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Schwartz idempotents and lattice projections

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

In the fiber at \(t\), the twisted algebra \(C_r^*(\bar E_{\mathbb Z},tc)\) must contain a nonzero projection of trace about \(|t|^{n_{\mathrm f}}\). The continuous group \(\bar E\) has a twisted representation on \(L^2(\mathfrak n)\) in which Schwartz functions on \(\bar E\) correspond exactly to Schwartz integral kernels, so the rank-one projection onto a Gaussian comes from a Schwartz idempotent for twisted convolution. Rescaling it by the dilations of the grading and sampling it on the lattice \(\bar E_{\mathbb Z}\) gives an almost idempotent element of the discrete twisted algebra, which functional calculus turns into a projection; its trace is controlled by the \(\ell^2\) norm of the samples, which is about the square root of the cell volume [OpenAI-KK, Section 8.4, Lemmas 8.5 to 8.8].

The notation of An affine nilpotent group is in force: \(E\), \(\bar E\), the cocycle \(c\), the dilations \(\Delta_y\), Haar measure in coordinates, the lattice \(L=\bar E_{\mathbb Z}\), and \(\mathbf e(v)=e^{2\pi iv}\). A Schwartz function on \(\bar E\) is one in the Euclidean coordinates \((\lambda,b-1)\in\mathfrak n^*\times\mathfrak n\). We use that the Fourier transform \(\hat f(\xi)=\int f(\lambda)\mathbf e(\lambda(\xi))\,d\lambda\) is a bijection of the Schwartz space onto itself and preserves \(L^2\) norms, also as a partial Fourier transform in some of the variables: in the Hermite basis it acts diagonally by unimodular factors, which preserves the rapidly decreasing coefficient sequences (Theorem 3.2, Proposition 2.2 and Corollary 4.2 of Hermite functions, tempered distributions and the Schwartz kernel theorem, after the change of variables \(x\mapsto2\pi x\) that relates the two normalizations).

1. A Schwartz idempotent

For a sign \(\epsilon\in\{1,-1\}\) and functions on \(\bar E\), define the twisted convolution and involution \[ \begin{aligned} &(J*_\epsilon K)(h)\\ &\quad=\int_{\bar E}J(a)K(a^{-1}h)\,\mathbf e\bigl(\epsilon c(a,a^{-1}h)\bigr)\,da,\\ &J^{*_\epsilon}(h)=\overline{J(h^{-1})}\,\mathbf e\bigl(-\epsilon c(h^{-1},h)\bigr) . \end{aligned} \]

Lemma 1.1 (Schwartz idempotents; [OpenAI-KK, Lemma 8.5]). For each sign \(\epsilon\) there is a Schwartz function \(J_\epsilon\) on \(\bar E\) with \[ \begin{gathered} J_\epsilon^{*_\epsilon}=J_\epsilon,\qquad J_\epsilon*_\epsilon J_\epsilon=J_\epsilon,\\ \|J_\epsilon\|_{L^2(\bar E)}=1 . \end{gathered} \]

Proof. For \(\gamma=(\lambda,b)\) define on \(L^2(\mathfrak n)\) \[ (\pi_\epsilon(\gamma)\phi)(x)=\mathbf e(\epsilon\lambda(x))\,\phi(A_{b^{-1}}x) . \] These operators are unitary, because \(A_b\) preserves Lebesgue measure, and since \(\mu(A_{b^{-1}}x)=c(\gamma,\eta)+(\mu\circ L_{b^{-1}})(x)\) for \(\eta=(\mu,b')\), \[ \pi_\epsilon(\gamma)\pi_\epsilon(\eta)=\mathbf e\bigl(\epsilon c(\gamma,\eta)\bigr)\pi_\epsilon(\gamma\eta) . \] For an integrable \(J\) let \(\pi_\epsilon(J)=\int J(\gamma)\pi_\epsilon(\gamma)\,d\gamma\), of norm at most \(\|J\|_1\). By Fubini's theorem and the left invariance of Haar measure, \(\pi_\epsilon(J)\pi_\epsilon(K)=\pi_\epsilon(J*_\epsilon K)\); and since \(\pi_\epsilon(a)^*=\mathbf e(-\epsilon c(a^{-1},a))\pi_\epsilon(a^{-1})\) and inversion preserves Haar measure, \(\pi_\epsilon(J)^*=\pi_\epsilon(J^{*_\epsilon})\).

