbf G_a\), since an invertible polynomial over the ground field is constant, and these root groups generate \(N\). Choose the pole bound also for the finite detecting module \(W\), its line, and the finite highest-line generators of J. Polynomial root representatives of elements of \(H\) have uniformly bounded negative orders, independently of their positive jet coefficients. Indeed every matrix entry is a finite sum of root monomials of bounded total height, and each negative order is bounded by \(m\) times that height. The inverse has the same property. Increase \(s\) in (O.1.4) beyond twice these pole bounds. The preimage lattice then uses only finite jets already present in the partial frame.

Patch the detecting line and its tensor maps together with \(E_V\). Maps patch algebraically: inside \(\widehat R[1/u]\) one has \(\widehat R\cap R[1/u]=R\), as comparison of Laurent coefficients shows. The same intersection holds for finite projective modules by writing them as direct summands of finite free modules. An outside map compatible with a regular formal map therefore extends uniquely. Applying this to the finite tensor maps makes the patched \(W\) the bundle associated to the patched \(GL(V)\)-frame torsor; the detecting line is its patched subbundle. On the outside and on completion they satisfy the full schematic \(GL(V)/G\) orbit condition. The faithfully flat cover just proved detects all equations and the nonvanishing conditions, so the algebraic patched data satisfy that condition too. M.3 therefore reconstructs the actual \(G\)-torsor, not a torsor for a reduced or adjoint substitute. Relations hold on that cover and hence hold algebraically. The standard highest vectors are fixed by \(N\); hence their source maps are unchanged in the new formal frame, and are primitive at \(y\). On the outside nothing changed. Their full zero ideals, quotient flatness where imposed, and higher divisibility equations consequently remain the same. Central character maps are unchanged because \(N\) acts trivially in every central character. These are schematic identities.

All these data are finite: a finite-type framed bundle, finitely many polynomial jet coefficients, and finite detecting maps. They descend to a finitely generated, hence Noetherian, parameter algebra. A compatible full formal \(N\)-frame exists over that model by the affine unipotent-torsor argument of Proposition O.1.1.1; a prescribed finite frame can be corrected by lifting its root coefficients. There is no assertion that arbitrary infinite coefficients descend. Only the finite lattice quotient is used, and (O.2.4) compares every subsequent base change.

Finally, replacing the formal lift \(\phi\) by \(t\phi\), \(t\in K_{m,k}\), changes the new frame by \(ntn^{-1}\in K_{m,k}\subset N(A[[u]])\). This gives an isomorphic new bundle and the identical partial frame. Replacing \(n\) by the same coset has the same effect. Thus the construction is independent of its choices. The products of formal frames give \(n_2(n_1\phi)=(n_2n_1)\phi\); uniqueness of the algebraic patch gives the action law and its identity, over every base. Naturality and representability make it an actual scheme action. ∎

O.3. Whole homogeneous fibres and their full isotropy

Theorem O.3.1.1. Fix a complete geometric defect divisor \(D=(D_i)\) away from \(y\), and put \(\mathcal T'=\mathcal T(\sum_i\alpha_i^\vee D_i)\). Increase \(m\) as necessary for the bounded degrees \(d'_\alpha\). The entire partially framed fibre over \(D\), including arbitrary parameter rings over that fixed divisor, is \[ \widehat M_D=H/O_D,\qquad O_D=H_m\cap \Gamma(X-y,N^{\mathcal T'}). \tag{O.3.1} \] The map \(H\to\widehat M_D\) is an \(O_D\)-torsor. Its full stabilizer \(O_D\) is smooth connected split unipotent of dimension \[ o_D=\sum_{\alpha>0}(d'_\alpha+mh_\alpha+1). \tag{O.3.2} \] It embeds as a closed subgroup of \(H\). No infinitesimal stabilizer is discarded.

Proof. The full saturation and division of K.5 identifies this fibre with genuine \(N^{\mathcal T'}\)-torsors carrying the transported partial frame. Both standard affine charts have trivial such torsors, by central root-height induction and affine quasi-coherent \(H^1=0\). A specified finite frame at \(y\) can be matched by lifting finite root coefficients on its affine chart. Their transition is therefore a Laurent element of \(N\), with outside gauges on the right and regular frame-preserving gauges on the left.

Choose degree-compatible line bases for \(\mathcal T'\). In these bases outside root sections are \(u^{d'_\alpha}A[u^{-1}]\). A different inherited formal \(T'\)-frame multiplies each root coordinate by a compatible formal unit. Such changes preserve every pole and \(a_\alpha\) threshold, and conjugate all descriptions below; the residue character uses the correspondingly transformed differential. Degree bases are therefore coordinates, not a replacement of the given \(T'\)-frame.

Normalize the transition by increasing height. At a central height, its Laurent coefficients outside the interval \[ d'_\alpha+1,\ldots,a_\alpha-1 \tag{O.3.3} \] are removed by the outside and the \(K_{m,k}\) gauges. These two coefficient submodules have zero intersection because \(a_\alpha>d'_\alpha\). Root commutator corrections have larger height and are handled at the next step. The algorithm uses coefficient subtraction and the root polynomials, with no division; it works over every \(A\), including rings with nilpotents. Every normalized root coordinate has order at least \(-mh_\alpha\), so the representative lies in \(H_m\). This proves transitivity of the finite \(H\)-action on the full functor.

At the trivial torsor a stabilizing transition is an outside section, up to \(K_{m,k}\). Its root coefficients are exactly \[ u^{-mh_\alpha},u^{-mh_\alpha+1},\ldots,u^{d'_\alpha}. \tag{O.3.4} \] The coefficients give an affine space of dimension (O.3.2). Multiplication and inversion stay in these intervals because both degree and height are additive in each root monomial. Successive height kernels are vector groups. Its image in the finite quotient is closed: in degree bases it is given by the missing coefficient equations, and in any other formal frame by their invertible finite linear transformations. Its intersection with \(K_{m,k}\) is the trivial group functor, because every root coefficient would have order both at most \(d'_\alpha\) and at least \(a_\alpha\).

Two representatives \(g_1,g_2\in H_m\) describe the same partial framed torsor precisely when \(g_2=kg_1o\) with \(k\in K_{m,k}\) and an outside gauge \(o\). Normality puts the resulting outside factor in \(H_m\), hence in \(O_D\). Conversely every such factor supplies the full isomorphism. Thus the functor is exactly the right quotient in (O.3.1), with its full torsor of representatives. This also proves the torsor assertion and smoothness. The dimension is \(kn-o_D=\sum_\alpha(a_\alpha-d'_\alpha-1)\); after quotient by \(B\) it is \(-\sum_\alpha(d'_\alpha+1)\), exactly N.5's fixed-divisor dimension. The argument proves the whole fixed-divisor functor; it does not force an ambient nonflat zero family into that functor. ∎

Proposition O.1.1.1 and Theorems O.2.1.1–O.3.1.1 construct the schemes and action over all parameter algebras. Exercise O.9.2 checks the displayed SL3 coefficient ranges, group law and dimensions, including characteristics two and three. Editable SVG source.

O.4. The residue character and the conductor obstruction

Theorem O.4.1.1. Define \[ \chi(n)=\sum_i\operatorname{Res}_y(\omega_{i,y}n_{\alpha_i}), \qquad n\in H_m. \tag{O.4.1} \] It factors through the finite group \(H\). Let \(\eta_x=\sum_j m_{j,x}\alpha_j^\vee\) be the full defect at \(x\), and \(c_{i,x}=\operatorname{ord}_x\omega_i\). For sufficiently large \(m\), uniformly on the bounded chart, \[ \chi|_{O_D}=0 \quad\Longleftrightarrow\quad c_{i,x}-\langle\alpha_i,\eta_x\rangle\ge0 \quad\hbox{for all }i,x. \tag{O.4.2} \] If an inequality fails, there is a root \(\mathbf G_a\subset O_D\) on which \(\chi\) is a nonzero linear map.

Proof. Height-one coordinates add under the root multiplication polynomials. Thus they define a group map to the simple-root additive groups, including in small characteristic. We do not identify this quotient with the full abelianization. The residue in (O.4.1) uses only the finite negative coefficients down to \(-m\); it vanishes on \(K_{m,k}\), whose simple-root threshold is positive. Hence it defines an algebraic additive character of \(H\).

The original \(\omega_i\) extends to a meromorphic differential-valued map \(\omega'_i:\mathcal L'_{\alpha_i}\dashrightarrow\Omega_X\). The source enlargement by the divisor gives its order \(c_{i,x}-\langle\alpha_i,\eta_x\rangle\). At \(y\) it agrees with the original differential in the transported source frame. If every order is nonnegative, \(\omega'_i\) is regular. For an outside root section \(\gamma_i\), \(\omega'_i\gamma_i\) is then a rational differential with its only possible pole at \(y\), so its residue there is zero. This residue assertion is elementary on \(\mathbf P^1\): partial fractions give the residues at finite points; substituting \(z=u^{-1}\), with \(dz=-u^{-2}du\), makes the residue at infinity their negative sum. The coefficient identity applies over arbitrary parameter rings as well. It proves the forward vanishing on the entire stabilizer, not just on points.

If \(\omega'_i\) has a pole of order \(r>0\) at \(x\ne y\), choose the outside root section whose local coefficient is \(-a^{-1}t^{r-1}\) modulo \(t^r\), where \(a t^{-r}dt\) is the leading differential, and whose jets vanish through the pole orders at all other poles. Polynomial Chinese remainders on \(X-y=\mathbf A^1\) provide this section of degree less than the sum \(R_i\) of these pole orders. Its residue at \(x\) is \(-1\), its other finite residues are zero, and therefore its residue at \(y\) is \(1\). In a degree basis its pole bound at \(y\) is at most \(R_i-1-d'_{\alpha_i}\). Choose \(m\ge R_i-1-d'_{\alpha_i}\). The section belongs to \(O_D\), and its scalar multiples give the required root \(\mathbf G_a\), with character \(t\mapsto t\).

The degrees \(d'_\alpha\) and the pole-order sums are uniformly bounded by the fixed fundamental bounds \(b_i\): a pole order is at most the positive part of \(\langle\alpha_i,\eta_x\rangle\), since \(c_{i,x}\ge0\). Their sums are bounded by the sum of the positive Cartan entries times those bounds. Thus one \(m\) works for every geometric divisor in the chart. For a fixed divisor, the zero-character identity is an identity of polynomial maps and persists on all parameter algebras. Conditions on ambient moving divisors retain their full equations; no test on geometric points replaces their moduli functor. ∎

O.5. Character-equivariant perverse sheaves on the fibres

Theorem O.5.1.1. Give \(\widehat M_D\) the left \(H\)-action above. If the character in (O.4.2) is nontrivial on \(O_D\), every \((H,\chi^*\mathcal L_\psi)\)-equivariant constructible complex on this fibre is zero. If it is trivial, \(\chi^*\mathcal L_\psi\) descends to a rank-one local system \(\mathscr L_D\) on \(H/O_D\), and its character-equivariant geometric perverse sheaves are exactly \[ \mathscr L_D[\dim\widehat M_D](\dim\widehat M_D/2)\otimes_E V, \tag{O.5.1} \] with \(V\) a finite-dimensional \(E\)-vector space. On the full quotient fibre the normalized generator is \[ \Psi_D=e_{\mathcal T',\omega'}^*\mathcal L_\psi[d_D](d_D/2), \qquad d_D=-\sum_{\alpha>0}(d'_\alpha+1). \tag{O.5.2} \]

Proof. Pull equivariance back along a stabilizer root line at the trivial framed torsor. Its action on that point is constant, so the pullback stalk complex \(V^\bullet\) is isomorphic to \(\mathcal L_\psi(c t)\otimes V^\bullet\), with \(c\ne0\). The Artin–Schreier cover \(z^p-z=ct\) is connected: over \(\kappa(t)\) this equation has no solution, since a pole of \(f^p-f\) has order divisible by \(p\), whereas \(ct\) has a simple pole at infinity. The nontrivial character sheaf therefore has a monodromy value different from one. On each nonzero finite stalk cohomology group the proposed isomorphism would identify the trivial monodromy with that scalar. Their difference is invertible in \(E\); hence every group is zero. Transitivity of \(H\) and conservative geometric stalks make the whole complex zero.

If \(\chi|_{O_D}=0\), its additive multiplicativity supplies descent for \(\chi^*\mathcal L_\psi\) along the actual \(O_D\)-torsor \(H\to H/O_D\). Tensoring with its inverse turns twisted equivariance into ordinary equivariance. Pull an ordinarily equivariant complex back to \(H\) along the orbit map; equivariance identifies it with the constant complex of its stalk at the base point. Smooth descent consequently makes all its cohomology sheaves locally constant. For a perverse complex on the smooth homogeneous scheme this means a local system shifted by its dimension: this follows from the ordinary support and dual support bounds on a smooth stratum, as in N.4.

