A unipotent support for irreducible representations
In this paper the author solves Problem II from his earlier work [Proc. Symp. Pure Math. 37, 313-317 (1980; Zbl 0453.20005)]. This problem has to do with the existence (and uniqueness) of a certain unipotent class in a connected algebraic group \(G\) associated with an irreducible complex character of \(G(\mathbf{F}_ q)\). This class is called the unipotent class for the character. In the paper two other maps from the set of irreducible representations of \(G(\mathbf{F}_ q)\) to unipotent classes in \(G\) are studied, namely the one defined by the author himself in his book [Characters of reductive groups over a finite field, Ann. Math. Stud. 107, Princeton (1984; Zbl 0556.20033)] and the so-called wavefront set proposed by \textit{N. Kawanaka} [see e.g. Invent. Math. 84, 575-616 (1986; Zbl 0596.20028)]. Kawanaka has conjectured (and proved in certain cases) that these two maps coincide. One of the results in the paper under review establishes this conjecture in general. The main tool in the paper is a decomposition theorem for the character of a generalized Gelfand-Graev representation in terms of certain intersection cohomology complexes. This is also used to prove a result stated in another paper by the author [J. Fac. Sci., Univ. Tokyo, Sect. IA 36, 297-328 (1989; Zbl 0688.20020)] on the support of character sheaves.
generalized Gelfand-Graev representation
scientific article; zbMATH DE number 97609
scientific article