DOCUMENTA MATHEMATICA, Vol. 1 (1996), 293-317

Meinolf Geck

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

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.

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