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An involution for Hecke algebras

Chuan Qin

TL;DR

The work introduces two duality generalizations for Hecke-algebra representations: an unequal-parameter involution for generalized affine Hecke algebras following S.-I. Kato, and a relative Howlett–Lehrer–style involution linking endomorphism algebras to the ACK duality on Harish-Chandra series. It proves D[M] = [M^{*}] for finite-dimensional modules under a Bernstein–Lusztig presentation, and establishes unitarity results under real parameters. Extending to the p-adic setting, the paper analyzes the Bernstein decomposition and shows an analogous involution on endomorphism algebras, compatible with Aubert–Zelevinsky duality at the level of Bernstein blocks. The results provide a framework to study involution in arbitrary Bernstein blocks for p-adic reductive groups and connect finite-, affine-, and p-adic representation theories through explicit endomorphism-algebra involutions and HL-type formulas.

Abstract

We give two generalizations of the Alvis-Curtis duality for Hecke algebras: an unequal parameter version for the affine Hecke algebras, based on S.-I. Kato's work, and a relative version for finite Hecke algebras, based on Howlett-Lehrer's work. Our results for the finite case focus on the involution theorem for finite Hecke algebras that appear in Howlett-Lehrer's theory, where they proved a version for characters of certain subgroups of a Weyl group. We hope that our results will serve as a stepping stone for the study of involution for an arbitrary Bernstein block in the p-adic reductive group case. We also prove their compatibility with the Alvis-Curtis-Kawanaka duality (Aubert-Zelevinsky duality) when restricted to some Harish-Chandra series (resp. Bernstein blocks). This article is part of the author's PhD thesis.

An involution for Hecke algebras

TL;DR

The work introduces two duality generalizations for Hecke-algebra representations: an unequal-parameter involution for generalized affine Hecke algebras following S.-I. Kato, and a relative Howlett–Lehrer–style involution linking endomorphism algebras to the ACK duality on Harish-Chandra series. It proves D[M] = [M^{*}] for finite-dimensional modules under a Bernstein–Lusztig presentation, and establishes unitarity results under real parameters. Extending to the p-adic setting, the paper analyzes the Bernstein decomposition and shows an analogous involution on endomorphism algebras, compatible with Aubert–Zelevinsky duality at the level of Bernstein blocks. The results provide a framework to study involution in arbitrary Bernstein blocks for p-adic reductive groups and connect finite-, affine-, and p-adic representation theories through explicit endomorphism-algebra involutions and HL-type formulas.

Abstract

We give two generalizations of the Alvis-Curtis duality for Hecke algebras: an unequal parameter version for the affine Hecke algebras, based on S.-I. Kato's work, and a relative version for finite Hecke algebras, based on Howlett-Lehrer's work. Our results for the finite case focus on the involution theorem for finite Hecke algebras that appear in Howlett-Lehrer's theory, where they proved a version for characters of certain subgroups of a Weyl group. We hope that our results will serve as a stepping stone for the study of involution for an arbitrary Bernstein block in the p-adic reductive group case. We also prove their compatibility with the Alvis-Curtis-Kawanaka duality (Aubert-Zelevinsky duality) when restricted to some Harish-Chandra series (resp. Bernstein blocks). This article is part of the author's PhD thesis.

Paper Structure

This paper contains 31 sections, 51 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. History and background
  3. Main results
  4. Involution for unequal parameter generalized affine Hecke algebras
  5. The relative version of involution via comparison: the finite case and the affine case
  6. General setup
  7. Notation and conventions
  8. Root systems and affine Weyl groups
  9. Affine Hecke algebras
  10. The Involution theorem d'après S-I. Kato
  11. Proof of the main Theorem
  12. A Key Lemma
  13. The involution using the Bernstein-Lusztig presentation
  14. Unitarity of the involution
  15. A relative version of the involution via the Howlett-Lehrer theory
  16. ...and 16 more sections

Key Result

Theorem 1.1

Define a twisted action $*$ on $\mathcal{H}$ as follows: for set where $q(w) = \prod_{k=1}^r q_{s_{i_k}}$. Given $\mathcal{H}$-module $(\pi, M)$, let $(\pi^{*},M^{*})$ be the $\mathcal{H}$-module such that $M^{*} = M$ as $K$-vector space, equipped with the $\mathcal{H}$-action $\pi^{*}(h)(m) := \pi (h^{*}) (m)$, $\forall m \in M$ and $\forall h \in \mathcal{H}$.

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Generic Hecke algebra
  • Definition 2.2: Finite Hecke algebra
  • Definition 2.3: Iwahori-Matsumoto
  • Definition 2.4: Bernstein-Lusztig
  • Theorem 2.5: Bernstein
  • Theorem 2.6
  • Definition 2.7
  • Proposition 3.1
  • ...and 71 more