Lecture 9: Finite Groups of Lie Type and Hecke Algebras

We continue to study the representation theory of finite groups of Lie type and its connection to Hecke algebras, now in a more general setting. We start by defining Hecke algebras of arbitrary Weyl groups. Then we introduce finite groups G of Lie type and relate the coinduced representations C[B \G] to Hecke algebras, similarly to what was done for GLn(Fq). In the second part of this lecture we discuss a way to produce representations of finite groups of Lie type that is of crucial importance for the classification of irreducibles and computation of their characters: the Deligne-Lusztig induction. We start by explaining the induction (or, in our conventions, co-induction) of a character of B lifted from a character of T . Then we describe maximal tori of G, since our field is not algebraically closed two maximal tori do not need to be conjugate. Finally, we discuss the Deligne-Lusztig induction, it is constructed using étale cohomology and produces a virtual representation of G starting with a character of a torus.