Marc Cabanes

Representation Theory of Finite Reductive Groups

At the crossroads of representation theory, algebraic geometry and finite group theory, this book blends together many of the main concerns of modern algebra, synthesizing the past 25 years of research, with full proofs of some of the most remarkable achievements in the area. Cabanes and Enguehard follow three main themes: first, applications of étale cohomology, leading, via notions of twisted induction, unipotent characters and Lusztig’s approach to the Jordan decomposition of characters, to the proof of the recent Bonnafé–Rouquier theorems. The second is a straightforward and simplified account of the Dipper–James theorems relating irreducible characters and modular representations, while introducing modular Hecke and Schur algebras. The final theme is local representation theory. One of the main results here is the authors’ version of Fong–Srinivasan theorems showing the relations between twisted induction and blocks of modular representations. Throughout, the text is illustrated by many examples; background is provided by several introductory chapters on basic results, and appendices on algebraic geometry and derived categories. The result is an essential introduction for graduate students and reference for all algebraists.

On Fusion in Unipotent Blocks

The context of this note is as follows. One considers a connected reductive group G and a Frobenius endomorphism F [ratio ] G → G defining G over a finite field of order q . One denotes by G F the associated (finite) group of fixed points. Let [lscr ] be a prime not dividing q . We are interested in the [lscr ]-blocks of the finite group G F . Such a block is called unipotent if there is a unipotent character (see, for instance, [ 6 , Definition 12.1]) among its representations in characteristic zero. Roughly speaking, it is believed that the study of arbitrary blocks of G F might be reduced to unipotent blocks (see [ 2 , Theoreme 2.3], [ 5 , Remark 3.6]). In view of certain conjectures about blocks (see, for instance, [ 9 ]), it would be interesting to further reduce the study of unipotent blocks to the study of principal blocks (blocks containing the trivial character). Our Theorem 7 is a step in that direction: we show that the local structure of any unipotent block of G F is very close to that of a principal block of a group of related type (notion of ‘control of fusion’, see [ 13 , §49]).

Local methods for blocks of reductive groups over a finite field

The aim of this paper is to show for general blocks of reductive groups over a finite field some analogues of the results in [CE.1] about unipotent blocks. This includes the distribution of ordinary characters into blocks (Theorem 3.3) and the structure of defect groups (Theorem 3.5), thus yielding the main results of [FS.1; FS.2; Br.1; CE.1]. It should be noted that, by a theorem of Broue [Br.2, 2.3], most blocks are in some sense “equivalent” to a unipotent block. This theorem provides an important idea one must keep in mind, but in addition to the restrictions on the blocks concerned, Broue’s equivalence (a perfect isometry) does not imply the isomorphism of defect groups (Remark 3.6) but just the equality of their orders.

On Okuyama’s Theorems about Alvis-Curtis Duality

The purpose of this paper is to report on the unpublished manuscript [O] by T. Okuyama where are proved some conjectures generalizing to homotopy categories the theorems of [CaRi] and [LS] holding in derived categories. We refer to the latter references and [CaEn]~\S 4 for a broader introduction to the subject. The main theme is the one of complexes related with the Coxeter complex and the action of parabolic subgroups on them, either for finite groups with BN-pairs or for finite dimensional Hecke algebras. Okuyamas contractions prove a quite efficient tool in a number of situations (see the proof of Solomon-Tits theorem in \S~6). We often stray away from Okuyamas proofs when it allows simplifications.

Representation Theory of Finite Reductive Groups: Frontmatter

At the crossroads of representation theory, algebraic geometry and finite group theory, this book blends together many of the main concerns of modern algebra, synthesizing the past 25 years of research, with full proofs of some of the most remarkable achievements in the area. Cabanes and Enguehard follow three main themes: first, applications of étale cohomology, leading, via notions of twisted induction, unipotent characters and Lusztig’s approach to the Jordan decomposition of characters, to the proof of the recent Bonnafé–Rouquier theorems. The second is a straightforward and simplified account of the Dipper–James theorems relating irreducible characters and modular representations, while introducing modular Hecke and Schur algebras. The final theme is local representation theory. One of the main results here is the authors’ version of Fong–Srinivasan theorems showing the relations between twisted induction and blocks of modular representations. Throughout, the text is illustrated by many examples; background is provided by several introductory chapters on basic results, and appendices on algebraic geometry and derived categories. The result is an essential introduction for graduate students and reference for all algebraists.

Representation Theory of Finite Reductive Groups: List of terminology

At the crossroads of representation theory, algebraic geometry and finite group theory, this book blends together many of the main concerns of modern algebra, synthesizing the past 25 years of research, with full proofs of some of the most remarkable achievements in the area. Cabanes and Enguehard follow three main themes: first, applications of étale cohomology, leading, via notions of twisted induction, unipotent characters and Lusztig’s approach to the Jordan decomposition of characters, to the proof of the recent Bonnafé–Rouquier theorems. The second is a straightforward and simplified account of the Dipper–James theorems relating irreducible characters and modular representations, while introducing modular Hecke and Schur algebras. The final theme is local representation theory. One of the main results here is the authors’ version of Fong–Srinivasan theorems showing the relations between twisted induction and blocks of modular representations. Throughout, the text is illustrated by many examples; background is provided by several introductory chapters on basic results, and appendices on algebraic geometry and derived categories. The result is an essential introduction for graduate students and reference for all algebraists.

Representation Theory of Finite Reductive Groups: Contents

At the crossroads of representation theory, algebraic geometry and finite group theory, this book blends together many of the main concerns of modern algebra, synthesizing the past 25 years of research, with full proofs of some of the most remarkable achievements in the area. Cabanes and Enguehard follow three main themes: first, applications of étale cohomology, leading, via notions of twisted induction, unipotent characters and Lusztig’s approach to the Jordan decomposition of characters, to the proof of the recent Bonnafé–Rouquier theorems. The second is a straightforward and simplified account of the Dipper–James theorems relating irreducible characters and modular representations, while introducing modular Hecke and Schur algebras. The final theme is local representation theory. One of the main results here is the authors’ version of Fong–Srinivasan theorems showing the relations between twisted induction and blocks of modular representations. Throughout, the text is illustrated by many examples; background is provided by several introductory chapters on basic results, and appendices on algebraic geometry and derived categories. The result is an essential introduction for graduate students and reference for all algebraists.

Representation Theory of Finite Reductive Groups: Index

At the crossroads of representation theory, algebraic geometry and finite group theory, this book blends together many of the main concerns of modern algebra, synthesizing the past 25 years of research, with full proofs of some of the most remarkable achievements in the area. Cabanes and Enguehard follow three main themes: first, applications of étale cohomology, leading, via notions of twisted induction, unipotent characters and Lusztig’s approach to the Jordan decomposition of characters, to the proof of the recent Bonnafé–Rouquier theorems. The second is a straightforward and simplified account of the Dipper–James theorems relating irreducible characters and modular representations, while introducing modular Hecke and Schur algebras. The final theme is local representation theory. One of the main results here is the authors’ version of Fong–Srinivasan theorems showing the relations between twisted induction and blocks of modular representations. Throughout, the text is illustrated by many examples; background is provided by several introductory chapters on basic results, and appendices on algebraic geometry and derived categories. The result is an essential introduction for graduate students and reference for all algebraists.