Representations of Finite Groups of Lie Type

François Digne; Jean Michel

doi:10.1017/9781108673655

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.

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Contents

Frontmatter
pp i-vi
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Contents
pp v-viii
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Introduction to the Second Edition
pp 1-1
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From the Introduction to the First Edition
pp 2-4
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1 - Basic Results on Algebraic Groups
pp 5-18
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2 - Structure Theorems for Reductive Groups
pp 19-38
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4 - Rationality, the Frobenius Endomorphism, the Lang–Steinberg Theorem
pp 59-78
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5 - Harish-Chandra Theory
pp 79-90
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6 - Iwahori–Hecke Algebras
pp 91-112
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7 - The Duality Functor and the Steinberg Character
pp 113-129
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8 - ℓ-Adic Cohomology
pp 130-136
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9 - Deligne–Lusztig Induction: The Mackey Formula
pp 137-147
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10 - The Character Formula and Other Results on Deligne–Lusztig Induction
pp 148-160
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11 - Geometric Conjugacy and the Lusztig Series
pp 161-195
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12 - Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters
pp 196-224
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13 - Green Functions
pp 225-241
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14 - The Decomposition of Deligne–Lusztig Characters
pp 242-248
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References
pp 249-254
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Index
pp 255-258
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Representations of Finite Groups of Lie Type

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Contents

Frontmatter
pp i-vi

Contents
pp v-viii

Introduction to the Second Edition
pp 1-1

From the Introduction to the First Edition
pp 2-4

1 - Basic Results on Algebraic Groups
pp 5-18

2 - Structure Theorems for Reductive Groups
pp 19-38

3 - (B,N)-Pairs; Parabolic, Levi, and Reductive Subgroups; Centralisers of Semi-simple Elements
pp 39-58

4 - Rationality, the Frobenius Endomorphism, the Lang–Steinberg Theorem
pp 59-78

5 - Harish-Chandra Theory
pp 79-90

6 - Iwahori–Hecke Algebras
pp 91-112

7 - The Duality Functor and the Steinberg Character
pp 113-129

8 - ℓ-Adic Cohomology
pp 130-136

9 - Deligne–Lusztig Induction: The Mackey Formula
pp 137-147

10 - The Character Formula and Other Results on Deligne–Lusztig Induction
pp 148-160

11 - Geometric Conjugacy and the Lusztig Series
pp 161-195

12 - Regular Elements; Gelfand–Graev Representations; Regular and Semi-Simple Characters
pp 196-224

13 - Green Functions
pp 225-241

14 - The Decomposition of Deligne–Lusztig Characters
pp 242-248

References
pp 249-254

Index
pp 255-258

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