The descent isomorphism for a constant local system is a locally constant invertible matrix on \(O_D\times H\). Both factors are geometrically connected, so that matrix is constant. The identity section and the descent unit force it to be the identity. Thus its only datum is \(V\), and every linear map of such data descends. This proves (O.5.1), including geometric semisimplicity of this local perverse category. It asserts no description of an arbitrary equivariant derived category.

When (O.4.2) holds, \(\omega'\) is regular and the evaluation cocycle is defined. Its change under \(n\) is exactly \(\chi(n)\): take the simple-root Čech coordinate, apply \(\omega'_i\), and use the coefficient-of-\(u^{-1}du\) trace of G.2 and L.1. The descended rank-one system is therefore the stated evaluation system, with its unit at the trivial torsor. Descent through the whole regular frame group \(B\), with normalized pullback \([\dim B](\dim B/2)\), changes the dimension to \(d_D\), giving (O.5.2).

In a Weil form all the maps just used are defined with their actual descent. \(V\) carries the resulting Frobenius operator, and Frobenius may permute conjugate divisor fibres. An arbitrary such operator may have Jordan blocks. The assertion of semisimplicity is geometric; no arithmetic splitting has been inferred. Higher root groups and all schematic stabilizers have participated in the torsor descent. ∎

O.6. Finitely many relevant geometric divisors

Proposition O.6.1.1. Call \(D\) relevant when (O.4.2) holds. Its geometric support is contained in the finite set \(C\) of zeros of the \(\omega_i\), and only finitely many relevant geometric divisors occur. In particular, if every \(\omega_i\) is an isomorphism, the only relevant divisor is zero.

Proof. Write \(2\rho=\sum_{\alpha>0}\alpha=\sum_i r_i\alpha_i\), with positive integral \(r_i\). For a defect \(\eta_x=\sum_jm_{j,x}\alpha_j^\vee\), the simple-reflection identity proved in N.5 gives \[ 2\sum_jm_{j,x} =\langle2\rho,\eta_x\rangle =\sum_i r_i\langle\alpha_i,\eta_x\rangle. \tag{O.6.1} \] Outside \(C\), all \(c_{i,x}=0\). Relevance would make every pairing on the right nonpositive, while a nonzero positive-coroot defect makes the left positive. Thus there is no defect there. At each \(x\in C\), relevance bounds \[ 2\sum_jm_{j,x}\le\sum_i r_i c_{i,x}. \tag{O.6.2} \] There are finitely many tuples of nonnegative integers satisfying this bound, and finitely many \(x\). These are inequalities in the original root and coroot lattices; central characters pair to zero and have not been removed. If \(C\) is empty they force \(D=0\). Finiteness concerns geometric support of sheaves. It neither removes infinitesimal divisor parameters from the compactification nor replaces K.5's full subfunctors by their point sets. ∎

O.7. Both ordinary and exceptional boundary restrictions

Theorem O.7.1.1. Let \(j\) be the genuine open and \(\Psi=e^*\mathcal L_\psi[d_0](d_0/2)\). Each of \(j_!\Psi,Rj_*\Psi,j_{!*}\Psi\) has zero ordinary and exceptional restriction to every irrelevant complete fixed-divisor fibre. On a relevant fibre every perverse cohomology object of either restriction is a finite sum of \(\Psi_D\). All ordinary cohomology sheaves of the corresponding normalized frame restrictions are locally constant on that whole fibre. In particular, for \(P=j_{!*}\Psi\), \[ H^q(s_D^*P)=0\quad(q\ge-d_D),\qquad H^q(s_D^!P)=0\quad(q\le-d_D). \tag{O.7.1} \] These are strict bounds, not a claim of general-conductor cleanness.

Proof. On the genuine frame scheme the evaluation satisfies \(e(nz)=e(z)+\chi(n)\), by the same simple-root coefficient trace used in Theorem O.5.1.1. It gives twisted \(H\)-equivariance with its actual unit and multiplication cocycle. The compactified action of Theorem O.2.1.1 preserves the open and each full defect functor. Its action map is smooth: the isomorphism \((n,z)\mapsto(n,nz)\) of \(H\times\widehat{\overline M}\) identifies it with projection. Smooth base change for the actual images and exceptional operations in Lesson5 Q.1, and tensoring by the invertible rank-one character system, therefore transport this equivariance to both extensions. N.3 makes them perverse, so their canonical map and its perverse image carry the same equivariance.

Restrict along the invariant whole fixed-divisor fibre. Smooth base change in the action square gives twisted equivariance of both ordinary and exceptional restrictions. Apply Theorem O.5.1.1. For an irrelevant divisor the stabilizer root line kills the entire complex, hence both restrictions vanish. For a relevant divisor, the orbit pullback makes all cohomology local systems; perverse cohomology has exactly the form (O.5.1). Full \(B\)-descent gives the finite sums of (O.5.2), with their actual Frobenius data.

The intermediate extension has no subobject or quotient supported in the closed boundary. N.4 therefore puts its ordinary boundary restriction in perverse degrees at most \(-1\) and its exceptional boundary restriction in degrees at least \(1\). These inequalities persist on locally closed fixed-divisor strata, by ordinary upper and exceptional lower perverse restriction. Their cohomology is locally constant by the preceding homogeneous calculation. On a smooth stack of dimension \(d_D\), translating the perverse shift gives precisely (O.7.1). Equivalently, on a smooth frame atlas of relative dimension \(r\), use the scheme dimension \(d_D+r\) and normalized pullback \([r]\); the two \(r\)'s cancel. This checks the sign even when \(d_D\) is negative.

The bounded charts cover geometric points, using a different \(y\) when necessary. The actual smooth comparisons on common frame atlases identify the restriction statements, so they hold intrinsically. If a moving degree stratum has only irrelevant geometric divisors, conservative geometric stalks on its entire atlas show that both restrictions there are zero. More generally any nonzero boundary support lies over the finite set of relevant divisors of Proposition O.6.1.1. This is a statement about constructible sheaf support on the full moduli space; all ambient nilpotents and nonflat families remain in that space. On a relevant fibre the finite multiplicity spaces and their degrees have not been computed. Their possible nonzero values are exactly why the strict bounds alone do not prove cleanness for general conductors. ∎

O.8. Cleanness at zero conductor

Theorem O.8.1.1. If every map \(\omega_i:\mathcal L_{\alpha_i}\to\Omega_X\) is an isomorphism, the canonical maps are isomorphisms \[ j_!\Psi\ \xrightarrow{\ \sim\ }\ j_{!*}\Psi\ \xrightarrow{\ \sim\ }\ Rj_*\Psi. \tag{O.8.1} \] Both ordinary and exceptional restrictions to the whole boundary are zero. The isomorphisms retain the actual Weil structures and normalized duality.

Proof. Every nonzero complete defect is irrelevant by Proposition O.6.1.1. Theorem O.7.1.1 kills the ordinary restriction of \(Rj_*\Psi\) at all its boundary geometric points, on conservative full frame atlases. Hence its entire closed boundary restriction is zero. The open-closed localization triangle for \(Q=Rj_*\Psi\) is \[ j_!j^*Q\longrightarrow Q\longrightarrow i_*i^*Q\longrightarrow. \tag{O.8.2} \] The third term is zero and \(j^*Q=\Psi\); its first arrow is the canonical \(j_!\Psi\to Rj_*\Psi\). Thus it is an isomorphism, and its perverse image is the same object. The exceptional vanishing was already proved pointwise in Theorem O.7.1.1 and follows also by adjunction. All triangles, units, equivariance and comparisons are the actual ones, so the isomorphism is Weil-compatible. Applying the intrinsic duality of L.4 interchanges the extensions and replaces \(\psi\) by \(\psi^{-1}\), giving the same normalized dual isomorphism.

This hypothesis is realizable without changing the group: \(2\rho^\vee=\sum_{\alpha>0}\alpha^\vee\) is an integral cocharacter of the original \(T\); induce its \(T\)-bundle from \(\mathcal O(-1)\). Its simple-root lines are \(\mathcal O(-2)\), since \(\langle\alpha_i,2\rho^\vee\rangle=2\), and can be identified with \(\Omega_{\mathbf P^1}\). The last pairing follows by the same simple-reflection proof as N.5, on the coroot system. No use of the possibly nonintegral \(\rho^\vee\) or adjoint replacement is required. Arbitrary allowed central factors and their specified character maps are retained. If \(G\) is a torus, there are no defects, the genuine and compactified fixed-\(T\) datum coincide, and (O.8.1) is the identity. ∎

Theorems O.4.1.1–O.5.1.1 and O.7.1.1 classify both restrictions; Theorem O.8.1.1 proves the canonical clean extension at zero conductor. Exercise O.9.1 distinguishes the actual SL2 divisor patterns. Relevant-fibre multiplicities remain to be computed. Editable SVG source.

O.9. Exercises

Exercise O.9.1 (medium). For \(SL_2\) take the actual torus line \(\mathcal L=\mathcal O(-r)\), \(r\ge1\), and a nonzero map \(\omega:\mathcal L^2\to\Omega_X\). Classify the relevant geometric divisors. Compare \(r=1\) with \(r=2\) when \(\omega\) has either a double zero or two simple zeros.

Solution. The map is a nonzero section of \(\mathcal O(2r-2)\); let its zero multiplicities be \(c_x\). A defect \(\eta_x=m_x\alpha^\vee\) has pairing \(\langle\alpha,\eta_x\rangle=2m_x\). Thus precisely \[ D=\sum_xm_xx,\qquad 0\le m_x\le\lfloor c_x/2\rfloor \tag{O.9.1} \] are relevant. This proves finiteness directly with the actual \(SL_2\) coroot, including its factor two. At \(r=1\) there are no zeros, so the basic sheaf is clean by Theorem O.8.1.1. At \(r=2\), a double zero at \(x\) allows \(D=0\) and \(D=x\); two simple zeros allow only \(D=0\). In the latter case Theorem O.7.1.1 and the localization proof give cleanness even though the original map has zeros.

In the double-zero case the nonzero relevant fibre has new root line \(\mathcal O(-2)\), hence fixed-divisor stack dimension \(1\), while the original genuine dimension is \(3\). Its character generator is nonzero. The intermediate-extension restrictions, if nonzero, can occur only in ordinary degrees at most \(-2\) and exceptional degrees at least \(2\), by (O.7.1). This calculation does not assert that those multiplicities are nonzero or that general cleanness fails; determining them requires the subsequent Hecke argument.

Exercise O.9.2 (advanced). Work with \(SL_3\), including characteristics two and three, and the actual diagonal torus bundle \(\mathcal O(-2)\oplus\mathcal O\oplus\mathcal O(2)\). Take \(m=4,k=15\). Compute \(H,B,O_0,\widehat M\), the partial-frame vector quotient, and the character obstruction for a defect \(\alpha_1^\vee x\).

Solution. The root heights are \(1,1,2\), and the root degrees are \(-2,-2,-4\). The thresholds are \(11,11,7\); \(s=7\), so \(2s=14\ge11\). In upper-unitriangular coordinates \[ (a,b,c)(a',b',c')=(a+a',b+b',c+c'+ab'). \tag{O.9.2} \] No factor two or three is inverted. The pole ranges of \(H_m\) begin at \(-4,-4,-8\). Its finite quotient has dimension \(45\). The regular frame group has dimension \(11+11+7=29\). The full stabilizer \(O_0\) has \(a,b\) exponents \(-4,-3,-2\), and \(c\) exponents \(-8,-7,-6,-5,-4\), hence dimension \(11\). The product \(ab'\) stays in the latter interval, which verifies the group condition explicitly. The homogeneous frame scheme has dimension \(45-11=34\), agreeing with \(12+12+10\) from (O.1.3). Its quotient stack has dimension \(34-29=5\).

The vector group \(J_7/K_{4,15}\) has four coefficients per simple root and none for the long root, hence rank eight. The ordinary level-seven genuine scheme has dimension \(8+8+10=26\); its partial-frame vector torsor has dimension \(34\), as required. Identify the two simple-root lines with \(\Omega_X\). The character is the sum of their \(u^{-1}du\) coefficients. Those coefficients are absent from \(O_0\), so its character is zero.

At a defect \(\alpha_1^\vee x\) the new degrees are \(0,-3,-3\), using the actual \(A_2\) Cartan pairings \(2,-1,1\). The first stabilizer interval now contains \(u^{-1}\); equivalently the first new differential has a pole of order two at \(x\). Theorem O.4.1.1 supplies its scalar root line with nontrivial residue character. Both boundary restrictions are zero. Normality can also be read from (O.9.2): a simple coefficient of order \(11\) multiplied by a pole coefficient of order \(-4\) has order \(7\), exactly the long-root threshold. The argument preserves the original \(SL_3\) torus and its full center \(\mu_3\), including the nonreduced center in characteristic three.

