Meinolf Geck

On the Average Values of the Irreducible Characters of Finite Groups of Lie 
Type on Geometric Unipotent Classes 

Documenta Mathematica 1 (1996), 293-317.

In 1980, Lusztig posed the problem of showing the existence of a unipotent 
support for the irreducible characters of a finite reductive group 
$G({\Bbb F}_q)$. 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. 

1991 Mathematics Subject Classification: Primary 20C33, secondary 20G40.