An involution for Hecke algebras

2025/05/23 by Qin, Chuan
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2505.17401

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.

Citations

• Some Unipotent Arthur Packets for Reductive p-adic Groups
• The Explicit Local Langlands Correspondence for G₂
• Lie groups and Lie algebras

Related

• The Aubert-Zelevinsky involution for G₂ and its associated Hecke algebras
• Involutions in Coxeter groups
• Asymptotic Hecke algebras and involutions
• An involution based left ideal in the Hecke algebra
• Involutions in Weyl groups and nil-Hecke algebras