Appendix P. Hecke generation, Whittaker extensions and full cohomology

This continuation keeps geometric finite \(E/\mathbf Q_\ell\) coefficients, the actual quotient enhancements of Appendices L–O, the original group and torus, and actual Weil maps. Until Theorem P.10.1.1, assume that the derived group is simply connected, as in J–M. The last proof transfers the local cohomology calculation to every original split connected reductive group; it does not assert a global compactification equivalence for that group. Geometric Ext vanishing is not an assertion of semisimplicity for arbitrary Frobenius operators.

Fix distinct marked points \(I=\{x_1,\ldots,x_s\}\) on \(X=\mathbf P^1\), and the zero-conductor background \(\mathcal T_0\) of O.8. For original dominant cocharacters \(\boldsymbol\mu=(\mu_i)\), put \(\mathcal T_{\boldsymbol\mu}=\mathcal T_0(-\sum_i\mu_i x_i)\). The background differentials induce the actual regular differentials on these root lines. Work in the meromorphic decoration functor with poles at \(I\), always using its finite regular compactification charts, finite bundle bounds and full frame nerves. Write \(\Psi_{\boldsymbol\mu}\) for the genuine evaluation generator, \(\Delta_{\boldsymbol\mu}\) and \(\nabla_{\boldsymbol\mu}\) for its actual extensions, and \(\Pi_{\boldsymbol\mu}\) for their perverse image. All are extended by their closed inclusions in a common sufficiently large regular chart when necessary. A chart for \(\boldsymbol\tau\) contains the one for \(\boldsymbol\mu\) when every \(\tau_i-\mu_i\) is a nonnegative integral coroot combination: multiply every weight map by its prescribed divisor section. The central inverse maps have divisor zero. The full finite-generator divisibility equations of K.5, or the section-coordinate equations of I.1–I.3, make this inclusion closed on every parameter ring. Within a fixed central component a sufficiently large integral multiple of \(2\rho^\vee\) provides a common dominant bound for finitely many such charts. Thus every construction below takes place on actual finite-type quotient charts; no enhancement of the whole meromorphic ind-stack is being assumed.

The genuine rigid and tensor inputs are proved in Convolution and rigidity, Lemma B.1.1.1 and Theorem B.2.1.1 and Identifying the dual group, Lemmas B.3.1.1–B.3.2.1 and Theorem B.4.1.1. Appendices G–O provide the actual schemes, local coefficient bounds, perverse operations and character restrictions used in these proofs. The free Whittaker patterns, §§5.4–7 supplies the mathematical context. The closed correspondence, filtration, original-lattice transfer and full proofs of the assertions used here are given below.

P.1. Intermediate extension detects all degree-one extensions

Lemma P.1.1.1. Let \(j:\mathcal U\hookrightarrow\mathcal Y\) be an open immersion of the finite-type quotient charts in Appendix N, and let \(S_1,S_2\) be perverse sheaves on \(\mathcal U\). The restriction map on Yoneda extensions is injective: \[ \operatorname{Ext}^1_{\operatorname{Perv}(\mathcal Y)} (j_{!*}S_2,j_{!*}S_1) \longrightarrow \operatorname{Ext}^1_{\operatorname{Perv}(\mathcal U)}(S_2,S_1). \tag{P.1.1} \] In particular every geometric Whittaker intermediate extension of a genuine rank-one evaluation generator has zero self-\(\operatorname{Ext}^1\).

Proof. A perverse sheaf \(P\) is the intermediate extension of its open restriction precisely when it has no subobject and no quotient supported in the complement. Here is the full argument. The canonical adjunction maps \[ {}^pH^0(j_!j^*P)\longrightarrow P \longrightarrow{}^pH^0(Rj_*j^*P) \tag{P.1.2} \] are identities after open restriction. The cokernel of the first and kernel of the second are therefore supported in the complement. If \(P\) has neither kind of boundary object, the first map is surjective and the second injective. Its image inside the last term is exactly the image of their composite, which defines \(j_{!*}j^*P\). Conversely that image has no boundary subobject or quotient, by open adjunction, as proved in N.4.

Restriction is fully faithful on these intermediate extensions. A morphism on \(\mathcal U\) induces a morphism between the two canonical image constructions, giving existence. If a morphism between intermediate extensions restricts to zero, its image is a boundary subobject of the target, and hence zero; this gives uniqueness. These statements use the actual adjunction maps on full frame nerves and therefore descend to the quotient charts.

Consider an extension \(0\to A\to P\to B\to0\), with \(A=j_{!*}S_1\) and \(B=j_{!*}S_2\). It too has neither boundary subobjects nor boundary quotients. Indeed a boundary subobject of \(P\) maps to a boundary subobject of \(B\), which is zero, and then lies in \(A\), where it is zero. A map from \(P\) to a boundary quotient is zero on \(A\); it factors through \(B\), which has no such quotient, and is zero. Consequently \(P=j_{!*}j^*P\). If its restricted extension is split, \(j^*P=S_1\oplus S_2\) with the given inclusion and projection. The image construction commutes with finite direct sums, and its full faithfulness identifies the inclusion and projection upstairs with those of \(j_{!*}S_1\oplus j_{!*}S_2\). Thus the original extension splits. This proves injectivity.

For the last assertion, the genuine evaluation generator \(\Psi=e^*\mathcal L_\psi[d](d/2)\) has actual derived self-Hom complex \(E\), by L.3 and the rank-one inverse tensor evaluation. Thus its derived degree-one self-Hom is zero. Yoneda degree-one extensions in the perverse heart equal these derived degree-one maps: an extension gives its triangle; conversely the perverse cohomology sequence of a triangle \(A\to P\to B\to A[1]\), with \(A,B\) in the heart, places \(P\) in the heart and supplies the corresponding short exact sequence. These constructions are inverse and preserve the zero class. Applying (P.1.1) gives the asserted vanishing on every bounded closure, and extension by its closed inclusion gives the same assertion in a larger ambient chart. No degree-two or higher restriction-injectivity assertion has been made. In a Weil category Frobenius can still create degree-one extensions of its multiplicity modules, so the conclusion is explicitly geometric. ∎

P.2. Sharp closed Hecke bounds and actual duality

Theorem P.2.1.1. Let \(A_\lambda=I_\lambda(d_\lambda/2)\) be a normalized simple Satake kernel and \(\lambda^*=-w_0\lambda\). For dominant \(\eta\), its Hecke image of \(\Pi_\eta\) is defined on the regular source chart for \(\tau=\eta+\lambda^*\) by a proper correspondence. Its normalized duality comparison is \[ D\mathsf H_{A_\lambda}(\Pi_\eta^\psi) \simeq\mathsf H_{A_\lambda}(\Pi_\eta^{\psi^{-1}}). \tag{P.2.1} \] The same statement holds at the distinct marked points, with all bounds componentwise.

Proof. First sharpen the absolute pole bound of H.1. In a geometric highest-weight module \(W_\chi\), every weight is \(\chi\) minus a nonnegative root combination. Since \(\lambda\) is dominant, every weight exponent is at most \(p_\chi=\langle\chi,\lambda\rangle\). Consequently every lattice in the actual Schubert closure satisfies \[ t^{p_\chi}W_\chi[[t]]\ \subset\ W_\chi(\mathcal E'). \tag{P.2.2} \] On the open orbit this follows on every parameter ring from its integral left and right frames and the diagonal exponents. In a finite lattice Grassmannian the assertion is a closed incidence condition, given by vanishing of the map from that fixed submodule to the universal quotient. It therefore holds on the scheme-theoretic Schubert closure. This argument allows a negative central exponent and does not replace the closure by its geometric points.

Apply this to every highest-map generator and to both generators for every invertible central character. Transport of a regular decoration for \(\mathcal T_\eta\) through the inverse modification, whose type is \(\lambda^*\) by H.1, becomes regular for \(\mathcal T_\tau\). Polynomial relations are preserved by the actual punctured isomorphism; the central inverse equations are preserved too. Conversely, start with a bounded regular source chart \(S\) for \(\mathcal T_\tau\). The whole modification scheme \(W\to S\) is projective by H.1–H.2. Its transported target decoration has the finite meromorphic bounds of H.2. In its finite section coordinates, being regular for \(\mathcal T_\eta\) is the vanishing of all polar coefficients of all generators. Thus the locus \(W_\eta\subset W\) is closed, including the central inverse equations, and its projection \(l\) to \(S\) is proper.

Viewed from the target regular compactification, \(W_\eta\) is the open part of the full inverse Schubert fibration where the transported source lies in \(S\). The sharp bound just proved supplies source regularity everywhere; the remaining source degree bounds and primitivity at the chosen frame point are open conditions. The finite image over \(S\) is covered by finitely many target bundle and section charts, so all the comparisons can be made on those charts. Choose a common frame point away from the marked points. On its actual frame schemes, the target Schubert fibration is étale locally a product with \(Z_{\lambda^*}\). H.3's chain-inversion comparison identifies the left vertical kernel with the normalized right vertical kernel \(A_{\lambda^*}\). That kernel is self-dual, by the normalized simple IC duality in Lesson 7.

On this product the coefficient is \(\Pi_\eta\boxtimes A_{\lambda^*}\). Actual external duality sends it to \(D\Pi_\eta\boxtimes A_{\lambda^*}\). Duality for an open restriction preserves this comparison on \(W_\eta\); duality for its closed inclusion and its proper projection preserves it in the image on \(S\). L.4 and N.4 identify \(D\Pi_\eta^\psi\) with \(\Pi_\eta^{\psi^{-1}}\). This proves (P.2.1) without assuming cleanness of \(\Pi_\eta\). The common smooth frame normalization \([r](r/2)\) occurs on both sides and cancels. Ordinary descent along the finite disc-frame group adds no dimension shift, by H.3. Its actual overlap cocycles identify these product comparisons, including the evaluation and trace maps. At several distinct points use the product of their disc-frame fibrations and impose the finitely many polar equations at each point. Properness, inversion and the same external duality then prove the stated componentwise version. ∎

P.3. Generation from the clean basic object

Theorem P.3.1.1. For \(\boldsymbol\lambda\) dominant, with kernels at the distinct marked points, \[ \mathsf H_{\boldsymbol A_{\boldsymbol\lambda}}\Pi_{\boldsymbol0} \simeq\Pi_{\boldsymbol\lambda^*}. \tag{P.3.1} \] This is the actual intermediate-extension identification, with its unit on the open stratum.

Proof. O.8 gives \(\Pi_{\boldsymbol0}=\Delta_{\boldsymbol0}=\nabla_{\boldsymbol0}\). By Theorem P.2.1.1 its Hecke image \(K\) is a proper image on the source chart for \(\boldsymbol\lambda^*\). Meromorphic gluing at the common frame point \(y\) commutes with modification at the disjoint \(x_i\). The source and target evaluation cocycles use the same background differential away from the marks. Therefore the correspondence, its vertical kernels and its actual proper image carry the twisted meromorphic action of O.2–O.5. On every irrelevant whole fixed-divisor fibre \(K\) is zero. On every relevant fibre its cohomology is locally constant.

All relevant divisors are supported at the marks, by O.6. Their saturated labels \(\boldsymbol\mu\) satisfy \(\lambda_i^*-\mu_i\in Q^\vee_+\), and relevance is exactly dominance of each \(\mu_i\). There are finitely many such labels. Source nilpotents or nonflat zero schemes remain in the compactification; the preceding assertion classifies the geometric support of the complex on its full atlas.

On the whole genuine fibre \(\mathcal B_{\boldsymbol\mu}\), the actual local-disc calculation G.4–G.5 identifies the restriction with \[ \Psi_{\boldsymbol\mu}\otimes \bigotimes_i C_{\lambda_i,\mu_i,-\mu_i}(D_i)[h_{-\mu_i}], \qquad D_i=(d_{\lambda_i}-h_{\mu_i})/2. \tag{P.3.2} \] Here \(C\) uses raw IC. Target boundary terms vanish by the already proved cleanness of \(\Pi_{\boldsymbol0}\). The tensor comparison is the actual compact Künneth comparison for the independent local modifications, with the sum of their residue functions; it is obtained in the common disc frames before descent. G.6 puts each coefficient in degrees at most zero, strictly below zero unless \(\mu_i=\lambda_i^*\), and identifies the entire extremal coefficient with its unit \(E\).

The differential in a chosen local coordinate can introduce invertible simple-root factors. These do not require surjectivity of \(T(A[[t]])\to T_{\rm ad}(A[[t]])\). The simple roots form the full character basis of \(T_{\rm ad}\), so their units give an actual adjoint-torus element. Its conjugation action on the original \(G\) descends along the fppf central presentation; it fixes the original torus cocharacters and every central map, and scales the root groups on all parameter rings. It preserves Schubert supports and their equivariant IC units. Thus it identifies this character complex with the one in G.6, even in bad characteristic. No fractional cocharacter or reduced center is introduced.

