李群
2009-8
世界图书出版公司
Daniel Bump
451
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This book aims to be a course in Lie groups that can be covered in one year with a gronp of good graduate students. I have attempted to address a problem that anyone teaching this subject must have， which is that the amount of essential material is too nmch to cover. One approach to this problem is to emphasize the beautiful representation theory of compact groups， and indeed this book can be used for a course of this type if after Chapter 25 one skips ahead to Part III. But I did not wantto omit important topics such as the Bruhat decomposition and the theory ofsymmetric spaces. For these subjects， compact groups are not sufficient. 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.
《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.
PrefacePart I: Compact Groups 1 Haar Measure 2 Schur Orthogonality 3 Compact Operators 4 The Peter-Weyl TheoremPart 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 GroupsPart III: Topics 34 Mackey Theory 35 Characters of GL(n,C) 36 Duality between Sk and GL(n,C) ……ReferencesIndex
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Bump的李群， 与调和分析与表示论有关系， 学数论的不妨一读
GTM225,我在新华书店看到的,内页用纸偏白,似乎没有前一本买的GTM222好(后者纸张偏黑一点,优点是纸面更光滑,摸起来更舒服一点)
D. Bump是表示论的大家，以前读过他的automorphic forms and representations, 非常易读。这本书应该也是一本很容易的入门书