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El. knyga: Representations of Finite Groups of Lie Type

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(Université de Picardie Jules Verne, Amiens), (Centre National de la Recherche Scientifique (CNRS), Paris)
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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.

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.

Recenzijos

' a useful resource for beginning graduate students in algebra as well as seasoned researchers.' Mathematical Reviews Clippings ' clearly written; there are useful examples, motivational comments, and exercises scattered throughout the text.' Mark Hunacek, The Mathematical Gazette

Daugiau informacijos

An up-to-date and self-contained introduction based on a graduate course taught at the University of Paris.
Introduction to the Second Edition 1(1)
From the Introduction to the First Edition 2(3)
1 Basic Results on Algebraic Groups
5(14)
1.1 Basic Results on Algebraic Groups
5(3)
1.2 Diagonalisable Groups, Tori, X(T), Y(T)
8(3)
1.3 Solvable Groups, Borel Subgroups
11(2)
1.4 Unipotent Groups, Radical, Reductive and Semi-Simple Groups
13(2)
1.5 Examples of Reductive Groups
15(4)
2 Structure Theorems for Reductive Groups
19(20)
2.1 Coxeter Groups
19(5)
2.2 Finite Root Systems
24(5)
2.3 Structure of Reductive Groups
29(6)
2.4 Root Data, Isogenics, Presentation of G
35(4)
3 (B, N)-Pairs; Parabolic, Levi, and Reductive Subgroups; Centralisers of Semi-Simple Elements
39(20)
3.1 (B, N)-Pairs
39(3)
3.2 Parabolic Subgroups of Coxeter Groups and of (B, N)-Pairs
42(3)
3.3 Closed Subsets of a Crystallographic Root System
45(6)
3.4 Parabolic Subgroups and Levi Subgroups
51(5)
3.5 Centralisers of Semi-Simple Elements
56(3)
4 Rationality, the Frobenius Endomorphism, the Lang-Steinberg Theorem
59(20)
4.1 k0-Varieties, Frobenius Endomorphisms
59(4)
4.2 The Lang-Steinberg Theorem; Galois Cohomology
63(7)
4.3 Classification of Finite Groups of Lie Type
70(5)
4.4 The Relative (B, N)-Pair
75(4)
5 Harish-Chandra Theory
79(12)
5.1 Harish-Chandra Induction and Restriction
79(4)
5.2 The Mackey Formula
83(3)
5.3 Harish-Chandra Theory
86(5)
6 Iwahori--Hecke Algebras
91(22)
6.1 Endomorphism Algebras
91(6)
6.2 Iwahori--Hecke Algebras
97(7)
6.3 Schur Elements and Generic Degrees
104(4)
6.4 The Example of G2
108(5)
7 The Duality Functor and the Steinberg Character
113(17)
7.1 F-rank
113(3)
7.2 The Duality Functor
116(7)
7.3 Restriction to Centralisers of Semi-Simple Elements
123(3)
7.4 The Steinberg Character
126(4)
8 Adic Cohomology
130(7)
8.1 Adic Cohomology
130(7)
9 Deligne--Lusztig Induction: The Mackey Formula
137(11)
9.1 Deligne--Lusztig Induction
137(3)
9.2 Mackey Formula for Lusztig Functors
140(6)
9.3 Consequences: Scalar Products
146(2)
10 The Character Formula and Other Results on Deligne--Lusztig Induction
148(13)
10.1 The Character Formula
148(5)
10.2 Uniform Functions
153(4)
10.3 The Characteristic Function of a Semi-Simple Class
157(4)
11 Geometric Conjugacy and the Lusztig Series
161(35)
11.1 Geometric Conjugacy
161(6)
11.2 More on Centralisers of Semi-Simple Elements
167(3)
11.3 The Lusztig Series
170(5)
11.4 Lusztig's Jordan Decomposition of Characters: The Levi Case
175(8)
11.5 Lusztig's Jordan Decomposition of Characters: The General Case
183(5)
11.6 More about Unipotent Characters
188(3)
11.7 The Irreducible Characters of GLFn and UFn
191(5)
12 Regular Elements; Gelfand--Graev Representations; Regular and Semi-Simple Characters
196(29)
12.1 Regular Elements
196(5)
12.2 Regular Unipotent Elements
201(6)
12.3 Gelfand--Graev Representations
207(7)
12.4 Regular and Semi-Simple Characters
214(6)
12.5 The Character Table of SL2(Fq)
220(5)
13 Green Functions
225(17)
13.1 Invariants
225(6)
13.2 Green Functions and the Springer Correspondence
231(5)
13.3 The Lusztig--Shoji Algorithm
236(6)
14 The Decomposition of Deligne--Lusztig Characters
242(7)
14.1 Lusztig Families and Special Unipotent Classes
242(2)
14.2 Split Groups
244(3)
14.3 Twisted Groups
247(2)
References 249(6)
Index 255
Franēois Digne is Emeritus Professor at the Université de Picardie Jules Verne, Amiens. He works on finite reductive groups, braid and Artin groups. He has also co-authored with Jean Michel the monograph Foundations of Garside Theory (2015) and several notable papers on DeligneLusztig varieties. Jean Michel is Emeritus Director of Research at the Centre National de la Recherche Scientifique (CNRS), Paris. His research interests include reductive algebraic groups, in particular DeligneLusztig varieties, and Spetses and other objects attached to complex reflection groups. He has also co-authored with Franēois Digne the monograph Foundations of Garside Theory (2015) and several notable papers on DeligneLusztig varieties.
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