{
  "id": "LG-LLC-02",
  "title": "Irreducible representations of general linear groups over a local field",
  "course": "The local Langlands correspondence for GL_n",
  "language": "en",
  "status": "draft",
  "original_text_license": "CC0-1.0",
  "source": "src/LG-LLC-02.md",
  "reader": "LG-LLC-02.html",
  "source_sha256": "1f4f86cee437a496d98eca1d3780021a80eaf17e377b9d3fd60c5cd39b4e9bcc",
  "download": "LG-LLC-02-source.zip",
  "proved_here": [
    "Jacquet exactness, admissibility, adjunction, block geometric lemma, cuspidal support and finite length (Section1)",
    "Rank-two segment dictionary, Jacquet modules and Steinberg coefficient summation (Sections3–5)",
    "Rank-three constituent structure, multiplicity, unique quotients and genericity (Section6)",
    "Normalized mirabolic derivatives, filtration, Leibniz rule, duality and cuspidal-pair criterion (AppendixA)",
    "All-rank Whittaker uniqueness and irreducible supercuspidal mirabolic restriction (AppendixB)",
    "Distinct-support classification and entire submodule lattice (TheoremsC.2–C.3)",
    "Segment construction and all block Jacquet modules, duals and derivatives (TheoremsC.4–C.5)",
    "Irreducibility of arbitrary unlinked Z-products (TheoremC.7)",
    "Adjacent disjoint Z/Q-pair extensions (PropositionC.8)",
    "Linked overlapping Z-pair theorem and complete two-segment criterion (TheoremC.9 and CorollaryC.10)"
  ],
  "not_proved_here": [
    "Full multisegment and Langlands classifications",
    "Necessity in general products of three or more Z-segments",
    "General Q-product irreducibility and generic classification",
    "All-rank square-integrable and tempered classifications, including the rank-three temperedness problem"
  ],
  "programme_proof_dependencies": [
    {
      "title": "Smooth local representations and the Hecke-module dictionary",
      "source": "../automorphic-forms-and-representations-of-gl2/src/smooth-local-representations-and-the-hecke-module-dictionary.md",
      "reader": "../automorphic-forms-and-representations-of-gl2/smooth-local-representations-and-the-hecke-module-dictionary.html",
      "source_sha256": "3d3a6b2e1de976b8706c5081fdb4b04f80b66b566032923b5bc4c60d6b6b0e69",
      "locators": [
        "Theorems 3.1 and 4.1"
      ]
    },
    {
      "title": "Normalized induction and Jacquet modules",
      "source": "../automorphic-forms-and-representations-of-gl2/src/normalized-induction-and-jacquet-modules.md",
      "reader": "../automorphic-forms-and-representations-of-gl2/normalized-induction-and-jacquet-modules.html",
      "source_sha256": "ff3fd6db78b77fbe924c2c3cadfd03d4fdc4599ef54d0d04fdaea4c865c47e43",
      "locators": [
        "Proposition 1.2"
      ]
    },
    {
      "title": "Whittaker models, Kirillov models and the local classification",
      "source": "../automorphic-forms-and-representations-of-gl2/src/whittaker-models-kirillov-models-and-the-local-classification.md",
      "reader": "../automorphic-forms-and-representations-of-gl2/whittaker-models-kirillov-models-and-the-local-classification.html",
      "source_sha256": "a7945905ab37cda60b64169f1fe3d6a98de670e9acbcfff78b13592a23981004",
      "locators": [
        "Theorem 2.3",
        "Theorem 4.2",
        "Section 5",
        "Theorem 6.1"
      ]
    }
  ],
  "download_scope": "Source and coverage metadata for this lesson only; the linked preceding programme proof modules are not bundled."
}