PREFACE TO THE PRINCETON LANDMARKS IN MATHEMATICS EDITION PREFACE ACKNOWLEDGMENTS CHAPTER Ⅰ.SCOPE OF THE THEORY §1.The Classical Groups §2.Cartan Decomposition §3.Representations §4.Concrete Problems in Representation Theory §5..Abstract Theory for Compact Groups §6.Application of the Abstract Theory to Lie Groups §7.Problems CHAPTER Ⅱ.REPRESENTATIONS OF SU（2）, SL（2, R）, AND SL（2, C） §1.The Unitary Trick §2.Irreducible Finite-Dimensional Complex-Linear Representations of el（2, C） §3.Finite-Dimensional Representations of el（2, C） §4.Irreducible Unitary Representations of SL（2, C） §5.Irreducible Unitary Representations of SL（2, R） §6.Use of SU（1, 1） §7.Plancherel Formula §8.Problems CHAPTER Ⅲ.C∞ VECTORS AND THE UNIVERSAL ENVELOPING ALGEBRA §1.Universal Enveloping Algebra §2.Actions on Universal Enveloping Algebra §3.C∞ Vectors §4.Garding Subspace §5.Problems CHAPTER Ⅳ.REPRESENTATIONS OF COMPACT LIE GROUPS §1.Examples of Root Space Decompositions §2.Roots §3.Abstract Root Systems and Positivity §4.Weyl Group, Algebraically §5.Weights and Integral Forms §6.Centalizers of Tori §7.Theorem of the Highest Weight §8.Verma Modules §9.Weyl Group, Analytically §10.Weyl Character Formula §11.Problems CHAPTER Ⅴ.STRUCTURE THEORY FOR NONCOMPACT GROUPS §1.Caftan Decomposition and the Unitary Trick §2.Iwasawa Decomposition §3.Regular Elements, Weyl Chambers, and the Weyl Group §4.Other Decompositions §5.Parabolic Subgroups §6.Integral Formulas §7.Borel-Weil Theorem §8.Problems CHAPTER Ⅵ.HOLOMORPHIC DISCRETE SERIES §1.Holomorphic Discrete Series for SU（1, 1） §2.Classical Bounded Symmetric Domains §3.Harish-Chandra Decomposition §4.Holomorphic Discrete Series §5.Finiteness of an Integral §6.Problems CHAPTER Ⅶ.INDUCED REPRESENTATIONS §1.Three Pictures §2.Elementary Properties §3.Bruhat Theory §4.Formal Intertwining Operators §5.Gindikin-Karpelevic Formula §6.Estimates on Intertwining Operators, Part Ⅰ §7.Analytic Continuation of Intertwining Operators,Part Ⅰ §8.Spherical Functions §9.Finite-Dimensional Representations and the H function §10.Estimates on Intertwining Operators, Part II §I 1.Tempered Representations and Langlands Quotients §12.Problems CHAPTER Ⅷ.ADMISSIBLE REPRESENTATIONS §1.Motivation §2.Admissible Representations §3.Invariant Subspaces §4.Framework for Studying Matrix Coefficients §5.Harish-Chandra Homomorphism §6.Infinitesimal Character §7.Differential Equations Satisfied by Matrix Coefficients §8.Asymptotic Expansions and Leading Exponents §9.First Application: Subrepresentation Theorem §10.Second Application: Analytic Continuation of Interwining Operators, Part Ⅱ §11.Third Application: Control of K-Finite Z（gC）-Finite Functions §12.Asymptotic Expansiqns near the Walls §13.Fourth Application: Asymptotic Size of Matrix Coefficients §14.Fifth Application: Identification of Irreducible Tempered Representations §15.Sixth Application: Langlands Classification of Irreducible Admissible Representations §16.Problems CHAPTER Ⅸ.CONSTRUCTION OF DISCRETE SERIES §1.Infinitesimally Unitary Representations §2.A Third Way of Treating Admissible Representations §3.Equivalent Definitions of Discrete Series §4.Motivation in General and the