An involution for Hecke algebras

TL;DR

This paper extends the Alvis-Curtis duality to affine and finite Hecke algebras, providing new involution theorems and compatibility results that could impact the study of p-adic reductive groups.

Contribution

It introduces two generalizations of the Alvis-Curtis duality for Hecke algebras, including an unequal parameter version and a relative version for finite Hecke algebras, with compatibility results.

Findings

Established an involution theorem for finite Hecke algebras.

Proved compatibility with Aubert-Zelevinsky duality in certain cases.

Extended duality concepts to affine Hecke algebras.

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.