Cabanes M., Engeuhard M. Representation Theory of Finite Reductive Groups

Cambridge University Press, 2004 — 456 p. — (New Mathematical Monographs: 1).At the crossroads of representation theory, algebraic geometry and finite group theory, this book brings together many of the main concerns of modern algebra, synthesizing the past twenty-five years of research, by including some of the most remarkable achievements in the field. The text is illustrated throughout by many examples, and background material is provided by several introductory chapters on basic results as well as appendices on algebraic geometry and derived categories. The result is an essential introduction for graduate students and a reference for all algebraists.Proofs of major and recent results in a very active field.
Simplifications from published material that make possible the inclusion of those results in a single book.
Emphasis on writing pedagogically with introductory chapters and appendices on background subjects.Notations and conventions
Representing Finite BN-Pairs:
Cuspidality in finite groups
Finite BN-pairs
Modular Hecke algebras for finite BN-pairs
Modular duality functor and the derived category
Local methods for the transversal characteristics
Simple modules in the natural characteristic
Deligne–Lusztig Varieties, Rational Series, and Morita Equivalences:
Finite reductive groups and Deligne–Lusztig varieties
Characters of finite reductive groups
Blocks of finite reductive groups and rational series
Jordan decomposition as a Morita equivalence, the main reductions
Jordan decomposition as a Morita equivalence, sheaves
Jordan decomposition as a Morita equivalence, modules
Unipotent Characters and Unipotent Blocks:
Levi subgroups and polynomial orders
Unipotent characters as a basic set
Jordan decomposition of characters
On conjugacy classes in type D
Standard isomorphisms for unipotent blocks
Decomposition Numbers and q-Schur Algebras:
Some integral Hecke algebras
Decomposition numbers and q-Schur algebras, general linear groups
Decomposition numbers and q-Schur algebras, linear primes
Unipotent Blocks and Twisted Induction:
Local methods. Twisted induction for blocks
Unipotent blocks and generalized Harish Chandra theory
Local structure and ring structure of unipotent blocks
Appendix 1: Derived categories and derived functors
Appendix 2: Varieties and schemes
Appendix 3: Etale cohomology