Construction in SU（1, 1） §5.Finite-Dimensional Spherical Representations §6.Duality in the General Case §7.Construction of Discrete Series §8.Limitations on K Types §9.Lemma on Linear Independence §10.Problems CHAPTER Ⅹ.GLOBAL CHARACTERS §1.Existence §2.Character Formulas for SL（2, R） §3.Induced Characters §4.Differential Equations §5.Analytieity on the Regular Set, Overview and Example §6.Analyticity on the Regular Set, General Case §7.Formula on the Regular Set §8.Behavior on the Singular Set §9.Families of Admissible Representations §10.Problems CHAPTER Ⅺ.INTRODUCTION TO PLANCHEREL FORMULA §1.Constructive Proof for SU（2） §2.Constructive Proof for SL（2, C） §3.Constructive Proof for SL（2, R） §4."Ingredients of Proof for General Case §5.Scheme of Proof for General Case §6.Properties of Ff §7.Hirai's Patching Conditions §8.Problems CHAPTER Ⅻ.EXHAUSTION OF DISCRETE SERIES §1.Boundedness of Numerators of Characters §2.Use of Patching Conditions §3.Formula for Discrete Series Characters §4.Schwartz Space §5.Exhaustion of Discrete Series §6.Tempered Distributions §7.Limits of Discrete Series §8.Discrete Series of M §9.Schmid's Identity §10.Problems CHAPTER ⅫⅠ.PLANCHEREL FORMULA §1.Ideas and Ingredients §2.Real-Rank-One Groups, Part I §3.Real-Rank-One Groups, Part II §4.Averaged Discrete Series §5.Sp （2, R） §6.General Case §7.Problems CHAPTER ⅪⅤ.IRREDUCIBLE TEMPERED REPRESENTATIONS §1.SL（2, R） from a More General Point of View §2.Eisenstein Integrals §3.Asymptotics of Eisenstein Integrals §4.The Functions for Intertwining Operators §5.First Irreducibility Results §6.Normalization of Intertwining Operators and Reducibility §7.Connection with Plancherel Formula when dim A=1 §8.Harish-Chandra's Completeness Theorem §9.R Group §10.Action by Weyl Group on Representations of M §11.Multiplicity One Theorem §12.Zuckerman Tensoring of Induced Representations §13.Generalized Schmid Identities §14.Inversion of Generalized Schmid Identities §15.Complete Reduction of Induced Representations §16.Classification §17.Revised Langlands Classification §18.Problems CHAPTER ⅩⅤ.MINIMAL K TYPES §1.Definition and Formula §2.Inversion Problem §3.Connection with Intertwining Operators §4.Problems CHAPTER ⅩⅥ.UNITARY REPRESENTATIONS §1.SL（2, R） and SL（2, C） §2.Continuity Arguments and Complementary Series §3.Criterion for Unitary Representations §4.Reduction to Real Infinitesimal Character §5.Problems APPENDIX A: ELEMENTARY THEORY OF LIE GROUPS §1.Lie Algebras §2.Structure Theory of Lie Algebras §3.Fundamental Group and Covering Spaces §4.Topological Groups §5.Vector Fields and Submanifolds §6.Lie Groups APPENDIX B: REGULAR SINGULAR POINTS OF PARTIAL DIFFERENTIAL EQUATIONS §1.Summary of Classical One-Variable Theory §2.Uniqueness and Analytic Continuation of Solutions in Several Variables §3.Analog of Fundamental Matrix §4.Regular Singularities §5.Systems of Higher Order §6.Leading Exponents and the Analog of the Indicial Equation §7.Uniqueness of Representation APPENDIX C: ROOTS AND RESTRICTED ROOTS FOR CLASSICAL GROUPS §1.Complex Groups §2.Noncompact Real Groups §3.Roots vs.Restricted Roots in Noncompact Real Groups NOTES REFERENCES INDEX OF NOTATION INDEX