Asymptotic Hecke algebras and involutions

TL;DR

This paper explores the relationship between a Hecke algebra module structure on involutions in Weyl groups and the asymptotic Hecke algebra, extending previous work to deepen understanding of their algebraic connections.

Contribution

It establishes a link between the involution-based Hecke module and the asymptotic Hecke algebra, advancing the theoretical framework in algebraic representation theory.

Findings

Established a connection between involution modules and asymptotic Hecke algebras

Extended previous work on Hecke algebra modules in Weyl groups

Provided new insights into the structure of involutions in algebraic groups

Abstract

In a previous paper the author and D. Vogan defined and studied a Hecke algebra module structure on a vector space spanned by the involutions in a Weyl group. In this paper this study is continued by relating it to the asymptotic Hecke algebra.