Action of longest element on a Hecke algebra cell module

TL;DR

This paper generalizes Mathas's result on the action of the longest element in a Coxeter group's Hecke algebra to cases with unequal parameters, providing explicit formulas for its action on cell modules and in the asymptotic algebra.

Contribution

It extends known results to unequal parameter Hecke algebras and derives a simple formula for the image of the longest element in the asymptotic algebra.

Findings

Action of T_{w_0} is a permutation matrix times a power of v in the canonical basis.

Generalization to unequal parameter case.

Explicit formula for the image in the asymptotic Hecke algebra.

Abstract

By a result of Mathas, the basis element T_{w_0} of the Hecke algebra of a finite Coxeter group acts in the canonical basis of a cell module as a permutation matrix times plus or minus a power of v. We generalize this result to the unequal parameter case,. We also show that the image of T_{w_0} in the corresponding asymptotic Hecke algebra is given by a simple formula.