Cells and the reflection representation of Weyl groups and Hecke algebras

Abstract

Let H \mathcal {H} be the generic algebra of the finite crystallographic Coxeter group W W , defined over the ring Q [ u 1 / 2 , u − 1 / 2 ] \mathbb {Q}[{u^{1/2}},{u^{ - 1/2}}] . First, the two-sided cell corresponding to the reflection representation of H \mathcal {H} is shown to consist of the nonidentity elements of W W having a unique reduced expression. Next, the matrix entries of this representation are computed in terms of certain Kazhdan-Lusztig polynomials. Finally, the Kazhdan-Lusztig polynomials just mentioned are described in case W W is of type A l − 1 {{\text {A}}_{l - 1}} or B l {{\text {B}}_l} .