The perverse upper test on the finite relevant support now gives \(K\in{}^pD^{\le0}\), with strict ordinary boundary bounds at every label different from \(\boldsymbol\lambda^*\). The dimensions used are those of the whole fixed fibres, \(d_{\boldsymbol\mu}=d_{\boldsymbol0}+\sum_i h_{\mu_i}\); the vanished moving-divisor directions add no support dimension. Applying Theorem P.2.1.1 and the same argument with inverse character gives \(K\in{}^pD^{\ge0}\) and strict exceptional boundary bounds. Thus \(K\) is perverse and has neither a boundary subobject nor a boundary quotient. Its restriction on the top fibre is exactly \(\Psi_{\boldsymbol\lambda^*}\), by the extremal unit comparison. Lemma P.1.1.1's intermediate-extension characterization supplies (P.3.1), uniquely with that open unit. The comparisons were actual Weil maps; uniqueness preserves them. The proof concerns fixed distinct marked points and does not assert a collision or Ran comparison. ∎

Theorems P.2.1.1–P.3.1.1 prove the closed proper comparison and both strict boundary bounds with actual units. The compactification hypotheses are retained. Editable SVG source.

P.4. Rigid adjunction and the coherent module action

Theorem P.4.1.1. The finite Hecke functors have both adjoints \(\mathsf H_{A^\vee}\), with the actual units, counits and triangle identities. If \(I\) denotes the covariant inverse-kernel functor and \[ \Phi(B)=\mathsf H_{I(B)}\Pi_{\boldsymbol0}, \tag{P.4.1} \] then \(\Phi(A_{\boldsymbol\mu})=\Pi_{\boldsymbol\mu}\) and \[ \mathsf H_A\Phi(B)\simeq\Phi(B*I(A)). \tag{P.4.2} \] All comparisons are coherent on common bounded charts.

Proof. H.3 constructs \(\mathsf H_A\mathsf H_B=\mathsf H_{A*B}\) by the actual proper image on the common chain of bundles. For several distinct points the same construction is the product chain. Forgetting the independent intermediate bundles in either order gives the same projection formula and proper base-change map. The unit, associator and their coherence diagrams are consequently the ones already proved on that chain.

Lesson 7 B.2 proves rigidity of the genuine Satake category, and Lesson 11 B.4 identifies it with \(\operatorname{Rep}_E(\check G)\). Apply the action to its coevaluation \(\mathbf1\to A^\vee*A\) and evaluation \(A*A^\vee\to\mathbf1\). With H.3's unit and composition these are respectively the unit and counit for \(\mathsf H_{A^\vee}\dashv\mathsf H_A\). The two rigid triangle identities become the adjunction triangles, because all four maps are images of those same diagrams. The other evaluation and coevaluation give \(\mathsf H_A\dashv\mathsf H_{A^\vee}\). This establishes adjunction on actual degree-one derived Hom as well as degree-zero maps, without an assumption that every Hecke functor is t-exact on the ambient category.

The inverse-kernel functor is covariant and reverses convolution: \(I(B*C)=I(C)*I(B)\), with \(I^2=1\), by chain reversal in Lesson 7 B.1. On a normalized simple \(A_\lambda\), actual normalized duality identifies \(A_\lambda^\vee=I(A_\lambda)=A_{\lambda^*}\). For a coefficient multiplicity space \(M\), contragredience replaces \(M\) by \(M^*\), whereas covariant inversion does not. This distinction is needed in (P.4.2). Formula (P.3.1) gives \(\Phi(A_{\boldsymbol\mu})=\Pi_{\boldsymbol\mu}\). Moreover \(I(B*I(A))=A*I(B)\); applying H.3 proves (P.4.2). Its coherence is again the common-chain coherence. Thus this is a Satake module action, not an unconstructed intrinsic convolution of Whittaker sheaves. ∎

P.5. The full central character and representation absorption

Lemma P.5.1.1. For \(R=\check G^I\), a simple \(V_{\boldsymbol\sigma}\) has trivial character of the full center exactly when each \(\sigma_i\) belongs to the original coroot lattice \(Q^\vee\). Every such simple is a direct summand of \(V_{\boldsymbol\lambda}\otimes V_{\boldsymbol\lambda}^*\) for some original dominant \(\boldsymbol\lambda\). The labels not in that lattice lie in components disjoint from \(\Pi_{\boldsymbol0}\).

Proof. The full diagonalizable center has character group \(X_*(T)/Q^\vee\), by the pinned root-datum construction in AG-RG-05. Every weight of a simple differs from its highest weight by an integral root combination; therefore its central character is the class of that highest weight, with precisely the stated triviality condition. This uses the entire quotient, including its finite part.

The representation assertion is the already proved complete argument in Lesson 11 B.3.1.1–B.3.2.1. Its precise inputs are the root-lattice zero-weight lemma proved by rank-one lowering and the Weyl-alternant multiplicity calculation for a sufficiently deep dominant weight. The first gives \(V_\sigma[0]\ne0\); the second gives a nonzero map \(V_\sigma\to\operatorname{End}(V_\lambda)\) with \(\lambda\) in the original character lattice of \(\check G\), not the weight lattice of a substituted cover. Simplicity makes that map injective, and the characteristic-zero complete reducibility proved in the earlier RT-LIE programme splits it. Apply that proof independently in each factor of \(R\) and take the external tensor of the embeddings and splittings. This gives the asserted summand for the product. For a torus the only center-trivial simple is the unit, and the assertion is its unit embedding.

For the last assertion use the actual abelianization \(C=G/G_{\rm der}\). Since the derived group is simply connected, its torus cocharacter lattice is \(Q^\vee\), and the central quotient presentation gives \(X_*(C)=X_*(T)/Q^\vee\). A meromorphic decoration induces the specified \(C\)-bundle isomorphism off the marks. Its relative loop index at each \(x_i\) is locally constant. One direct check uses every character of this split torus: the difference of the two finite lattice lengths in a sufficiently large truncation is the integer valuation, and the ranks of the finite projective lattice quotients are locally constant on the parameter scheme. Their character-additive integers give the torus cocharacter index. Nilpotent Laurent units remain in the lattice functor and do not alter these ranks. The indices are therefore open and closed component conditions in our full finite charts. The index of \(\Pi_{\boldsymbol\sigma}\) is the image of each \(\sigma_i\), while that of \(\Pi_{\boldsymbol0}\) is zero. Disjoint open and closed supports have zero derived Hom: restriction to the components splits the sheaf category as their product. Hence every degree-one extension between these objects is zero when any index differs. ∎

P.6. Geometric semisimplicity

Theorem P.6.1.1. Let \(\mathcal W_I^\psi\) be the Serre category generated by the \(\Pi_{\boldsymbol\mu}^\psi\) on our finite bounded charts. Every object is a finite direct sum of these simples, with geometric finite-dimensional coefficient spaces.

Proof. The objects are simple: their genuine rank-one generators are simple by L.3, and the intermediate-extension characterization preserves simplicity. By Lemma P.1.1.1 their self-\(\operatorname{Ext}^1\) is zero. Using Theorem P.3.1.1 and the adjunction of Theorem P.4.1.1 gives \[ \operatorname{Ext}^1(\Pi_{\boldsymbol\mu},\Pi_{\boldsymbol\nu}) =\operatorname{Ext}^1(\Pi_{\boldsymbol0}, \mathsf H_{A_{\boldsymbol\mu}}\Pi_{\boldsymbol\nu}). \tag{P.6.1} \] The equality means the actual derived Hom in degree one, which equals heart Yoneda Ext as proved in Lemma P.1.1.1. Formula (P.4.2), the genuine Satake equivalence and complete reducibility express the right-hand Hecke object as a finite sum of \(\Pi_{\boldsymbol\sigma}\), with the multiplicities in \(V_{\boldsymbol\nu}\otimes V_{\boldsymbol\mu}^*\). Thus it suffices to kill \(\operatorname{Ext}^1(\Pi_{\boldsymbol0},\Pi_{\boldsymbol\sigma})\).

It is zero on different central components by Lemma P.5.1.1. On the zero component that lemma makes \(\Pi_{\boldsymbol\sigma}\) a direct summand of \(\mathsf H_{A_{\boldsymbol\lambda}}\Pi_{\boldsymbol\lambda}\), since the corresponding representation is \(V_{\boldsymbol\lambda}\otimes V_{\boldsymbol\lambda}^*\). But \[ \operatorname{Ext}^1(\Pi_{\boldsymbol0}, \mathsf H_{A_{\boldsymbol\lambda}}\Pi_{\boldsymbol\lambda}) =\operatorname{Ext}^1(\Pi_{\boldsymbol\lambda},\Pi_{\boldsymbol\lambda})=0 \tag{P.6.2} \] by the other rigid adjunction and Theorem P.3.1.1. Ext is additive on finite direct summands, so the desired group is zero. Every pair of simples therefore has zero degree-one extension. Induction on the finite length of an object in the generated Serre category splits each successive extension. This proves the theorem. Arbitrary Weil multiplicity spaces may still have a Jordan block; no semisimplicity assertion about those operators follows from this geometric proof. ∎

P.7. An actual finite filtration for the standard extensions

Theorem P.7.1.1. Both \(\Delta_{\boldsymbol\nu}\) and \(\nabla_{\boldsymbol\nu}\) belong to \(\mathcal W_I\). This assertion is proved with actual perverse subobjects and quotients.

Proof. Induct on the nonnegative integer \(H(\boldsymbol\nu)=\sum_i\langle2\rho,\nu_i\rangle\). If it is zero, dominance makes every simple-root pairing zero. The induced differentials have zero conductor, so O.8 gives \(\Delta_{\boldsymbol\nu}=\Pi_{\boldsymbol\nu}=\nabla_{\boldsymbol\nu}\). Central cocharacters are included in this base case.

For a positive value, N.3 makes \(\Delta_{\boldsymbol\nu}\) perverse. Its canonical map onto \(\Pi_{\boldsymbol\nu}\) is surjective by the no-boundary-quotient characterization in Lemma P.1.1.1. Let \(K\) be its actual perverse kernel. It is supported in the boundary and carries the twisted meromorphic action, because the canonical map and perverse truncations preserve the smooth action cocycle, as in O.7. That theorem makes its restriction zero on every irrelevant fibre and locally constant of character type on every relevant fibre. The latter have finitely many labels \(\boldsymbol\mu\), with \(\nu_i-\mu_i\in Q^\vee_+\). A proper boundary label has \[ H(\boldsymbol\mu)=H(\boldsymbol\nu)-2\sum_{i,j}m_{i,j} <H(\boldsymbol\nu), \qquad \nu_i-\mu_i=\sum_jm_{i,j}\alpha_j^\vee. \tag{P.7.1} \] The equality follows from \(\langle2\rho,\alpha_j^\vee\rangle=2\), already proved in N.5. The induction hypothesis therefore puts all lower standards \(\Delta_{\boldsymbol\mu}\) in \(\mathcal W_I\).

Here is the needed finite-support argument. Let \(F\) be any boundary perverse object with the same finite relevant support and twisted action. Choose the union of its maximal relevant strata; this is open in its closed support, and on it \(F\) is a sum of \(\Psi_{\boldsymbol\mu}\otimes M_{\boldsymbol\mu}\), by the whole-orbit classification O.5. Closed-support perverse descent and open restriction make these restrictions perverse, so the spaces \(M_{\boldsymbol\mu}\) occur in degree zero there. Locally closed adjunction gives an actual map \[ \bigoplus_{\text{maximal }\boldsymbol\mu} \Delta_{\boldsymbol\mu}\otimes M_{\boldsymbol\mu}\longrightarrow F \tag{P.7.2} \] whose restriction there is the identity. The domain is perverse by N.3 and its closed inclusion, and belongs to \(\mathcal W_I\) by induction. Its perverse image belongs to that Serre category. Its perverse cokernel has strictly fewer relevant strata in its support. The adjunction map is compatible with the character cocycle; hence the cokernel is again twisted equivariant and O.5 applies to it. Induction on the finite number of support strata puts this cokernel in \(\mathcal W_I\). The short exact sequence with the actual image then puts \(F\) in \(\mathcal W_I\).

All strata and closed inclusions used in this argument are the full fixed-divisor subfunctors of K.5 in a common finite chart. Even if a nilpotent family does not factor through a stratum, its geometric stalks are among the retained points of that chart; zero outside the finite relevant support is conservative on the full atlas. No parameter is replaced by its point set. The locally closed extension in (P.7.2) is the standard extension on its regular compactification followed by the closed inclusion, since the actual divisor multiplication identifies that regular subfunctor; both descriptions have the same extension-by-zero adjunction.

