Representation Theory of Finite Reductive Groups
Finite reductive groups form, together with derived subgroups of symmetric groups and sporadic groups, the class of non-commutative finite simple groups. Their representation theory has gone through an enormous development during the last 30 years. The main goal of the monograph is an exposition of the recent Bonnafé-Rouquier affirmative answer to one of the Broué conjectures. Morita equivalence between blocks of finite reductive groups and blocks defined by unipotent characters motivates the second essential aim of the book - an investigation of modular aspects of representation theory by means of unipotent blocks and unipotent characters.
The book is divided into five parts. Part I includes a description of a general concept of finite BN-pairs and Hecke algebras, an auto-equivalence of the derived category of the category of modules over a group algebra and a determination of simple modules in the natural characteristic. The second part is devoted to algebraic-geometric aspects of representation theory. An exposition of Deligne-Lusztig methods is completed by the Bonnafé-Rouquier proof of the Morita equivalence between the category of blocks of finite reductive groups and the category of unipotent blocks. Part III deals with unipotent characters. Here the reader can find a complete proof of the Lusztig theorem on restrictions of irreducible characters to a commutator in the case of connected reductive groups. The main theme of Part IV is the author's exposition of the Dipper-James theory describing modular aspects of representation theory. The final part of the book is devoted to further development of local methods, which makes it possible, for example, to prove a version of Fong-Srinivasan theorems on defect groups and ordinary characters of unipotent blocks. The monograph ends with three very compact appendices describing basic categorical and algebraic tools used in the book (e.g., étale cohomology, derived categories, sheaves and varieties). Material in the monograph is well-arranged and developed in a logical order, which makes the book very suitable as a reference for researchers. The text contains a lot of examples and every chapter is followed by a rich collection of exercises. The monograph can hence also be used as an introduction to the representation theory of finite reductive groups for graduate students.