Automorphisms of Coxeter groups and Lusztig's conjectures for Hecke algebras with unequal parameters

Abstract

Let $(W, S)$ be a Coxeter system, let $G$ be a finite solvable group of automorphisms of $(W, S)$ and let $\varphi$ be a weight function which is invariant under $G$. Let $\varphi_{G}$ denote the weight function on $W^{G}$ obtained by restriction from $\varphi$. The aim of this paper is to compare the $\mathbf{a}$-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated with $(W, \varphi)$ and to $(W^{G}, \varphi_{G})$, provided that Lusztig's Conjectures hold.