Harish-Chandra Vertices, Green Correspondence in Hecke Algebras, and Steinbergs Tensor Product Theorem in Nondescribing Characteristic

Abstract

This is a survey on some recent results in representation theory of finite groups of Lie type which were derived in joint work with J. Du. We begin with two generalizations of Green’s theory of vertices and sources: First by allowing as vertex of an indecomposable representation of a finite group G not only subgroups of G but also subfactors. In addition we derive a relative version of the notion of a vertex (and a source) by considering socalled Mackey systems of subfactors. As a consequence one gets for example the Harish-Chandra theory for irreducible ordinary representations of finite groups of Lie type as special case. Secondly we extend the classical theory of vertices and sources to Hecke algebras associated with Coxeter groups. We apply this to the representation theory of finite general linear groups G in nondescribing characteristic. In particular if we take as Mackey system the set of Levi subgroups of G,(considered as factor groups of the corresponding parabolic subgroups), we explain how vertices with respect to this Mackey system correspond to vertices in the associated Hecke algebra of type A. Using a version of Steinberg’s tensor product theorem in the nondescribing characteristic case the vertices of the irreducible representation with respect to the Mackey system of Levi subgroups (Harish-Chandra vertices) are described.