Cabanes, Marc; Enguehard, Michel Representation theory of finite reductive groups. (English) Zbl 1069.20032 New Mathematical Monographs 1. Cambridge, MA: Cambridge University Press (ISBN 0-521-82517-2/hbk). xviii, 436 p. (2004). This monograph treats the representation theory of finite reductive groups mostly in transversal characteristic, i.e. in a characteristic that differs from the natural characteristic \(p\) of the group. The book consists of five parts and three appendices. In part one the group is viewed simply as a finite group with split BN-pair of characteristic \(p\). It introduces Harish-Chandra induction, cuspidality, Hecke algebras, derived categories, Alvis-Curtis-Deligne-Lusztig duality, Brauer’s main theorems…. In later parts the finite reductive group is viewed as the fixed point group \({\mathbf G}^F\) under the action of a Frobenius endomorphism \(F\) on a reductive algebraic group \(\mathbf G\) in characteristic \(p\). Part two is the heart of the book. It treats Deligne-Lusztig theory and culminates in the Bonafé-Rouqier proof of a conjecture of Broué concerning the existence of a Morita equivalence underlying Lusztig’s Jordan decomposition of characters. Next the representations become modular, in transversal characteristic. Part three is about unipotent characters and unipotent blocks. It includes the proof of a theorem of Lusztig showing that the restriction to \([\mathbf{G,G}]^F\) of an irreducible character of \({\mathbf G}^F\) is multiplicity free. Part four is about decomposition numbers and the \(q\)-Schur algebras of Dipper-James. Part five gives the authors’ version of Fong-Srinivasan theory with ‘\(e\)-generalized’ Harish-Chandra theory. The appendices gather some background material. Reviewer: Wilberd van der Kallen (Utrecht) Cited in 2 ReviewsCited in 121 Documents MSC: 20G05 Representation theory for linear algebraic groups 20G40 Linear algebraic groups over finite fields 20C33 Representations of finite groups of Lie type 20-02 Research exposition (monographs, survey articles) pertaining to group theory 20C08 Hecke algebras and their representations 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20C15 Ordinary representations and characters 20C20 Modular representations and characters Keywords:finite reductive groups; blocks; BN-pairs; cuspidal representations; Deligne-Lusztig varieties; reductive algebraic groups; Harish-Chandra theory; ordinary irreducible characters; virtual characters; modular representations; Morita equivalences PDFBibTeX XMLCite \textit{M. Cabanes} and \textit{M. Enguehard}, Representation theory of finite reductive groups. Cambridge, MA: Cambridge University Press (2004; Zbl 1069.20032)