Abstract

Finite groups of Lie type are finite groups arising as rational points over a finite field of reductive linear algebraic groups (e.g. the group of invertible matrices with entries in a finite field). In the study of the ordinary representation theory of this kind of groups, inductive methods have a great relevance: in particular, it is possible to define a functor, called “parabolic induction”, that allows to construct representations of a finite groups of Lie type starting from representations of suitable subgroups (called “Levi subgroups”) that are finite groups of Lie type themselves.

After a brief introduction on finite groups of Lie type and parabolic induction, I will talk about the main method to study this functor and the decomposition of parabolically induced representations, that consists in the study of the endomorphism algebras of these representations. Thanks to the work to Howlett and Lehrer (that has been completed by some subsequent works of Lusztig and Geck), the structure of these endomorphism algebras for finite groups of Lie type arising from a connected linear algebraic group is completely described, as they turn out to be Iwahori-Hecke algebras. After outlining the main points of this classical result, I will briefly talk about a possible generalization in the case of a finite group of Lie type arising from a disconnected groups.