Geodesic
On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes
Documenta mathematica, Tome 1 (1996), pp. 293-317.

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Summary: 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.
Classification : 20C33, 20G40 
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     author = {Geck, Meinolf},
     title = {On the average values of the irreducible characters of finite groups of {Lie} type on geometric unipotent classes},
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     url = {http://geodesic.mathdoc.fr/item/DOCMA_1996__1__a5/}
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Geck, Meinolf. On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes. Documenta mathematica, Tome 1 (1996), pp. 293-317. http://geodesic.mathdoc.fr/item/DOCMA_1996__1__a5/