Unipotent representations: changing q to -q

TL;DR

This paper introduces an involution on unipotent representations of certain Chevalley groups over finite fields, revealing a symmetry in their degree polynomials when changing q to -q.

Contribution

It constructs a novel involution on unipotent representations that relates their degree polynomials via q to -q substitution.

Findings

Degree polynomials are related by sign change when q is replaced with -q.

The involution provides new insights into the structure of unipotent representations.

The results apply to Chevalley groups with a central longest Weyl group element.

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 $-q$.