Harish-Chandra theory, q-Schur algebras, and decomposition matrices for finite classical groups

Summary

Abstract

We give a survey on the present state of knowledge in the representation theory of finite groups of Lie type in the non-defining characteristic case. A systematic use of Harish-Chandra induction leads to the definition of q-Schur algebras. Their representation theory yields explicit information on the decomposition matrices for classical groups.

Introduction

This paper is an extended version of my talk given at the conference. It is intended as a brief survey on recent developments in the representation theory of finite groups of Lie type in non-defining characteristic.

Let G = G(q) denote a finite group of Lie type. Examples are classical groups such as GL_n(q) or Sp_n(q), but also exceptional groups such as E₈(q). Let κ be an algebraically closed field of characteristic l ≥ 0 and assume that l does not divide q. We are interested in the category κG-mod of finitely generated left κG-modules.

If l = 0 (or, more generally, l ⊥ |G|, the order of G), every finite dimensional κG-module is completely reducible. Hence it suffices to find the simple κG-modules. In 1955, Green determined the ordinary character table for the general linear groups GL_n(q) [26], thus solving the problem for these groups. Beginning with the seminal paper [7] of Deligne and Lusztig in 1976, Lusztig arrived at a complete classification of the simple κG-modules for all finite groups of Lie type G (see Lusztig [36]).

The representation theory for these groups in characteristics l ≠ 0, l ⊥ q, started in 1982 with the fundamental paper [16] by Fong and Srinivasan, in which these authors determined the l-blocks of the general linear groups and the unitary groups.