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Representation Theory of Finite Reductive Groups

Marc Cabanes and Michel Enguehard
Cambridge University Press
Publication Date: 
Number of Pages: 
New Mathematical Monographs
[Reviewed by
Fernando Q. Gouvêa
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Representation Theory of Finite Reductive Groups is the first book in a new series from Cambridge University Press called New Mathematical Monographs. The series is dedicated to publishing "books containing an in-depth discussion of a substantial area of mathematics." The statement of purpose also makes a very serious promise: "As well as being detailed, [the books in this series] will be readable and contain the motivational material necessary for those entering a field." The latter task is not at all easy to pull off in a monograph at a high level. One hopes that Cambridge will be able to stick to that promise.

In this first book, Marc Cabanes and Michel Enguehard introduce us to the study of the representations of a particular class of finite groups. These groups, which can be described as the groups you get by taking the points over a finite field of a reductive algebraic groups, include (in a sense) most of the finite simple groups. The approach is very high-powered: by page 55 we are reading about derived categories and duality functors. The book comes with appendices recalling the theories of derived categories, varieties and schemes, and étale cohomology.

The style is, as one might expect, quite dense. The first sentence of Chapter I, for example, is "The main functors in representation theory of finite groups are the restriction to subgroups and its adjoint, called induction." That sends, I think, the correct signal about the authors' approach: we're doing serious work with serious pre-requisites. Whether they keep the promise of including "motivational material" depends on the reader's background. This is not the place to start learning about group representations or even about reductive groups, I think. On the other hand, the book does try to lead the reader into the material, and in particular it tries to show why the heavy theoretical apparatus is necessary and helpful. For experts, then, but not a bad start to the series.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College and is the editor of MAA Reviews.

Introduction; Notations and conventions; Part I. Representing Finite BN-Pairs: 1. Cuspidality in finite groups; 2. Finite BN-pairs; 3. Modular Hecke algebras for finite BN-pairs; 4. Modular duality functor and the derived category; 5. Local methods for the transversal characteristics; 6. Simple modules in the natural characteristic; Part II. Deligne-Lusztig Varieties, Rational Series, and Morita Equivalences: 7. Finite reductive groups and Deligne-Lusztig varieties; 8. Characters of finite reductive groups; 9. Blocks of finite reductive groups and rational series; 10. Jordan decomposition as a Morita equivalence, the main reductions; 11. Jordan decomposition as a Morita equivalence, sheaves; 12. Jordan decomposition as a Morita equivalence, modules; Part III. Unipotent Characters and Unipotent Blocks: 13. Levi subgroups and polynomial orders; 14. Unipotent characters as a basic set; 15. Jordan decomposition of characters; 16. On conjugacy classes in type D; 17. Standard isomorphisms for unipotent blocks; Part IV. Decomposition Numbers and q-Schur Algebras: 18. Some integral Hecke algebras; 19. Decomposition numbers and q-Schur algebras, general linear groups; 20. Decomposition numbers and q-Schur algebras, linear primes; Part V. Unipotent Blocks and Twisted Induction: 21. Local methods. Twisted induction for blocks; 22. Unipotent blocks and generalized Harish Chandra theory; 23. Local structure and ring structure of unipotent blocks; Appendix 1: Derived categories and derived functors; Appendix 2: Varieties and schemes; Appendix 3: Etale cohomology; References; Index.