Representation Theory of Finite Reductive Groups

A research level synthesis and reference in a key branch of modern algebra, first published in 2004.

Marc Cabanes (Author), Michel Enguehard (Author)

9780521825177, Cambridge University Press

Hardback, published 29 January 2004

456 pages, 144 exercises
22.9 x 15.2 x 2.9 cm, 0.84 kg

'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.' Zentralblatt MATH

At the crossroads of representation theory, algebraic geometry and finite group theory, this 2004 book blends together many of the main concerns of modern algebra, 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 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. The final theme is local representation theory. One of the main results here is the authors' version of Fong–Srinivasan theorems. Throughout the text is illustrated by many examples and 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.

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.

Subject Areas: Algebra [PBF]