The kernel. Let \(\hat J_+(\xi,b)=\int_{\mathfrak n^*}J(\lambda,b)\mathbf e(\lambda(\xi))\,d\lambda\) be the partial Fourier transform. Integrating first in \(\lambda\), \[ (\pi_\epsilon(J)\phi)(x)=\int_U\hat J_+(\epsilon x,b)\,\phi(A_{b^{-1}}x)\,db , \] and the substitution \(y=b^{-1}(1+x)-1\), that is \(b=(1+x)(1+y)^{-1}\), which for fixed \(x\) has Jacobian one, gives the integral kernel \[ \begin{aligned} &\mathcal K_J(x,y)\\ &=\hat J_+\bigl(\epsilon x,\\ &\qquad(1+x)(1+y)^{-1}\bigr) . \end{aligned}\tag{1.1} \] The map \((x,y)\mapsto(\epsilon x,(1+x)(1+y)^{-1}-1)\) and its inverse are polynomial, with Jacobian of absolute value one. Composition with such a map preserves Schwartz functions: derivatives of the composite are polynomial combinations of derivatives of the function, and a polynomial weight in one set of coordinates is bounded by a polynomial weight in the other, because the inverse map is polynomial. Together with the properties of the partial Fourier transform, \(J\mapsto\mathcal K_J\) is a bijection from Schwartz functions on \(\bar E\) onto Schwartz functions on \(\mathfrak n\times\mathfrak n\), and it preserves \(L^2\) norms. An operator determines its continuous kernel: if a continuous kernel gives the zero operator, its pairings with all products of test functions vanish, and so does the kernel.

Twisted convolution and involution preserve Schwartz functions: \((a,h)\mapsto J(a)K(a^{-1}h)\mathbf e(\epsilon c(a,a^{-1}h))\) is Schwartz on \(\bar E\times\bar E\), since \((a,h)\mapsto(a,a^{-1}h)\) is polynomial with polynomial inverse and the phase has polynomial derivatives, and integrating in \(a\) keeps rapid decay of all derivatives in \(h\).

Let \(\phi(x)=2^{n_{\mathrm f}/4}e^{-\pi|x|^2}\), of norm one, so that the orthogonal projection onto its span has the Schwartz kernel \(\phi(x)\overline{\phi(y)}\), and let \(J_\epsilon\) be the Schwartz function with this kernel. The operator of \(J_\epsilon*_\epsilon J_\epsilon\) is the square of the projection and that of \(J_\epsilon^{*_\epsilon}\) its adjoint, so both equal the projection, and by injectivity both functions equal \(J_\epsilon\). Finally \(\|J_\epsilon\|_2^2=\iint|\phi(x)|^2|\phi(y)|^2\,dx\,dy=1\). \(\square\)

2. Weighted Riemann sums

Lemma 2.1 (weighted Riemann sums; [OpenAI-KK, Lemma 8.6]). Let \(w_1,\ldots,w_k\) be positive integers, and for \(0<y\le1\) let \(\mathcal L_y=\prod_ay^{w_a}\mathbb Z\subseteq\mathbb R^k\), with cell volume \(v_y=y^{w_1+\cdots+w_k}\). For a Schwartz function \(f\) on \(\mathbb R^k\), \(v_y\sum_{a\in\mathcal L_y}f(a)\to\int f\) as \(y\to0\), and \(v_y\sum_{a\in\mathcal L_y}|f(a)|\) is bounded. For a Schwartz function \(G(a,h)\) on \(\mathbb R^k\times\mathbb R^k\) and every \(M\), \[ \begin{aligned} &\sup_h(1+|h|)^M\\ &\quad\cdot\Bigl|v_y\sum_{a\in\mathcal L_y}G(a,h)-\int G(a,h)\,da\Bigr|\\ &\quad\le C_{G,M}\,y . \end{aligned} \]

