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The Aubert-Zelevinsky involution for $G_2$ and its associated Hecke algebras

Chuan Qin

TL;DR

This work computes the Aubert–Zelevinsky duality for the exceptional group $G_2$ within Bernstein blocks, focusing on principal and intermediate series. By constructing progenerators for Levi subgroups, it transfers the duality to endomorphism algebras $\mathcal{H}^{\mathfrak{s}}(G)$ and derives explicit involutions on the corresponding affine Hecke algebras. The paper simultaneously verifies numerous instances of Bernstein’s conjecture for $G_2$, distinguishing unitarizable from non-unitarizable duals and providing concrete dual correspondences between $G_2$-representations and Hecke-algebra modules. These results extend Muic’s methodology to the exceptional group $G_2$, offering a detailed, case-by-case atlas of dualities across principal and intermediate blocks and illuminating the Langlands-quotient structure in this setting.

Abstract

Motivated by the recent work of Aubert-Xu and the techniques in G. Muic's article, we provide examples of computations of the Aubert-Zelevinsky duality functor for the principal and mediate series of the exceptional group $G_2$, and deduce corresponding results regarding the involution on the Hecke algebra side. These computations also allow us to confirm several instances of the Bernstein conjecture for $G_2$. This article is developed from part of the author's PhD thesis.

The Aubert-Zelevinsky involution for $G_2$ and its associated Hecke algebras

TL;DR

This work computes the Aubert–Zelevinsky duality for the exceptional group within Bernstein blocks, focusing on principal and intermediate series. By constructing progenerators for Levi subgroups, it transfers the duality to endomorphism algebras and derives explicit involutions on the corresponding affine Hecke algebras. The paper simultaneously verifies numerous instances of Bernstein’s conjecture for , distinguishing unitarizable from non-unitarizable duals and providing concrete dual correspondences between -representations and Hecke-algebra modules. These results extend Muic’s methodology to the exceptional group , offering a detailed, case-by-case atlas of dualities across principal and intermediate blocks and illuminating the Langlands-quotient structure in this setting.

Abstract

Motivated by the recent work of Aubert-Xu and the techniques in G. Muic's article, we provide examples of computations of the Aubert-Zelevinsky duality functor for the principal and mediate series of the exceptional group , and deduce corresponding results regarding the involution on the Hecke algebra side. These computations also allow us to confirm several instances of the Bernstein conjecture for . This article is developed from part of the author's PhD thesis.

Paper Structure

This paper contains 31 sections, 32 theorems, 139 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Notation and preliminaries
  4. Representations of $p$-adic groups
  5. The Aubert-Zelevinsky duality
  6. The Bernstein decomposition
  7. Hecke algebras associated with Bernstein blocks
  8. Principal blocks
  9. Computations of the duality for Bernstein blocks of $G_2$
  10. Preliminaries on $G_2$
  11. Some definitions and abbreviations
  12. Representation theory of $L_{\gamma}$, $\gamma = \alpha$ or $\beta$
  13. Involution for affine Hecke algebras
  14. Case by case discussion
  15. The intermediate case $\mathfrak{s} = [L_{\alpha}, \sigma]_G$ and $\mathfrak{s}=[L_{\beta}, \sigma]_G$
  16. ...and 16 more sections

Key Result

Proposition 3.1

Suppose that $\chi, \chi_1$ and $\chi_2$ are characters of $F^{\times}$, and $\gamma \in$$\{\alpha, \beta\}$. Then we have the following.

Figures (1)

  • Figure :

Theorems & Definitions (47)

  • Definition 2.1
  • Remark
  • Proposition 3.1
  • Definition 3.2: Iwahori-Matsumoto
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • Lemma 3.6
  • Corollary 3.7
  • ...and 37 more