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Hecke algebras and local Langlands correspondence for non-singular depth-zero representations

Maarten Solleveld, Yujie Xu

Abstract

Let G be a connected reductive group over a non-archimedean local field. We say that an irreducible depth-zero (complex) G-representation is non-singular if its cuspidal support is non-singular. We establish a Local Langlands Correspondence for all such representations. We obtain it as a specialization from a categorical version: an equivalence between the category of finite-length non-singular depth-zero G-representations and the category of finite-length right modules of a direct sum of twisted affine Hecke algebras constructed from Langlands parameters. We also show that our LLC and our equivalence of categories have several nice properties, for example compatibility with parabolic induction.

Hecke algebras and local Langlands correspondence for non-singular depth-zero representations

Abstract

Let G be a connected reductive group over a non-archimedean local field. We say that an irreducible depth-zero (complex) G-representation is non-singular if its cuspidal support is non-singular. We establish a Local Langlands Correspondence for all such representations. We obtain it as a specialization from a categorical version: an equivalence between the category of finite-length non-singular depth-zero G-representations and the category of finite-length right modules of a direct sum of twisted affine Hecke algebras constructed from Langlands parameters. We also show that our LLC and our equivalence of categories have several nice properties, for example compatibility with parabolic induction.

Paper Structure

This paper contains 22 sections, 78 theorems, 469 equations.

Table of Contents

  1. Introduction
  2. Deligne--Lusztig packets for p-adic groups
  3. Embeddings of tori and extensions
  4. Supercuspidal L-parameters of depth zero
  5. Preliminaries
  6. Extensions related to enhancements of L-parameters
  7. An LLC for non-singular depth-zero supercuspidal representations
  8. Some subquotients of the Iwahori--Weyl group
  9. q-parameters for Hecke algebras
  10. The Hecke algebra of a non-singular depth-zero Bernstein block
  11. The supercuspidal case
  12. The non-supercuspidal case
  13. Hecke algebras for non-singular depth-zero Langlands parameters
  14. Preliminaries
  15. Comparison of q-parameters
  16. ...and 7 more sections

Key Result

Theorem 1

(all results in § sec:LLC) There exists a bijection such that:

Theorems & Definitions (150)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 140 more