An involution for Hecke algebras

Chuan Qin

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arXiv · 2505.17401

An involution for Hecke algebras

Chuan Qin

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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.

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Chuan Qin. 2025-05-23. An involution for Hecke algebras. https://arxiv.org/abs/2505.17401

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