Apply this finite-support argument to \(K\). The exact sequence \(0\to K\to\Delta_{\boldsymbol\nu}\to\Pi_{\boldsymbol\nu}\to0\) proves the assertion for \(\Delta\). Intrinsic duality replaces the character by its inverse and exchanges the two extensions; the same argument applies to that character. Thus it proves the assertion for \(\nabla\) as well. In particular this step supplies an actual finite filtration and does not infer membership from a Grothendieck-group equality. ∎

P.8. Marked cleanness and the geometric Whittaker category

Theorem P.8.1.1. For every original dominant marked tuple, \[ \Delta_{\boldsymbol\mu}\simeq\Pi_{\boldsymbol\mu} \simeq\nabla_{\boldsymbol\mu}. \tag{P.8.1} \] Moreover \(\Phi\) is an exact geometric \(E\)-linear equivalence from the genuine Satake category at the distinct marked points to \(\mathcal W_I\), with the module comparison (P.4.2).

Proof. Theorem P.7.1.1 and geometric semisimplicity split the canonical surjection \(\Delta_{\boldsymbol\mu}\to\Pi_{\boldsymbol\mu}\). Write its kernel as a complementary direct summand. Its support lies in the boundary. But \(\Delta\) is extension by zero and has zero ordinary boundary restriction, so the complementary summand has zero restriction there too. It is zero: its support was there, and geometric stalks of the actual frame atlas are conservative. Thus the canonical surjection is an isomorphism. Applying actual duality gives the other canonical isomorphism. The maps themselves were defined with the actual adjunctions, so this conclusion preserves Weil structures, even though the splitting used to prove it was geometric.

The genuine Satake category and \(\mathcal W_I\) are semisimple. Their simples are respectively \(A_{\boldsymbol\mu}\) and \(\Pi_{\boldsymbol\mu}\), and Theorem P.3.1.1–Theorem P.4.1.1 send one to the other. The endomorphisms of each \(\Pi\) are exactly \(E\) by Lemma P.1.1.1's open full faithfulness and L.3's rank-one self-Hom. Distinct simples have zero Hom. The functor sends a scalar to that same scalar by its open unit. It is therefore fully faithful on finite sums, and every object of \(\mathcal W_I\) is such a sum. This proves essential surjectivity and exactness, with the coherent module comparison already constructed.

The conclusion also covers every finite bounded character-perverse object whose support has these finite relevant strata: once all standards are available, the actual image/cokernel induction in Theorem P.7.1.1 puts it in the same Serre category. The assertion is geometric and concerns fixed distinct marks. Neither a moving-point factorization equivalence nor a half-twisted derived or Ran equivalence has been constructed here. ∎

P.9. Full concentration and the tensor multiplicity

Theorem P.9.1.1. Let \(\lambda,\mu\) be original dominant cocharacters, and let \(\eta=\mu+\nu\) be dominant. For the actual descended conductor character and raw IC, put \(D=(d_\lambda+h_\nu)/2\). Then \[ R\Gamma_c(Z_\lambda\cap S_\nu, I_\lambda\otimes(\chi_\mu^\nu)^*\mathcal L_\psi) \simeq \operatorname{Hom}_{\check G} (V_\lambda\otimes V_\mu,V_\eta)(-D)[-h_\nu]. \tag{P.9.1} \] In particular all other degrees vanish. For split finite-field data the raw operator on the displayed degree is \(q^D\operatorname{id}\), with no Jordan part. If \(\mu\) is not dominant, the entire complex is zero whenever the character is defined.

Proof. Theorem P.8.1.1 and its module comparison give the actual geometric finite decomposition \[ \mathsf H_{A_\lambda}\Pi_\eta =\bigoplus_\sigma\Pi_\sigma\otimes \operatorname{Hom}_{\check G}(V_\lambda\otimes V_\sigma,V_\eta). \tag{P.9.2} \] Indeed the corresponding representation is \(V_\eta\otimes V_\lambda^*\), and rigid evaluation identifies its \(V_\sigma\) multiplicity with the stated Hom. This identification follows from the actual module and rigid maps, rather than a numerical character calculation.

Restrict to the whole fibre \(\mathcal B_\mu\). Cleanness (P.8.1) kills the restriction of every summand except \(\sigma=\mu\); that remaining restriction is \(\Psi_\mu\) times the displayed Hom space in degree zero. On the other hand G.4–G.5, now with the clean target \(\Pi_\eta\), identify precisely this restriction with \(\Psi_\mu\otimes C_{\lambda\mu\nu}(D)[h_\nu]\). Target boundary terms vanish by cleanness. The rank-one inverse tensor and the unit in L.3 cancel \(\Psi_\mu\). This gives \(C_{\lambda\mu\nu}(D)[h_\nu]=\operatorname{Hom}(V_\lambda\otimes V_\mu,V_\eta)[0]\), which is (P.9.1), including its raw signs and twist.

Nonempty slices have integral \(D\) by the root-lattice dimension calculation in Lesson 5. For finite-field data E.5 identifies the actual operator on the surviving top raw group with \(q^D\operatorname{id}\) through its trace-oriented character-zero component lines. Since every other group has now vanished, this supplies the whole Weil statement and shows that the normalized multiplicity space has identity operator. No inference from a trace or a purity weight is involved. E.3 already kills the entire complex for nondominant \(\mu\), independently of this comparison. For several distinct marks use the tensor of (P.9.1): the degrees and twists add, and the multiplicity is the \(\check G^I\) Hom. This asserts no collision limit. ∎

P.10. Transfer to the full original group

Theorem P.10.1.1. The local concentration and multiplicity formula (P.9.1) holds for every original split connected reductive \(G\), with its full lattices and center, in every geometric characteristic different from \(\ell\). The actual Weil comparison is unchanged.

Proof. Use the smooth central torus cover \(q:\widetilde G=G_{\rm sc}\rtimes T\to G\) of H.4. It has simply connected derived group. For a nonempty slice put \(\delta=\lambda-\nu\in Q^\vee\), let \(\delta_{\rm sc}\) be its unique lift, and take \[ \widetilde\lambda=(0,\lambda),\quad \widetilde\mu=(0,\mu),\quad \widetilde\nu=(-\delta_{\rm sc},\lambda),\quad \widetilde\eta=(-\delta_{\rm sc},\mu+\lambda). \tag{P.10.1} \] Their images are the prescribed original labels; every root pairing is the original one, so dominance and the numbers \(d_\lambda,h_\nu,D\) are unchanged. H.7 gives the actual nilpotent closed slice comparison, the equality of descended functions, and the IC, compact-coefficient and Weil comparisons on all parameter bases. Apply Theorem P.9.1.1 to these lifted labels. It remains to compare the actual representation Hom spaces.

Here is that root-datum comparison. In \(\widetilde R=\check{\widetilde G}\), the subtorus \(T_R\) with cocharacter lattice \(q^*X^*(T)\subset P\oplus X^*(T)\) is primitive, since H.4 gives its free quotient \(P\). It contains all coroots of \(\widetilde R\), namely \((\alpha_{\rm sc},\alpha)=q^*\alpha\), hence their saturation, the torus of \(\widetilde R_{\rm der}\). The inverse image in \(\widetilde R\) of the image of \(T_R\) in its abelianization is a connected normal reductive subgroup \(R\) with maximal torus \(T_R\) and derived subgroup \(\widetilde R_{\rm der}\). This follows directly from the reductive central presentation of AG-RG-05: it is the quotient of the derived factor times that subtorus by their actual central intersection. Restriction of a root \((\alpha^\vee_{\rm sc},0)\) of \(\widetilde R\) is the original \(\alpha^\vee\), while its coroot is \(q^*\alpha\). Thus its pinned root datum is exactly the original dual datum, and the programme classification identifies \(R=\check G\).

Restriction of a simple \(\widetilde R\)-module to \(R\) is simple with highest weight its image under \(q_*\): it has the same full derived factor and its remaining central torus acts by scalars. The map \(R\times Z(\widetilde R)\to\widetilde R\) is fppf surjective by that same central presentation. The difference \[ \widetilde\lambda+\widetilde\mu-\widetilde\eta =(\delta_{\rm sc},0) \tag{P.10.2} \] is in the full root lattice of \(\widetilde R\). Consequently source and target of our tensor Hom have the same character of its entire center. Every \(R\)-equivariant map then commutes with that center and, by the fppf product, is \(\widetilde R\)-equivariant. Restriction therefore gives the actual isomorphism \[ \operatorname{Hom}_{\widetilde R} (V_{\widetilde\lambda}\otimes V_{\widetilde\mu},V_{\widetilde\eta}) \simeq\operatorname{Hom}_{\check G}(V_\lambda\otimes V_\mu,V_\eta). \tag{P.10.3} \] This proves (P.9.1) for nonempty slices, with the full original center retained. In particular the canonical lift \((0,\nu)\) must not replace the adjusted lift in (P.10.1).

For completeness an empty slice also has zero claimed multiplicity. If \(V_\eta\) occurs in \(V_\mu\otimes V_\lambda\), a highest tensor vector has nonzero projection to the highest-weight line of \(V_\mu\). To prove this, decompose its first factor by weight and choose the smallest positive height below \(\mu\) occurring, if the top projection were zero. Applying each simple raising operator, the components one height higher receive no contribution from raising the second factor, since those first-factor components were absent. Thus every coefficient in that first-factor weight is killed by all simple raisings. In an irreducible highest-weight module such a vector is a highest vector and is confined to weight \(\mu\), by the earlier RT-LIE highest-weight proof. This contradicts the chosen positive height. A nonzero top projection therefore supplies weight \(\eta-\mu=\nu\) in \(V_\lambda\). The genuine Satake weight functor, already proved in Lesson 11 B.4, identifies that weight space with the untwisted top cohomology of \(Z_\lambda\cap S_\nu\); it is zero when the slice is empty. Hence the tensor multiplicity is zero. Nondominant \(\mu\) is handled for the original group directly by E.3.

This transfer uses the actual nilpotent slice immersion and coefficient invariance proved in H.6–H.7; it does not reduce the moduli functor or its center. It transfers the local formula and its Frobenius. A global compactification or global Whittaker-category transfer for arbitrary \(G\) is a distinct assertion and is not claimed here. ∎

Theorems P.6.1.1–P.10.1.1 give the extension, filtration, cleanness and full cohomology arguments. Exercises P.11.1–P.11.2 check the raw shifts, operators and adjusted central lift. Editable SVG source.

P.11. Exercises

Exercise P.11.1 (medium). For the original group \(SL_2\), write \(V_m\) for the dual \(PGL_2\)-module of highest weight \(m\alpha^\vee\), \(m\ge0\). Compute every local Whittaker group with \(\lambda=m\alpha^\vee\), \(\mu=n\alpha^\vee\), and \(\eta=e\alpha^\vee\). Give the actual Frobenius operators for \(m=n=1\).

Solution. The rank-one highest-weight proof gives weights \(m,m-1,\ldots,-m\), each once, in coroot units. Multiplying the two finite geometric sums and comparing coefficients gives \[ V_m\otimes V_n=\bigoplus_{e=|m-n|}^{m+n}V_e. \tag{P.11.1} \] Indeed at weight \(r\) the product counts pairs in \([-m,m]\times[-n,n]\) with sum \(r\). For \(r\ge0\) that count equals the number of integers \(e\) in the displayed interval with \(e\ge r\); symmetry treats negative \(r\). Complete reducibility and the triangular highest characters prove the claimed direct sum, with multiplicity one. The cocharacters and tensor summands remain those of the original dual \(PGL_2\); a half-coroot summand has not been admitted.

Here \(\nu=(e-n)\alpha^\vee\), \(h_\nu=2(e-n)\), \(d_\lambda=2m\) and \(D=m+e-n\). Theorem P.9.1.1–Theorem P.10.1.1 give precisely one group \(H_c^{2(e-n)}=E(-(m+e-n))\) when \(|m-n|\le e\le m+n\), and zero in every degree otherwise. For \(m=n=1\), the labels \(e=0,1,2\) give respectively \(H^{-2}=E\), \(H^0=E(-1)\), \(H^2=E(-2)\), with operators \(1,q,q^2\). Negative cohomological degree is possible because raw IC is already shifted. These are full operators, not just their traces.

Exercise P.11.2 (advanced). Take the original \(PGL_2\), let \(\omega^\vee\) be its fundamental cocharacter, and set \(\lambda=\mu=\omega^\vee\), \(\eta=0\). Determine the adjusted cover labels and compare the local group with the dual \(SL_2\) tensor product.

Solution. Here \(\nu=-\omega^\vee\) and \(\delta=2\omega^\vee=\alpha^\vee\). The actual labels are \[ \widetilde\nu=(-\alpha^\vee_{\rm sc},\omega^\vee),\qquad \widetilde\eta=(-\alpha^\vee_{\rm sc},2\omega^\vee). \tag{P.11.2} \] The source semidirect abelianization index of the slice is \(\omega^\vee\), the same as that of \(\widetilde\lambda\). The unadjusted lift \((0,-\omega^\vee)\) has the wrong index and gives an empty source intersection. The tensor difference with the adjusted lift is \((\alpha^\vee_{\rm sc},0)\), in the source-dual root lattice, as required by (P.10.2).