Proof. The cells \(Q_a=a+\prod_j[0,y^{w_j})\), \(a\in\mathcal L_y\), tile \(\mathbb R^k\), have volume \(v_y\) and diameter at most \(\sqrt ky\). For \(x\in Q_a\), the mean value theorem and the Schwartz bounds give \[ \begin{aligned} &|G(a,h)-G(x,h)|\\ &\quad\le\sqrt ky\sup_{z\in Q_a}|\nabla_aG(z,h)|\\ &\quad\le C_{K,M}\,y(1+|a|)^{-K}(1+|h|)^{-M} , \end{aligned} \] because \(1+|z|\) and \(1+|a|\) are comparable on a cell. For \(K>k\), \(v_y\sum_a(1+|a|)^{-K}\le C\int(1+|x|)^{-K}dx\), which is finite. Integrating the difference over each cell and summing gives the estimate; the other statements follow in the same way. \(\square\)

3. Discrete twisted projections

For real \(\theta\), let \(C_r^*(L,\theta c)\) be the norm closure of the linear span of the unitaries \(u_\theta(\gamma)\), \(\gamma\in L\), on \(\ell^2(L)\). For \(a\in\ell^1(L)\) the series \(\sum_\gamma a_\gamma u_\theta(\gamma)\) converges in norm, with norm at most \(\sum_\gamma|a_\gamma|\), the coefficient \(\ell^1\) norm. Since \(c(\gamma,1)=0\), it maps \(\delta_1\) to \(\sum_\gamma a_\gamma\delta_\gamma\), whose norm is the coefficient \(\ell^2\) norm. Its identity trace \(\operatorname{tr}_\theta\), the coefficient \(a_1\), satisfies \(\operatorname{tr}_\theta(A^*A)=\|A\delta_1\|^2\).

Proposition 3.1 (nonzero discrete twisted projections; [OpenAI-KK, Proposition 8.7]). There are \(\theta_0>0\) and \(C<\infty\), depending only on \(m\) and the functions \(J_\pm\), such that the following holds for \(0<|\theta|<\theta_0\). Put \(\epsilon=\operatorname{sign}\theta\), \(y=|\theta|^{1/m}\), \(\nu=y^{mn_{\mathrm f}}=|\theta|^{n_{\mathrm f}}\), and \[ K_\theta=\sum_{\gamma\in L}\nu J_\epsilon(\Delta_y\gamma)\,u_\theta(\gamma) . \] Then \(K_\theta\) is self-adjoint, its coefficient \(\ell^1\) norm is at most \(C\), its coefficient \(\ell^2\) norm lies between \(C^{-1}\sqrt\nu\) and \(C\sqrt\nu\), and \(K_\theta^2-K_\theta\) has coefficient \(\ell^1\) norm at most \(Cy\) and coefficient \(\ell^2\) norm at most \(Cy\sqrt\nu\). The continuous functional calculus gives a projection \(p_\theta\in C_r^*(L,\theta c)\) with \[ \begin{gathered} \|p_\theta-K_\theta\|\le Cy,\\ 0<\operatorname{tr}_\theta(p_\theta)\le C|\theta|^{n_{\mathrm f}} . \end{gathered} \] The coefficients of \(K_\theta\) and the operators \(p_\theta\) depend measurably on \(\theta\).

Proof. In the coordinates of \(\bar E\), \(\Delta_yL\) is the rectangular lattice with spacing \(y^w\) in each coordinate of weight \(w\), of cell volume \(\nu\) by Lemma 2.1 of the previous lesson. Lemma 2.1, applied to \(J_\epsilon\) and \(|J_\epsilon|^2\), bounds the coefficient \(\ell^1\) norm and shows that \[ \begin{aligned} \nu^{-1}\sum_\gamma|\nu J_\epsilon(\Delta_y\gamma)|^2&=\nu\sum_{a\in\Delta_yL}|J_\epsilon(a)|^2\\ &\to\|J_\epsilon\|_2^2=1 , \end{aligned} \] which gives the \(\ell^2\) bounds for small \(\theta\).

