本书作者采取了与许多教材以紧李群的表示论作为理论基础不同的安排，并精心挑选一系列材料，以给予读者更广阔的视野。为介绍紧李群，本书涵盖了 Peter-weyl定理、极大环面的共轭性（提供了两组证明），Weyl特征标公式等内容。随后本书研究了复分析群，一般非紧李群，内容包括：Weyl 群的Coxeter表示、Iwasawa及Bruhat分解、Cartan分解、对称空间、Cayley变换、相对根系、Satake图形，扩展的 Dyakin图以及李群嵌入的方式综述。本书通过介绍表示论在多种领域中的应用（这些领域有：随机矩阵论、Toeplitz矩阵的子式、对称代数分解、 Gelfand对、Hecke代数、有限一般线性群的表示及Grassmann簇与旗簇的上同调），并将对称群的表示论与酋群间的Frobenius- Schur对偶作为统一的主题处理，使读者能够对表示理论有更加深刻地理解。
目录：
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 Cartan's 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
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