李群

前言

　　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.

內容概要

　　《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.

書籍目錄

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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