Action of longest element on a Hecke algebra cell module

Abstract

By a result of Mathas, the basis element T[subscript w0] 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[subscript w0] in the corresponding asymptotic Hecke algebra is given by a simple formula.