The original dual is \(SL_2\); its standard two-dimensional module satisfies \(V\otimes V=\operatorname{Sym}^2V\oplus\bigwedge^2V\), by the actual symmetrizer and antisymmetrizer over characteristic-zero \(E\). The second summand is its unit and occurs once. On the covering dual it is the corresponding central determinant character, whose restriction is the unit; replacing the adjusted target by a canonical zero lift would lose precisely this character. The extremal slice is a point, \(d_\lambda=1\), \(h_\nu=-1\), and \(D=0\). Thus its raw complex is \(E[1]\), its only group is \(H^{-1}=E\), and Frobenius is the identity. This also illustrates that the smooth central cover and its adjusted labels work in characteristic two, retaining the nonreduced \(\mu_2\) in the simply connected central presentation. ∎

Appendix Q. Arbitrary regular conductors and the original group

Keep the finite rational coefficient field, pinning, raw IC and actual Weil conventions of Appendix P. The local assertions below apply to every split connected reductive group. The global assertions assume that the original derived group is simply connected, as in J–P, and use \(X=\mathbf P^1\). Global differentials are nonzero regular maps of line bundles. Their conductor records their zero divisors; it need not lift point by point to the original cocharacter lattice.

The free Whittaker patterns, §§4.1–4.2 formulates the conductor in the adjoint lattice and uses an enlargement with connected centre. Here we prove the precise group, full root datum, all-base decoration comparison and resulting assertions. The inputs already proved in this lesson are the full highest-line algebra and equations (J.3–J.4), finite quotient presentations and perverse operations (M–N), and marked cleanness and local cohomology (P.8–P.10).

Q.1. A split enlargement with the original group as exact kernel

Write \(X_T=X^*(T)\), \(Y_T=X_*(T)\), and let \(Q\subset X_T\) be the original root lattice. The torus \(T_{\rm ad}=T/Z(G)\) has character lattice \(Q\), with the simple roots as a basis, and cocharacter lattice \(Y_{\rm ad}=\operatorname{Hom}(Q,\mathbf Z)\). Bars indicate the actual quotient map.

Proposition Q.1.1.1. Adjoint conjugation defines an action of \(T_{\rm ad}\) on the original \(G\). The group \[ G^+=G\rtimes T_{\rm ad},\qquad T^+=T\times T_{\rm ad}, \qquad q_a(g,a)=a \tag{Q.1.1} \] is split connected reductive, with derived subgroup \(G_{\rm der}\). The kernel of \(q_a:G^+\to T_{\rm ad}\) is the actual original \(G\). Its roots and coroots are \[ \alpha^+=(\alpha,\alpha)\in X_T\oplus Q,\qquad (\alpha^\vee)^+=(\alpha^\vee,0)\in Y_T\oplus Y_{\rm ad}. \tag{Q.1.2} \] Its full centre is the smooth torus \[ S=\{(t,\bar t^{-1}):t\in T\}\simeq T. \tag{Q.1.3} \] Its schematic intersection with the original kernel is exactly \(Z(G)\), including its nonreduced part.

Proof. Use the original central presentation proved in AG-RG-05 and used in H.4. Lift \(a\in T_{\rm ad}(A)\) fppf locally to \(t\in T(A')\). Conjugation by \(t\) is unchanged when the lift is multiplied by any element of the entire \(Z(G)\). Descent of morphisms therefore gives an automorphism of \(G_A\), and multiplication of lifts proves its action law. This also holds for nilpotent \(A\).

The semidirect product has underlying smooth connected affine scheme \(G\times T_{\rm ad}\). Its connected normal unipotent radical projects trivially into the torus, by the original reductive-group result that a homomorphism from a connected unipotent group to a torus is trivial. It consequently lies in \(G\) and is normal there, so is trivial. Thus \(G^+\) is reductive. Adjoint conjugation acts trivially on the original abelianization; hence \(G^+/G_{\rm der}=(G/G_{\rm der})\times T_{\rm ad}\) is a torus. This proves equality of the derived subgroups.

Conjugation of \(u_\alpha(z)\) by \((t,a)\) multiplies \(z\) by \(\alpha(t)\alpha(a)\). The pinned rank-one group remains inside \(G\). These facts give (Q.1.2), the split maximal torus and every root subgroup. In particular its root pairing with \((y,d)\) is \(\langle\alpha,y\rangle+\langle\alpha,d\rangle\).

The map \(G^+\to G_{\rm ad}\), \((g,a)\mapsto\bar g a\), is a group map. The full inverse image of \(T_{\rm ad}\subset G_{\rm ad}\) is \(T\subset G\), so its kernel is precisely (Q.1.3). This kernel commutes with the root groups and \(T^+\). Conversely every central element maps to the trivial centre of the original adjoint group, by its root datum. Thus this is the whole centre. Intersecting its equation with \(a=1\) gives \(\bar t=1\), the entire original \(Z(G)\). All assertions concern equations of group schemes, rather than their field-valued points. ∎

Q.2. The entire highest-line datum and its quotient frame

Let \(W_\lambda\) be the geometric Weyl module of J.3 for an original dominant character \(\lambda\). Its weights differ from \(\lambda\) by elements of \(Q\). Its whole centre acts by \(\lambda\); the module need not be simple in positive characteristic.

Theorem Q.2.1.1. The canonical extension to \(G^+\) of highest character \((\lambda,0)\) has action \[ (g,a)w=\rho_\lambda(g) \frac{\rho_\lambda(t)}{\lambda(t)}w, \qquad \bar t=a, \tag{Q.2.1} \] interpreted by fppf descent. For \(b\in Q\), tensor this extension with \(b\circ q_a\) to obtain \(W^+_{(\lambda,b)}\). The full highest-line algebra satisfies \[ B_{G^+}=B_G\otimes k[Q] \tag{Q.2.2} \] as an algebra, with its genuine action and \(X_T\oplus Q\) grading. For source torus bundle \((\mathcal T,1)\), the entire regular decoration stack is canonically the original \(G\) decoration stack. The comparison retains all maps, inverse maps, higher relations, injection conditions, parameter-flat quotients and fixed-defect subfunctors on arbitrary bases.

Proof. Multiplying \(t\) by \(z\in Z(G)\) multiplies numerator and denominator in (Q.2.1) by the same actual character. The operator descends, and its multiplication law follows from \(\rho(t)\rho(h)\rho(t)^{-1}=\rho(\operatorname{Ad}_a h)\). It fixes the specified highest vector. On weight \(\beta\) its additional torus character is \(\beta-\lambda\in Q\).

The actual flag scheme \(G^+/B^+\) is \(G/B\): the extra torus is removed by right multiplication in \(B^+=B\rtimes T_{\rm ad}\). Its line of weight \((\lambda,b)\) has the same underlying line as that of weight \(\lambda\), with the indicated linearization. Thus its entire section module and its dual are exactly the displayed modules. Cartan multiplication commutes with the normalized operators, since the denominators for \(\lambda,\mu\) multiply to the one for \(\lambda+\mu\). This proves (Q.2.2). All the finite generators and higher equations of J.3–J.4 remain, together with a basis of \(Q\) and its inverse units.

For a \(G^+\)-bundle \(\mathcal E^+\) on \(X_A\), the decoration maps of weights \((0,b)\), with their inverse and product equations, identify the quotient torsor \(\mathcal E^+/G\) with the trivial source \(T_{\rm ad}\)-bundle. This specifies an actual frame: for a basis of \(Q\) the line maps and their inverses are units. Take the inverse image of the identity section in that quotient. It is a principal bundle \(\mathcal E\) for the exact kernel \(G\). Extension of structure group recovers \(\mathcal E^+\) with its specified quotient frame. The constructions are inverse on objects and isomorphisms by torsor descent.

The maps of weights \((\lambda,0)\) now restrict exactly to the original \(\kappa_\lambda\). Every \((\lambda,b)\) map is recovered by tensoring with its specified unit map, by the full product equations. No data are lost. Tensoring with invertible lines preserves injections and parameter-flat cokernels. The finite relations therefore identify the whole parameter schemes on fixed bundle charts, with their nilpotents. The primitive condition, relative zero ideals and every higher divisibility equation agree, identifying the genuine open and the entire relative fixed-defect functors.

Choose compatible finite bundle frames whose quotient frame is the specified one. Their remaining structure group is exactly \(G\). On genuine torsor charts the twisted unipotent group, its jet frame groups and nerves are the same, since the original \(N\) is unchanged. Thus the quotient enhancements and perverse normalizations agree. In a meromorphic ambient chart impose the same actual auxiliary framing as a closed subfunctor. The regular inverse-unit equations put the compared charts inside it. Nilpotent meromorphic torus directions outside this subfunctor are not identified with its identity. Closed support and its fully faithful pushforward are retained by N.3. ∎

Proposition Q.1.1.1 and §Q.7 give the exact lattice, root pairing, quotient and central graph in this \(SL_2\) example. The green region is \(2m+c\ge0\); the horizontal blue lattice is the original cocharacter lattice. The finite schematic intersection is displayed separately because cocharacter lattices cannot detect its nonreduced points. Editable SVG source.

Q.3. General regular differentials and a central twist

Fix \(\mathcal T_0\) and its simple-root isomorphisms with \(\Omega_X\), as in O.8. Let \(\mathcal T\) be the original torus bundle and let the nonzero maps \(\omega_i:\mathcal L_{\alpha_i}(\mathcal T)\to\Omega_X\) have zero divisors \(C_i=\sum_xc_{i,x}[x]\). Define the integral adjoint labels by \[ \langle\alpha_i,\delta_x\rangle=c_{i,x},\qquad \delta_x\in Y_{\rm ad},\qquad \widehat\delta_x=(0,\delta_x)\in Y_T\oplus Y_{\rm ad}. \tag{Q.3.1} \] The simple-root basis makes this integral. Each enlarged label is dominant, regardless of the original lifting obstruction.

Theorem Q.3.1.1. If the original derived group is simply connected, the genuine evaluation extension for every such regular differential on \(\mathbf P^1\) is clean: \[ j_!\Psi_\omega\simeq j_{!*}\Psi_\omega \simeq Rj_*\Psi_\omega. \tag{Q.3.2} \] These are the canonical actual maps on the full regular compactification. Ordinary and exceptional boundary restrictions vanish. The comparison preserves evaluation, perverse normalization, inverse character and actual Weil maps.

Proof. In \(G^+\), twist the zero-conductor background \((\mathcal T_0,1)\) at the distinct conductor points by \(\widehat\delta_x\). The source becomes \[ \mathcal T^+_{\widehat\delta} =\left(\mathcal T_0, T_{\rm ad}\left(-\sum_x\delta_x[x]\right)\right). \tag{Q.3.3} \] Its root \((\alpha_i,\alpha_i)\) has line \(\Omega_X(-C_i)\), with its specified inclusion into \(\Omega_X\). Theorem P.8.1.1 applies to this actual group and these actual dominant labels. Its derived group is the original simply connected group. Thus its full regular chart has canonical cleanness.

Set \(\mathcal R=\mathcal T\otimes\mathcal T_0^{-1}\), an actual original \(T\)-bundle. Dividing \(\omega_i\) by its Cartier zero divisor gives \[ \mathcal L_{\alpha_i}(\mathcal R) \simeq\mathcal O_X(-C_i). \tag{Q.3.4} \] The simple-root basis, with its tensor inverses, therefore supplies a specified torus-bundle isomorphism \(\mathcal R_{\rm ad}\simeq T_{\rm ad}(-\sum_x\delta_x[x])\). No pointwise lift of \(\delta_x\) to \(Y_T\) is involved.

Extend \(\mathcal R\) along the central graph \(T\simeq S\subset G^+\). Tensoring torsors by this central bundle is invertible, with inverse using \(\mathcal R^{-1}\). It sends (Q.3.3) to \[ (\mathcal T_0,\mathcal R_{\rm ad}) \otimes(\mathcal R,\mathcal R_{\rm ad}^{-1}) =(\mathcal T,1). \tag{Q.3.5} \] On \(W^+_{(\lambda,b)}\), the graph centre acts by the scalar character \(\lambda-b|_T\). Source and target of each decoration map consequently tensor with the same invertible line. All maps, inverse maps and full polynomial relations are transported exactly. This proves an isomorphism of entire decoration functors on every parameter base.

Every root vanishes on \(S\). Hence the twisted unipotent group, its simple-root quotients and evaluation map remain the same. Equation (Q.3.4) identifies the root-line inclusion with the actual prescribed \(\omega_i\), including its unit. Compatible frames and normalizations agree by Q.2. That theorem now identifies the twisted regular compactification for \((\mathcal T,1)\) with the full original one. The enhanced open diagrams are isomorphic, so both adjunction extensions and their perverse image transport. Applying this to the clean maps of P.8 proves (Q.3.2) and both boundary vanishings.

