Representations of Finite Groups of Lie Type (London Mathematical Society Student Texts, Series Number 95)

On its original publication, this book provided the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to reflect the continuous evolution of the subject, including entirely new chapters on Hecke algebras, Green functions and Lusztig families. The authors cover the basic theory of representations of finite groups of Lie type, such as linear, unitary, orthogonal and symplectic groups. They emphasise the Curtis-Alvis duality map and Mackey's theorem and the results that can be deduced from it, before moving on to a discussion of Deligne-Lusztig induction and Lusztig's Jordan decomposition theorem for characters. The book contains the background information needed to make it a useful resource for beginning graduate students in algebra as well as seasoned researchers. It includes exercises and explicit examples. Contents Introduction to the Second Edition From the Introduction to the First Edition 1 Basic Results on Algebraic Groups 1.1 Basic Results on Algebraic Groups 1.2 Diagonalisable Groups, Tori, X(T),Y(T) 1.3 Solvable Groups, Borel Subgroups 1.4 Unipotent Groups, Radical, Reductive and Semi-Simple Groups 1.5 Examples of Reductive Groups 2 Structure Theorems for Reductive Groups 2.1 Coxeter Groups 2.2 Finite Root Systems 2.3 Structure of Reductive Groups 2.4 Root Data, Isogenies, Presentation of G 3 (B,N)-Pairs; Parabolic, Levi, and Reductive Subgroups; Centralisers of Semi-Simple Elements 3.1 (B,N)-Pairs 3.2 Parabolic Subgroups of Coxeter Groups and of (B,N)-Pairs 3.3 Closed Subsets of a Crystallographic Root System 3.4 Parabolic Subgroups and Levi Subgroups 3.5 Centralisers of Semi-Simple Elements 4 Rationality, the Frobenius Endomorphism, the Lang–Steinberg Theorem 4.1 k0-Varieties, Frobenius Endomorphisms 4.2 The Lang–Steinberg Theorem; Galois Cohomology 4.3 Classification of Finite Groups of Lie Type 4.4 The Relative (B,N)-Pair 5 Harish-Chandra Theory 5.1 Harish-Chandra Induction and Restriction 5.2 The Mackey Formula 5.3 Harish-Chandra Theory 6 Iwahori–Hecke Algebras 6.1 Endomorphism Algebras 6.2 Iwahori–Hecke Algebras 6.3 Schur Elements and Generic Degrees 6.4 The Example of G2 7 The Duality Functor and the Steinberg Character 7.1 F-rank 7.2 The Duality Functor 7.3 Restriction to Centralisers of Semi-Simple Elements 7.4 The Steinberg Character 8 l-Adic Cohomology 8.1 l-Adic Cohomology 9 Deligne–Lusztig Induction: The Mackey Formula 9.1 Deligne–Lusztig Induction 9.2 Mackey Formula for Lusztig Functors 9.3 Consequences: Scalar Products 10 The Character Formula and Other Results on Deligne–Lusztig Induction 10.1 The Character Formula 10.2 Uniform Functions 10.3 The Characteristic Function of a Semi-Simple Class 11 Geometric Conjugacy and the Lusztig Series 11.1 Geometric Conjugacy 11.2 More on Centralisers of Semi-Simple Elements 11.3 The Lusztig Series 11.4 Lusztig’s Jordan Decomposition of Characters: The Levi Case 11.5 Lusztig’s Jordan Decomposition of Characters: The General Case 11.6 More about Unipotent Characters 11.7 The Irreducible Characters of GLnF and UFn 12 Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters 12.1 Regular Elements 12.2 Regular Unipotent Elements 12.3 Gelfand–Graev Representations 12.4 Regular and Semi-Simple Characters 12.5 The Character Table of SL2(Fq) 13 Green Functions 13.1 Invariants 13.2 Green Functions and the Springer Correspondence 13.3 The Lusztig–Shoji Algorithm 14 The Decomposition of Deligne–Lusztig Characters 14.1 Lusztig Families and Special Unipotent Classes 14.2 Split Groups 14.3 Twisted Groups References Index The original edition of this book, written for beginning graduate students, was the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to reflect the continuous evolution of the subject, including chapters on Hecke algebras and Green functions.