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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.

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
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