For finite-field data, a single finite extension splits the finite conductor support if necessary. All displayed group and torsor maps descend with the original data and commute with Frobenius. The canonical clean maps are isomorphisms over the original field, since their cones are zero if their pullbacks to that extension are zero, by conservative geometric stalks. This retains Weil maps without claiming semisimplicity for arbitrary Weil objects. ∎

Q.4. Fixed conductor and the correct representation block

Let \(I\) be a finite set of distinct allowed modification points and \(J=I\cup\operatorname{supp}(C)\), with \(\delta_x=0\) elsewhere. At \(x\in I\), an original label \(\mu_x\in Y_T\) is regular when \[ \langle\alpha_i,\mu_x\rangle+c_{i,x}\ge0 \quad\text{for all simple roots}. \tag{Q.4.1} \] This is exactly dominance of \(\widehat\mu_x=(\mu_x,\delta_x)\). At \(J\setminus I\), keep the fixed label \((0,\delta_x)\). Let \(\mathcal W_{\omega,I}\) be the geometric Serre category generated by the corresponding original evaluation intermediate extensions, on common finite regular charts with actual closed inclusions as in P.

Theorem Q.4.1.1. Under the global hypotheses of Q.3, this category is semisimple and all its generating extensions are clean. There is a central quotient \[ R^+=\check{G^+}\longrightarrow\check G,\qquad Z_a=\check T_{\rm ad}\subset Z(R^+),\qquad R^+/Z_a\simeq\check G. \tag{Q.4.2} \] The category is equivalent to the product over \(x\in I\) of the finite-dimensional \(R^+\)-representation blocks with \(Z_a\)-character \(\delta_x\). The original Satake action is pullback along this quotient, with exactly P.4's inverse and dual conventions. The factors at \(J\setminus I\) are fixed, rather than free variables.

Proof. By Q.1 and Lesson 11 B.4's full dual root-datum theorem, the character and cocharacter lattices of \(R^+\) are \(Y_T\oplus Y_{\rm ad}\) and \(X_T\oplus Q\). Its roots are \((\alpha^\vee,0)\), its coroots \((\alpha,\alpha)\). The primitive sublattice \(0\oplus Q\) specifies a torus \(Z_a\), on which every root is zero. The quotient has character lattice \(Y_T\oplus0\), cocharacter lattice \(X_T\), roots \(\alpha^\vee\) and coroots \(\alpha\). Pinned classification identifies it with the original \(\check G\). This quotient retains the original lattice; it is not a quotient by the entire centre of \(R^+\).

The highest weight \((\mu,d)\) restricts to \(Z_a\) as \(d\). Thus the simples in the \(\delta_x\) block are precisely the labels satisfying (Q.4.1). Apply P.8 to \(G^+\) at \(J\). It gives semisimplicity and the equivalence with these marked intermediate extensions. Tensoring with the fixed simples outside \(I\) is fully faithful, since each has endomorphism field \(E\). In a semisimple category their finite sums give exactly the stated Serre subcategory.

The central twist of Q.3 is fixed across all these labels. Its original source is \(\mathcal T(-\sum_{x\in I}\mu_x[x])\), and its auxiliary source becomes trivial. Q.2's actual quotient frame therefore identifies each full regular support with the original one, including extensions and its closed support in a common meromorphic chart. It does not replace a nilpotent auxiliary torus modification by a fixed point.

An original label \(\lambda\) lifts as \((\lambda,0)\); its representation is the pullback of \(V_\lambda\) from \(\check G\). It is trivial on \(Z_a\), so the tensor operations and specified contragredients of P.4 preserve each block. The actual Schubert and modification diagram agrees with the original one by Q.5 below; it preserves the quotient frame. Hence the rigid module maps restrict and transport to precisely the original action. ∎

A nonlifting \(\delta_x\) need not admit any one-dimensional object in this block. Calling it the usual untwisted representation category of \(\check G\) would lose that obstruction.

Q.5. Actual original slices inside the enlargement

Lemma Q.5.1.1. For every original derived group, the original Grassmannian is the exact identity fibre \[ \mathrm{Gr}_G= \mathrm{Gr}_{G^+}\times_{\mathrm{Gr}_{T_{\rm ad}}}\{1\}. \tag{Q.5.1} \] For \(\lambda\) original dominant and \(\nu\) arbitrary, \[ \mathrm{Gr}_G^\lambda=\mathrm{Gr}_{G^+}^{(\lambda,0)}, \quad Z_\lambda=Z^+_{(\lambda,0)},\quad Z_\lambda\cap S_\nu= Z^+_{(\lambda,0)}\cap S^+_{(\nu,0)}. \tag{Q.5.2} \] These are full scheme comparisons. They retain nilpotent parameters and identify IC and compact coefficient diagrams. Integral adjoint-torus conjugation acts on them with its canonical IC comparison.

Proof. Locally in the sheaf quotient, a representative \((g,a)\in G^+(A((t)))\) in the identity fibre has \(a\in T_{\rm ad}(A[[t]])\). Right multiply by \((1,a^{-1})\in L^+G^+\). The result \((g,1)\) is unique modulo \(L^+G\), since its possible discrepancy has auxiliary coordinate one. These operations commute with descent. Equivalently take the identity fibre of the trivialized quotient torsor on the formal disc. The unit in the torus Grassmannian is a closed subfunctor: its Laurent coefficient charts impose membership in the positive unit group, with every negative coefficient zero, also for nilpotent parameters. This proves the exact closed fibre.

For \(h\in L^+G\), \(a\in L^+T_{\rm ad}\), the class of \((h,a)t^{(\lambda,0)}\) is \(h t^\lambda\): \(a\) fixes \(t^\lambda\) and can be removed on the right. This proves equality of positive-loop orbit functors, with the actual stabilizer. Their schematic closures agree, since the identity fibre is closed and the closure's ideal is the kernel of restriction to that same orbit. The unipotent radical is the original \(N\), and its stabilizer at \((\nu,0)\) is exactly \(N(A((t)))\cap t^\nu G(A[[t]])t^{-\nu}\). Thus the whole semi-infinite orbit agrees. Its fibre product with the closed Schubert gives the entire slice.

Conjugation by an integral adjoint-torus element preserves the integral original group, fixes \(t^\lambda,t^\nu\), and acts on \(N(A((t)))\). It preserves these diagrams. On bounded supports the action factors through finite jets. The smooth orbit's constant IC generator has its tautological comparison, whose intermediate extension gives the comparison on the full Schubert. Uniqueness of intermediate extension proves the cocycle law and Frobenius compatibility. Scheme equality now identifies character pullback and compact pushforward as well. Integral adjoint-torus elements have not been assumed to lift to original integral torus elements. ∎

Q.6. Every degree for an arbitrary local regular conductor

Let \(c_i\ge0\), \(u_i\in k[[t]]^\times\), and define \(\delta\in Y_{\rm ad}\) by \(\langle\alpha_i,\delta\rangle=c_i\). In the pinned simple-root coordinates of \(n\), the character is \[ \chi_{\mu,c,u}(n)= \operatorname{Res}\sum_i t^{\langle\alpha_i,\mu\rangle+c_i}u_i(t)n_i(t)\,dt. \tag{Q.6.1} \] It descends to \(S_\nu\) when every \(\langle\alpha_i,\mu+\nu\rangle+c_i\ge0\). Set \[ \widehat\lambda=(\lambda,0),\quad \widehat\mu=(\mu,\delta),\quad \widehat\nu=(\nu,0),\quad \widehat\eta=(\mu+\nu,\delta),\qquad D=(d_\lambda+h_\nu)/2. \tag{Q.6.2} \] The dimensions use the original \(d_\lambda=\langle2\rho,\lambda\rangle\) and \(h_\nu=\langle2\rho,\nu\rangle\).

Theorem Q.6.1.1. For every original split connected reductive \(G\), when \(\widehat\mu,\widehat\eta\) are dominant, \[ R\Gamma_c(Z_\lambda\cap S_\nu, I_\lambda\otimes(\chi_{\mu,c,u}^\nu)^*\mathcal L_\psi) \simeq \operatorname{Hom}_{R^+} (V_{\widehat\lambda}\otimes V_{\widehat\mu}, V_{\widehat\eta}) (-D)[-h_\nu]. \tag{Q.6.3} \] All other degrees vanish. For split finite-field data the sole raw group has actual operator \(q^D\operatorname{id}\). If \(\widehat\mu\) is not dominant, the entire complex is zero whenever the character descends. The multiplicity is zero for an empty slice; \(D\) is integral for a nonempty one.

If an actual original \(\xi\in Y_T\) satisfies \(\bar\xi=\delta\), the displayed multiplicity is canonically \[ \operatorname{Hom}_{\check G} (V_\lambda\otimes V_{\mu+\xi},V_{\mu+\nu+\xi}). \tag{Q.6.4} \] An original lift is a hypothesis of this last formula, not of (Q.6.3).

Proof. The simple-root basis of \(Q\) specifies \(a\in T_{\rm ad}(k[[t]])\) with \(\alpha_i(a)=u_i\). Adjoint conjugation multiplies \(n_i\) by \(u_i\), identifying (Q.6.1) with the character whose units are one. Q.5 supplies its actual original-slice automorphism and canonical IC comparison. Compact pushforward therefore gives the cohomology comparison, with Weil compatibility for finite-field data. This works in bad characteristic without lifting \(a\) to \(T(k[[t]])\).

With units one, this is exactly the conductor character for \(G^+\) with label \(\widehat\mu\), by the root-pairing equality in Q.1. Its descent condition is dominance of \(\widehat\eta\). Q.5 identifies the entire character, IC and scheme diagram. The sum of positive enlarged roots pairs with \((\lambda,0),(\nu,0)\) as the original sum does with \(\lambda,\nu\); hence \(d_\lambda,h_\nu,D\) are unchanged. Apply P.10.1.1 to the actual reductive \(G^+\). That result applies to every derived group with its full lattices, and supplies all-degree concentration, the complete operator, nondominant vanishing and the empty-slice assertion. This proves (Q.6.3).

For an original lift, \(l=(-\xi,\delta)\) pairs to zero with every coroot \((\alpha_i,\alpha_i)\) of \(R^+\). It is an actual character of its abelianization, giving an invertible module \(L_l\). Thus \[ V_{(\mu,\delta)}=L_l\otimes V_{(\mu+\xi,0)},\qquad V_{(\mu+\nu,\delta)} =L_l\otimes V_{(\mu+\nu+\xi,0)}. \tag{Q.6.5} \] Cancel this same line in the tensor Hom. Every remaining module descends through the exact quotient (Q.4.2), giving (Q.6.4). Changing the lift by a central cocharacter tensors both original modules by the same character, which cancels too.

For a nonlifting conductor, all labels in (Q.6.3) are genuine weights of \(R^+\). Their difference \(\widehat\lambda+\widehat\mu-\widehat\eta=(\lambda-\nu,0)\) is in its full root lattice for a nonempty original slice. Their full central characters match. No fractional original label or representation of the wrong original dual group is admitted. ∎

Q.7. Exact rank-one models

For original \(SL_2\), write a cocharacter as \(m\alpha^\vee\) and its adjoint conductor as \(c\). Then \[ Y^+=\mathbf Z^2,\quad \langle\alpha^+,(m,c)\rangle=2m+c,\quad (\alpha^\vee)^+=(1,0),\quad S_*\mathbf Z=\mathbf Z(1,-2). \tag{Q.7.1} \] The actual isomorphism \[ SL_2\rtimes\mathbf G_m\longrightarrow GL_2,\qquad (g,u)\longmapsto g\operatorname{diag}(1,u^{-1}) \tag{Q.7.2} \] has inverse \(u=\det(A)^{-1}\), \(g=A\operatorname{diag}(1,u)\). Its multiplication follows from the specified adjoint action, on every ring. The original group is \(\det=1\). The centre graph sends \(a\) to \(aI_2\), with auxiliary coordinate \(a^{-2}\), and intersects the kernel in the schematic \(\mu_2\). In characteristic two, \(A=k[\epsilon]/\epsilon^2\) has the nontrivial such point \(a=1+\epsilon\). Its square is one, its original scalar matrix is not the identity, and its auxiliary coordinate is one.

The dual root datum identifies highest weights with those of \(GL_2\) by \[ (m,c)\longmapsto(m+c,-m). \tag{Q.7.3} \] The root \((1,0)\) becomes \((1,-1)\); the coroot pairing is the difference \(2m+c\). Auxiliary degree \(c\) is the full scalar central character. Thus \(V_{(0,1)}\) is standard, while \(V_{(1,0)}\) is the three-dimensional adjoint module. Only the latter descends to the original dual \(PGL_2\).