Self-adjointness. \[ \begin{aligned} K_\theta^*&=\sum_\gamma\nu\overline{J_\epsilon(\Delta_y\gamma)}\\ &\qquad\cdot\mathbf e(-\theta c(\gamma^{-1},\gamma))u_\theta(\gamma^{-1}) . \end{aligned} \] Since \(\Delta_y\) is an automorphism and \(\epsilon c(\Delta_y\gamma^{-1},\Delta_y\gamma)=\epsilon y^mc(\gamma^{-1},\gamma)\), which is \(\theta c(\gamma^{-1},\gamma)\), the identity \(J_\epsilon=J_\epsilon^{*_\epsilon}\) says exactly that the coefficient of \(u_\theta(\gamma)\) in \(K_\theta^*\) is \(\nu J_\epsilon(\Delta_y\gamma)\).

The defect. By the multiplier formula, the coefficient of \(u_\theta(\gamma)\) in \(K_\theta^2\), divided by \(\nu\), is, with \(h=\Delta_y\gamma\), \[ \nu\sum_{a\in\Delta_yL}J_\epsilon(a)J_\epsilon(a^{-1}h)\,\mathbf e\bigl(\epsilon c(a,a^{-1}h)\bigr), \] a Riemann sum for \((J_\epsilon*_\epsilon J_\epsilon)(h)=J_\epsilon(h)\). Its difference \(D_y(h)\) from \(J_\epsilon(h)\) satisfies \(|D_y(h)|\le C_My(1+|h|)^{-M}\) by Lemma 2.1, so the coefficients \(\nu D_y(\Delta_y\gamma)\) of \(K_\theta^2-K_\theta\) have \(\ell^1\) norm at most \(Cy\) and \(\ell^2\) norm at most \(Cy\sqrt\nu\).

The projection. Since the coefficient \(\ell^1\) norm bounds the operator norm, \(\|K_\theta^2-K_\theta\|\le Cy\), and every \(\lambda\) in the spectrum of the self-adjoint \(K_\theta\) satisfies \(|\lambda^2-\lambda|\le Cy\). Outside \([-\frac14,\frac14]\cup[\frac34,\frac54]\) one has \(|\lambda^2-\lambda|>\frac3{16}\) (Exercise 5.3), so for small \(y\) the spectrum lies in these two intervals. Let \(\chi\) be continuous, \(0\) on the first interval and \(1\) on the second, and \(p_\theta=\chi(K_\theta)\), a projection in \(C_r^*(L,\theta c)\). On the two intervals \(\chi(\lambda)-\lambda=(\lambda^2-\lambda)f(\lambda)\) with \(f(\lambda)=(1-\lambda)^{-1}\), respectively \(f(\lambda)=-\lambda^{-1}\), and \(|f|\le2\), so \(\|p_\theta-K_\theta\|\le2Cy\).

If \(p_\theta=0\), the spectrum lies in \([-\frac14,\frac14]\), so \(\|K_\theta\|\le\frac14\) and \(\|(K_\theta^2-K_\theta)\delta_1\|\ge\frac34\|K_\theta\delta_1\|\); for small \(y\) this contradicts \(\|(K_\theta^2-K_\theta)\delta_1\|\le Cy\sqrt\nu\) and \(\|K_\theta\delta_1\|\ge C^{-1}\sqrt\nu\). On the two intervals \(0\le\chi(\lambda)\le4\lambda^2\), so \(0\le p_\theta\le4K_\theta^2\) and \[ \operatorname{tr}_\theta(p_\theta)\le4\operatorname{tr}_\theta(K_\theta^2)=4\|K_\theta\delta_1\|^2\le C\nu . \] A nonzero projection commuting with the right operators \(v_\theta\) has positive identity trace (Lemma 2.1(3) of Separated bands and equivariant traces). The coefficients are continuous in \(\theta\); norm-convergent coefficient sums give measurable fields, and so does \(\chi\) applied to them, by uniform polynomial approximation of \(\chi\) on \([-2,2]\). \(\square\)

