Unipotent representations: changing q to -q

Abstract

Consider a Chevalley group over a finite field $F_q$ such that the longest element of the Weyl group is central. We construct an involution $\xi\mapsto\xi^!$ of the set of unipotent representations of this group such that the degree polynomial of a unipotent representation $\xi$ is obtained up to sign from the degree polynomial of $\xi^!$ by changing $q$ to $-q$.