Take original \(\lambda=\alpha^\vee,\mu=\nu=0,c=1\). The slice is the affine line \(n(a t^{-1})\), as calculated in Lesson 4 and §4 here. Its character is \(\operatorname{Res}(t a t^{-1}dt)=0\), and its raw IC is \(E[2]\) by §2's quadratic-cone proof. Thus its compact complex is \(E(-1)[0]\), by Semi-infinite orbits and weight functors, Lemma Q.1.2.1. For \(c=0\) the character is \(a\), and the whole complex is zero by E.1–E.2. Formula (Q.6.3) agrees: at \(c=1\) its multiplicity is the one-dimensional \(\operatorname{Hom}_{GL_2}(\operatorname{Ad}\otimes\operatorname{Std},\operatorname{Std})\); at \(c=0\) it is \(\operatorname{Hom}_{PGL_2}(\operatorname{Ad},1)=0\). The raw operator in the first case is \(q\). The standard enlarged module has not become an original dual module.

Globally let the original fundamental torus line be \(L=\mathcal O(-2)\). Its root line is \(L^2=\mathcal O(-4)\); choose its differential to \(\Omega=\mathcal O(-2)\) with simple zeros at distinct \(x,y\). The background is \(L_0=\mathcal O(-1)\), so the central-twist line is \(\mathcal O(-1)\). The enlarged source in Q.3 has first line \(\mathcal O(-1)\) and auxiliary line \(\mathcal O(-x-y)\). Under (Q.7.2) its diagonal lines are \(\mathcal O(-1),\mathcal O(3)\). Its genuine bundles are \[ 0\longrightarrow\mathcal O(-1)\longrightarrow E^+ \longrightarrow\mathcal O(3)\longrightarrow0. \tag{Q.7.4} \] Their determinant is \(\mathcal O(2)\). Tensoring the whole bundle by \(\mathcal O(-1)\) recovers original determinant-one extensions with lines \(\mathcal O(-2),\mathcal O(2)\). The root line stays \(\mathcal O(-4)\), so its extension parameter dimension is \(h^1(\mathcal O(-4))=3\). The inverse-unit maps and Q.3.4 retain the determinant frame, not just its degree.

Theorems Q.2.1.1–Q.3.1.1 transport every highest-line map and its inverse units. The bundle lines and extension dimension are the ones computed in §Q.7, and Theorem Q.6.1.1 gives the raw local degree, twist and actual Frobenius. The arrows are torsor tensoring and exact coefficient comparisons; no unidentified geometric deformation is depicted. Editable SVG source.

Q.8. Exercises

Exercise Q.8.1 (medium). For original \(SL_2\), let \(\lambda=m\alpha^\vee,\mu=n\alpha^\vee,\nu=r\alpha^\vee,c\ge0\), with \(m\ge0\), \(2n+c\ge0\), \(2(n+r)+c\ge0\). Compute every group and the obstruction to untwisting its block.

Solution. Put \(a=2m,b=2n+c,e=2(n+r)+c\). Coefficient comparison of the finite rank-one weight sums, as proved in P.11.1, gives derived highest weights \(a+b,a+b-2,\ldots,|a-b|\), each once. Source and target have the same full central character \(c\). Hence the multiplicity is one precisely when \[ |2m-(2n+c)|\le2(n+r)+c\le2m+2n+c. \tag{Q.8.1} \] The step-two parity is automatic. Otherwise every group is zero. Here \(h_\nu=2r,D=m+r\); the only possible group is \(H_c^{2r}=E(-(m+r))\), with actual operator \(q^{m+r}\). An original lift exists exactly for even \(c=2s\). The determinant character in Q.6.5 then gives the original \(PGL_2\) Hom with labels \(m,n+s,n+r+s\). For odd \(c\) no one-dimensional block object exists: (Q.7.3)'s two coordinates are equal only when \(2m+c=0\), impossible for integral \(m\). A half-coroot cannot remove the obstruction.

Exercise Q.8.2 (advanced). In characteristic two retain the entire original centre using dual numbers, and compare both full central characters in the conductor-one tensor Hom.

Solution. In \(A=k[\epsilon]/\epsilon^2\), \(1+\epsilon\) is its own inverse and has square one. The matrix \((1+\epsilon)I_2\) is a determinant-one scalar distinct from the identity. Its auxiliary quotient is one, so it remains in the original kernel; its geometric point would erase it. In the characteristic-zero coefficient group the auxiliary central torus acts with degree zero on \(\operatorname{Ad}\), degree one on \(\operatorname{Std}\). Source \(\operatorname{Ad}\otimes\operatorname{Std}\) and target \(\operatorname{Std}\) thus have the same full character, and root-lattice difference \((1,0)\). Their Hom is one-dimensional by Q.8.1. The nonreduced geometric centre and the coefficient-field central character are distinct objects, retained in their respective exact comparisons.

Exercise Q.8.3 (advanced). For original \(SL_3\), use diagonal torus lines \(\mathcal O(-4),\mathcal O,\mathcal O(4)\). Let \(\omega_1\) have a double zero at \(y\), \(\omega_2\) at distinct \(x\). Determine the conductor lifts and the dimensions of the genuine stratum, the two individual coroot defects and their sum. Explain why relevance alone does not kill these boundaries.

Solution. The simple-root degrees are \(-4,-4\), so both zero divisors have degree two. Adjoint coordinates are \(\delta_x=(0,2),\delta_y=(2,0)\). Solving the original Cartan equations gives simple-coroot coordinates \((2/3,4/3),(4/3,2/3)\). Neither is integral; their sum is \((2,2)=2\rho^\vee\). Both \((0,\delta_x),(0,\delta_y)\) are integral dominant enlarged labels. The background diagonal lines are \(\mathcal O(-2),\mathcal O,\mathcal O(2)\). Thus the central-twist bundle has these same degrees, namely \(-2\rho^\vee\), as Q.3.4 requires.

The positive-root degrees are \(-4,-4,-8\), with \(H^1\) dimensions \(3,3,7\), giving genuine dimension thirteen and no \(H^0\). Defect \(\alpha_2^\vee[x]\) changes them to \(-5,-2,-7\), giving dimension eleven; \(\alpha_1^\vee[y]\) gives \(-2,-5,-7\), also eleven. Together they give \(-3,-3,-6\), hence dimension nine, by N.5's entire unipotent tower calculation. O.4 shows all three defects are relevant: the remaining simple-root orders are nonnegative at their supports, including the negative off-diagonal Cartan pairing. Thus the stabilizer character alone does not force vanishing. Q.3 does: it transports the marked clean extension with actual enlarged dominant labels. The original \(SL_3\) and its full \(\mu_3\), including in characteristic three, remain the exact kernel. ∎

The global statements concern regular differentials on \(\mathbf P^1\) in the simply-connected-derived compactification already constructed. The local formula has no derived-group restriction. These proofs do not construct the characteristic-zero half-twisted derived Whittaker category, a full moving-point category equivalence, or its Ran enhancement.

11. What this lesson does not prove

Theorem 2.0 proves the actual plane-origin and exceptional-line supported comparisons, including their coefficient-system and Frobenius compatibility. Proposition 2.1 and §4 consequently prove the quadratic-cone étale IC and the Frobenius-compatible exceptional-curve splitting. The \(GL_2\) indicator product follows from lattice-chain counts and the bounded proper-fibre trace calculation, and the torus case is proved directly. General geometric IC stalk and costalk parity over every algebraically closed ground field with \(\ell\) invertible is proved in Identifying the dual group, Appendices C–D. Appendix A proves the actual scalar-Frobenius operator on all IC weight spaces and the canonical normalized symmetric tensor Weil lift. Appendix B proves the actual compact-support trace identity, the full spherical Satake algebra isomorphism, and the convolution identity for normalized IC traces. Appendix C proves split Tate Frobenius on every IC stalk and costalk. Appendix D proves the full orbit-indicator formula and identifies every graded IC stalk multiplicity with the polynomial (6.2). Appendix E proves geometric character vanishing, every nondominant-conductor case of (7.2), its upper-degree bound and zero-character top-component criterion, the actual top Frobenius scalar, and every highest-slice case. Appendix F proves the general last-residue top obstruction and its complete lowest-slice case. Appendix G constructs the relevant global strata and evaluation, their geometric self-maps, the actual finite Hecke chart and normalized coefficient comparisons, and its strict relevant-stratum bound. Appendix H constructs both projective modification projections, coherent Hecke transport on meromorphic decoration families, and a smooth central-torus reduction with actual Schubert, slice, coefficient and Weil comparisons. Appendix I proves actual finite section and polynomial decoration schemes, their arbitrary-base-change comparisons and closed pole transitions, and the open sheaf-injection/parameter-flat-quotient condition. Appendix J constructs the full finite highest-line algebra, its affine flag cone and actual flag-torsor open, finite equations for the entire datum, and canonical all-root saturation on geometric curve points. Appendix K constructs the full fixed-bundle injection parameter scheme, the actual projective auxiliary torus quotient and relative fixed-defect strata with every higher divisibility equation on arbitrary parameter rings. Appendix L constructs the genuine-stratum finite jet presentations, their entire unipotent quotient stacks and explicit dimensions, actual cohomology and self-extensions, and intrinsic normalized stratum duality. Appendix M constructs the separated compactified global jet schemes and actual quotient stack presentations, and proves that the entire genuine-reduction open immersion is representable and affine. Appendix N proves actual rational affine perverse exactness, compactified quotient perverse/duality descent, strict intermediate-extension boundary inequalities and the complete defect-stratum dimensions. Appendix O constructs the actual all-base meromorphic unipotent action and full homogeneous fibres and stabilizers, classifies both boundary restrictions by the residue character, proves finite relevant support, and establishes basic zero-conductor cleanness. Appendix P proves sharp closed Hecke transport and duality, basic generation, rigid adjunction, geometric semisimplicity, an actual finite standard filtration and marked cleanness/category. It proves full local Whittaker concentration and tensor multiplicities and transfers that local formula to every original split connected reductive group.

The algebraic character transition (6.11) is proved for every dual root datum and lattice in §§6.1–6.3. Appendix D proves the general stalk identity (6.3) in the arithmetic setting of §1. The compactified global jet presentations and affine genuine-open embeddings in Appendix M extend the fixed-bundle construction in the stated simply-connected-derived, regular setting. The actual rational coefficient perverse-image theorem and strict ordinary/exceptional intermediate-extension bounds are proved in Appendix N. The actual meromorphic action and full relevant/irrelevant boundary restrictions are proved in Appendix O, together with basic zero-conductor cleanness. Appendix P proves Hecke semisimplicity and cleanness for the dominant fixed-marked twists of the zero-conductor background, in the simply-connected-derived compactification setting. Appendix Q proves cleanness for every nonzero regular differential conductor in this global setting, retains the exact original-group kernel and all-base decoration data, and identifies the fixed-conductor geometric category with its actual enlarged-dual central-character blocks. It also proves the full arbitrary-conductor local cohomology formula for every original split connected reductive group, including conductors that do not lift to its original lattice. A global compactification/category transfer to an arbitrary derived group remains separate. Genuine-stratum duality is constructed in Appendix L. The relative defect strata have their exact scheme equations; a nilpotent family may cross them and remains in the ambient compactification. The full local Whittaker cohomology theorem (7.2), including all degrees and the original-group tensor multiplicity with its actual Weil operator, is proved in Appendix P. The derived D-module factorization equivalence (7.4) and the arithmetic Fargues–Scholze equivalence remain to be proved. Their statements have the displayed hypotheses and free locators. They are not used in the proofs of the rank-one or torus calculations. A full Ran-category construction and a six-functor theory on \(\mathrm{Bun}_G\) also remain unfinished.

References

• Edward Frenkel, Lectures on the Langlands program and conformal field theory, §§5.3–5.6: normalized functions, geometric Satake and Hecke kernels.
• Daniel Juteau, Carl Mautner and Geordie Williamson, Parity sheaves, Corollaries 2.8–2.9: the odd-Hom and open-restriction lifting framework.
• Thomas J. Haines, Robert E. Kottwitz and Amritanshu Prasad, Iwahori-Hecke Algebras, §§1.3–1.15: the universal unramified module and the rank-one intertwiner.
• George Lusztig, Singularities, character formulas, and a q-analogue of weight multiplicities, free author copy, equations (9.3)–(9.4), §11(c), printed pp. 225–227: polynomial and IC conventions.
• Dmitri Panyushev, On Lusztig's q-analogues of all weight multiplicities of a representation, §1, equation (1.1): the positive-power partition formula.
• Edward Frenkel, Dennis Gaitsgory and Kari Vilonen, Whittaker patterns in the geometry of moduli spaces of bundles on curves, Theorem 1, equation (1.7); Definition 4.2.2, Theorems 3–4 and §5.4.4: the twisted cohomology theorem and the globalized abelian Whittaker category.
• Dima Arinkin and collaborators, Proof of the geometric Langlands conjecture II: Kac–Moody localization and the FLE, §§1.3–1.4, especially Theorem 1.4.2 and Remark 1.4.6: the derived half-twisted factorization statement.
• Laurent Fargues and Peter Scholze, Geometrization of the local Langlands correspondence, VI.9–VI.11: relative arithmetic fusion, Weil actions and the reconstructed dual group.