On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes

Abstract

In 1980, Lusztig posed the problem of showing the existence of a unipotent support for the irreducible characters of a finite reductive group $G({\mathbb{F}}_q)$. This is defined in terms of certain average values of the irreducible characters on unipotent classes. The problem was solved by Lusztig [16] for the case where $q$ is a power of a sufficiently large prime. In this paper we show that, in general, these average values can be expressed in terms of the Green functions of $G$. In good characteristic, these Green functions are given by polynomials in $q$. Combining this with Lusztig's results, we can then establish the existence of unipotent supports whenever $q$ is a power of a good prime.