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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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