This book is an introduction to the study of representations of a special class of finite groups, called finite reductive groups. These are the groups of rational points over a finite field in reductive groups. According to the classification of finite simple groups, the alternating groups and the finite reductive groups yield all finite non-abelian simple groups, apart from 26 “sporadic” groups.

Representation theory, when applied to a given finite group G, traditionally refers to the program of study defined by R. Brauer. Once the ordinary characters of G are determined, this consists of expressing the Brauer characters as linear combinations of ordinary characters, thus providing the “decomposition” matrix and Cartan matrix of group algebras of the form k[G] where k is some algebraically closed field of prime characteristic ℓ. One may add to the above a whole array of problems:

• blocks of k[G] and induced partitions of characters,

• relations with ℓ-subgroups,

• computation of invariants controlling the isomorphism type of these blocks,

• checking of finiteness conjectures on blocks,

• study of certain indecomposable modules,

• further information about the category k[G]–mod and its derived category D(k[G]).

In the case of finite reductive groups, only parts of this program have been completed but, importantly, more specific questions or conjectures have arisen. For this reason, the present book may not match Brauer's program on all points. It will generally follow the directions suggested by the results obtained during the last 25 years in this area.