李群

Preface

Part Ⅰ： Compact Groups

1 Haar Measure

2 Schur Orthogonality

3 Compact Operators

4 The Peter-Weyl Theorem

Part Ⅱ： Lie Group Fundamentals

5 Lie Subgroups of GL（n， C）

6 Vector Fields

7 Left-Invariant Vector Fields

8 The Exponential Map

9 Tensors and Universal Properties

10 The Universal Enveloping Algebra

11 Extension of Scalars

12 Representations of S1（2， C）

13 The Universal Cover

14 The Local Frobenius Theorem

15 Tori

16 Geodesics and Maximal Tori

17 Topological Proof of Cartans Theorem

18 The Weyl Integration Formula

19 The Root System

20 Examples of Root Systems

21 Abstract Weyl Groups

22 The Fundamental Group

23 Semisimple Compact Groups

24 Highest-Weight Vectors

25 The Weyl Character Formula

26 Spin

27 Complexification

28 Coxeter Groups

29 The Iwasawa Decomposition

30 The Bruhat Decomposition

31 Symmetric Spaces

32 Relative Root Systems

33 Embeddings of Lie Groups

Part Ⅲ： Topics

34 Mackey Theory

35 Characters of GL（n，C）

36 Duality between Sk and GL（n，C）

37 The Jacobi-Trudi Identity

38 Schur Polynomials and GL（n，C）

39 Schur Polynomials and Sk

40 Random Matrix Theory

41 Minors of Toeplitz Matrices

42 Branching Formulae and Tableaux

43 The Cauchy Identity

44 Unitary Branching Rules

45 The Involution Model for Sk

46 Some Symmetric Algebras

47 Gelfand Pairs

48 Hecke Algebras

49 The Philosophy of Cusp Forms

50 Cohomology of Grassmannians

References

Index

《李群（英文版）》Part I covers standard general properties of representations of compactgroups （including Lie groups and other compact groups， such as finite or p-adie ones）. These include Schur orthogonality， properties of matrix coefficientsand the Peter-Weyl Theorem.
Part II covers the fundamentals of Lie gronps， by which I mean those sub-jects that I think are most urgent for the student to learn. These include thefollowing topics for compact groups： the fundamental group， the conjngacyof maximal tori （two proofs）， and the Weyl character formula. For noncom-pact groups， we start with complex analytic groups that are obtained bycomplexification of compact Lie groups， obtaining the lwasawa and Bruhatdecompositions. These arc the reductive complex groups. They are of course aspecial case， bnt a good place to start in the noncompact world. More generalnoncompact Lie groups with a Cartan decomposition are studied in the lastfew chapters of Part II. Chapter 31， on symmetric spaces， alternates exampleswith theory， discussing the embedding of a noncompact symmetric space inits compact dnal， the boundary components and Bergman-Shilov boundaryof a symmetric tube domain， anti Cartans classification. Chapter 32 con-structs the relative root system， explains Satake diagrams and gives examplesillustrating the various phenomena that can occur， and reproves the Iwasawadecomposition， formerly obtained for complex analytic groups， in this moregeneral context. Finally， Chapter 33 surveys the different ways Lie groups canbe embedded in oue another.