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On the average values of the irreducible characters of finite groups of Lie type on geometric unipotent classes. (English) Zbl 0873.20011
Summary: In 1980, G. Lusztig [Proc. Symp. Pure Math. 37, 313-317 (1980; Zbl 0453.20005)] 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 G. Lusztig [Adv. Math. 94, No. 2, 139-179 (1992; Zbl 0789.20042)] 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.

##### MSC:
 20C33 Representations of finite groups of Lie type 20G40 Linear algebraic groups over finite fields 20G05 Representation theory for linear algebraic groups
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