本書作者采取了與許多教材以緊李群的表示論作為理論基礎(chǔ)不同的安排，并精心挑選一系列材料，以給予讀者更廣闊的視野。為介紹緊李群，本書涵蓋了 Peter-weyl定理、極大環(huán)面的共軛性（提供了兩組證明），Weyl特征標(biāo)公式等內(nèi)容。隨后本書研究了復(fù)分析群，一般非緊李群，內(nèi)容包括：Weyl 群的Coxeter表示、Iwasawa及Bruhat分解、Cartan分解、對稱空間、Cayley變換、相對根系、Satake圖形，擴(kuò)展的 Dyakin圖以及李群嵌入的方式綜述。本書通過介紹表示論在多種領(lǐng)域中的應(yīng)用（這些領(lǐng)域有：隨機(jī)矩陣論、Toeplitz矩陣的子式、對稱代數(shù)分解、 Gelfand對、Hecke代數(shù)、有限一般線性群的表示及Grassmann簇與旗簇的上同調(diào)），并將對稱群的表示論與酋群間的Frobenius- Schur對偶作為統(tǒng)一的主題處理，使讀者能夠?qū)Ρ硎纠碚撚懈由羁痰乩斫狻?
目錄：
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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