4. Compact templates

Lemma 4.1 (compact templates and scale changes; [OpenAI-KK, Lemma 8.8]). For every \(\varepsilon_0>0\) there are smooth compactly supported \(J_\pm^\flat\) such that, for small \(|\theta|\), replacing \(J_\epsilon\) by \(J_\epsilon^\flat\) in \(K_\theta\) changes the coefficients by \(\ell^1\) norm at most \(\varepsilon_0\). For fixed such \(J^\flat\), and \(\alpha,\beta>0\) small with \(|\log(\alpha/\beta)|\le a\le1\), the coefficient families \(\gamma\mapsto\alpha^{n_{\mathrm f}}J^\flat(\Delta_{\alpha^{1/m}}\gamma)\) and \(\gamma\mapsto\beta^{n_{\mathrm f}}J^\flat(\Delta_{\beta^{1/m}}\gamma)\) differ by \(\ell^1\) norm at most \(Ca\). These coefficient bounds bound the norm of the corresponding sums of any unitaries.

Proof. Multiply \(J_\epsilon\) by a smooth function equal to \(1\) on a large ball; by Lemma 2.1 and the rapid decay of \(J_\epsilon\), the normalized lattice sum of the discarded part is small, uniformly for small mesh. For the second statement put \(r=\alpha/\beta\) and \(h=\Delta_{\beta^{1/m}}\gamma\); after the factor \(\beta^{n_{\mathrm f}}\), the difference is \(r^{n_{\mathrm f}}J^\flat(\Delta_{r^{1/m}}h)-J^\flat(h)\). For \(e^{-1}\le r\le e\) it is supported in one compact set, and by the mean value theorem in \(r\) it is at most \(C|\log r|\); the number of points of \(\Delta_{\beta^{1/m}}L\) in a compact set, times \(\beta^{n_{\mathrm f}}\), is bounded. \(\square\)

5. Exercises

5.1 (easy). Check that \(\phi(x)=2^{n_{\mathrm f}/4}e^{-\pi|x|^2}\) has \(L^2\) norm one on \(\mathbb R^{n_{\mathrm f}}\).

5.2 (medium). For \(m=2\), so that \(\bar E\cong\mathbb R^2\) with \(c((\lambda,\beta),(\mu,\beta'))=-\mu\beta\), write \(\pi_\epsilon(\lambda,\beta)\) on \(L^2(\mathbb R)\) and recognize the Schrödinger representation.

5.3 (medium). Show that for a self-adjoint operator \(K\) with \(\|K^2-K\|\le\delta<\frac3{16}\), the spectrum lies in \([-\frac14,\frac14]\cup[\frac34,\frac54]\).

5.4 (medium). For the one-dimensional lattice \(y\mathbb Z\) and \(f(x)=e^{-\pi x^2}\), compare \(y\sum_{a\in y\mathbb Z}f(a)\) with \(\int f=1\) for \(y=1\) and \(y=\frac12\).

6. Solutions

5.1. \(\int2^{n_{\mathrm f}/2}e^{-2\pi|x|^2}dx=2^{n_{\mathrm f}/2}(2^{-1/2})^{n_{\mathrm f}}=1\).

5.2. \(A_{b^{-1}}x=x-\beta\) for \(b=1+\beta\), so \((\pi_\epsilon(\lambda,\beta)\phi)(x)=\mathbf e(\epsilon\lambda x)\phi(x-\beta)\): modulations and translations, the Schrödinger representation of the Heisenberg group with Planck constant \(\epsilon\).

5.3. For \(\lambda\) in the spectrum, \(|\lambda^2-\lambda|\le\delta\). If \(\lambda\) were in \((\frac14,\frac34)\), or below \(-\frac14\), or above \(\frac54\), then \(|\lambda||\lambda-1|>\frac14\cdot\frac34=\frac3{16}\).

5.4. By Poisson summation \(y\sum_nf(ny)=\sum_ke^{-\pi k^2/y^2}\). For \(y=1\) this is \(1+2e^{-\pi}+\cdots\approx1.0864\); for \(y=\frac12\), \(1+2e^{-4\pi}+\cdots\approx1.0000070\). The error decreases much faster than the bound \(Cy\) of Lemma 2.1, which holds for all Schwartz functions.

References