On the values of unipotent characters in bad characteristic

TL;DR

This paper advances the computation of unipotent character values in Chevalley groups over finite fields, especially in bad characteristic, by reducing the problem to prime fields and enabling computer algebra methods for complex cases.

Contribution

It reduces the scalar determination problem to prime fields, facilitating the calculation of unipotent character values in exceptional groups.

Findings

Successfully computed unipotent character values in previously inaccessible cases

Reduced complex scalar determination to prime characteristic cases

Enabled computer algebra methods for exceptional groups

Abstract

Let $G(q)$ be a Chevalley group over a finite field $F_q$. By Lusztig's and Shoji's work, the problem of computing the values of the unipotent characters of $G(q)$ is solved, in principle, by the theory of character sheaves; one issue in this solution is the determination of certain scalars relating two types of class functions on $G(q)$. We show that this issue can be reduced to the case where $q$ is a prime, which opens the way to use computer algebra methods. Here, and in a sequel to this article, we use this approach to solve a number of cases in groups of exceptional type which seemed hitherto out of reach.