Cells and the Reflection Representation of Weyl Groups and Hecke Algebras

Abstract

Let $\mathcal {H}$ be the generic algebra of the finite crystallographic Coxeter group $W$, defined over the ring $\mathbb {Q}[{u^{1/2}},{u^{ - 1/2}}]$. First, the two-sided cell corresponding to the reflection representation of $\mathcal {H}$ is shown to consist of the nonidentity elements of $W$ 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 $W$ is of type ${{\text {A}}_{l - 1}}$ or ${{\text {B}}_l}$.