Action of longest element on a Hecke algebra cell module. (Q903172)

Mathas has proved that the basis element \(T_{w_0}\) of the one-parameter Hecke algebra (labelled by the longest element \(w_0\)) 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 \(v\). In this paper the author generalizes this result to the unequal parameter case. He also gives a simple formula for the image of \(T_{w_0}\) in the corresponding asymptotic Hecke algebra.