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  • Print publication year: 2020
  • Online publication date: February 2020

2 - Lusztig’s Classification of Irreducible Characters

Summary

After some general remarks about characters of finite groups (possibly twisted by an automorphism), this chapter focuses on the generalised characters $R(T,\theta)$ which where introduced by Deligne and Lustzig using cohomological methods. We refer to the books by Carter and Digne-Michel for proofs of some fundamental properties, like orthogonality relations and degree formulae. Based on these results, we develop in some detail the basic formalism of Lusztig's book, which leads to a classification of the irreducible characters of finite groups of Lie type in terms of a fundamental Jordan decomposition. Using the general theory about regular embeddings in Chapter 1, we state and discuss that Jordan decomposition in complete generality, that is, without any assumption on the center of the underlying algebraic group. The final two sections give an introduction to the problems of computing Green functions and characteristic functions of character sheaves.