| Preface to the Princeton Landmarks in Mathematics Edition |
| Preface |
| Acknowledgments |
| Scope of the Theory / Chapter I.： |
| The Classical Groups / 1.： |
| Cartan Decomposition / 2.： |
| Representations / 3.： |
| Concrete Problems in Representation Theory / 4.： |
| Abstract Theory for Compact Groups / 5.： |
| Application of the Abstract Theory to Lie Groups / 6.： |
| Problems / 7.： |
| Representations of SU(2), SL(2, R), and SL(2, C) / Chapter II.： |
| The Unitary Trick |
| Irreducible Finite-Dimensional Complex-Linear Representations of sl(2, C) |
| Finite-Dimensional Representations of sl(2, C) |
| Irreducible Unitary Representations of SL(2, C) |
| Irreducible Unitary Representations of SL(2, R) |
| Use of SU(1, 1) |
| Plancherel Formula |
| C[superscript [infinity] Vectors and the Universal Enveloping Algebra / 8.： |
| Universal Enveloping Algebra |
| Actions on Universal Enveloping Algebra |
| C[superscript [infinity] Vectors |
| Garding Subspace |
| Representations of Compact Lie Groups / Chapter IV.： |
| Examples of Root Space Decompositions |
| Roots |
| Abstract Root Systems and Positivity |
| Weyl Group, Algebraically |
| Weights and Integral Forms |
| Centalizers of Tori |
| Theorem of the Highest Weight |
| Verma Modules |
| Weyl Group, Analytically / 9.： |
| Weyl Character Formula / 10.： |
| Structure Theory for Noncompact Groups / 11.： |
| Cartan Decomposition and the Unitary Trick |
| Iwasawa Decomposition |
| Regular Elements, Weyl Chambers, and the Weyl Group |
| Other Decompositions |
| Parabolic Subgroups |
| Integral Formulas |
| Borel-Weil Theorem |
| Holomorphic Discrete Series / Chapter VI.： |
| Holomorphic Discrete Series for SU(1, 1) |
| Classical Bounded Symmetric Domains |
| Harish-Chandra Decomposition |
| Finiteness of an Integral |
| Induced Representations / Chapter VII.： |
| Three Pictures |
| Elementary Properties |
| Bruhat Theory |
| Formal Intertwining Operators |
| Gindikin-Karpelevic Formula |
| Estimates on Intertwining Operators, Part I |
| Analytic Continuation of Intertwining Operators, Part I |
| Spherical Functions |
| Finite-Dimensional Representations and the H function |
| Estimates on Intertwining Operators, Part II |
| Tempered Representations and Langlands Quotients |
| Admissible Representations / 12.： |
| Motivation |
| Invariant Subspaces |
| Framework for Studying Matrix Coefficients |
| Harish-Chandra Homomorphism |
| Infinitesimal Character |
| Differential Equations Satisfied by Matrix Coefficients |
| Asymptotic Expansions and Leading Exponents |
| First Application: Subrepresentation Theorem |
| Second Application: Analytic Continuation of Interwining Operators, Part II |
| Third Application: Control of K-Finite Z(g[superscript C])-Finite Functions |
| Asymptotic Expansions near the Walls |
| Fourth Application: Asymptotic Size of Matrix Coefficients / 13.： |
| Fifth Application: Identification of Irreducible Tempered Representations / 14.： |
| Sixth Application: Langlands Classification of Irreducible Admissible Representations / 15.： |
| Construction of Discrete Series / 16.： |
| Infinitesimally Unitary Representations |
| A Third Way of Treating Admissible Representations |
| Equivalent Definitions of Discrete Series |
| Motivation in General and the Construction in SU(1, 1) |
| Finite-Dimensional Spherical Representations |
| Duality in the General Case |
| Limitations on K Types |
| Lemma on Linear Independence |
| Global Characters / Chapter X.： |
| Existence |
| Character Formulas for SL(2, R) |
| Induced Characters |
| Differential Equations |
| Analyticity on the Regular Set, Overview and Example |
| Analyticity on the Regular Set, General Case |
| Formula on the Regular Set |
| Behavior on the Singular Set |
| Families of Admissible Representations |
| Introduction to Plancherel Formula / Chapter XI.： |
| Constructive Proof for SU(2) |
| Constructive Proof for SL(2, C) |
| Constructive Proof for SL(2, R) |
| Ingredients of Proof for General Case |
| Scheme of Proof for General Case |
| Properties of F[subscript f] |
| Hirai's Patching Conditions |
| Exhaustion of Discrete Series / Chapter XII.： |
| Boundedness of Numerators of Characters |
| Use of Patching Conditions |
| Formula for Discrete Series Characters |
| Schwartz Space |
| Tempered Distributions |
| Limits of Discrete Series |
| Discrete Series of M |
| Schmid's Identity |
| Ideas and Ingredients / Chapter XIII.： |
| Real-Rank-One Groups |
| Real-Rank-One Groups, Part II |
| Averaged Discrete Series |
| Sp (2, R) |
| General Case |
| Irreducible Tempered Representations / Chapter XIV.： |
| SL(2, R) from a More General Point of View |
| Eisenstein Integrals |
| Asymptotics of Eisenstein Integrals |
| The [eta] Functions for Intertwining Operators |
| First Irreducibility Results |
| Normalization of Intertwining Operators and Reducibility |
| Connection with Plancherel Formula when dim A = 1 |
| Harish-Chandra's Completeness Theorem |
| R Group |
| Action by Weyl Group on Representations of M |
| Multiplicity One Theorem |
| Zuckerman Tensoring of Induced Representations |
| Generalized Schmid Identities |
| Inversion of Generalized Schmid Identities |
| Complete Reduction of Induced Representations |
| Classification |
| Revised Langlands Classification / 17.： |
| Minimal K Types / 18.： |
| Definition and Formula |
| Inversion Problem |
| Connection with Intertwining Operators |
| Unitary Representations / Chapter XVI.： |
| SL(2, R) and SL(2, C) |
| Continuity Arguments and Complementary Series |
| Criterion for Unitary Representations |
| Reduction to Real Infinitesimal Character |
| Elementary Theory of Lie Groups / Appendix A： |
| Lie Algebras |
| Structure Theory of Lie Algebras |
| Fundamental Group and Covering Spaces |
| Topological Groups |
| Vector Fields and Submanifolds |
| Lie Groups |
| Regular Singular Points of Partial Differential Equations / Appendix B： |
| Summary of Classical One-Variable Theory |
| Uniqueness and Analytic Continuation of Solutions in Several Variables |
| Analog of Fundamental Matrix |
| Regular Singularities |
| Systems of Higher Order |
| Leading Exponents and the Analog of the Indicial Equation |
| Uniqueness of Representation |
| Roots and Restricted Roots for Classical Groups / Appendix C： |
| Complex Groups |
| Noncompact Real Groups |
| Roots vs. Restricted Roots in Noncompact Real Groups |
| Notes |
| References |
| Index of Notation |
| Index |
| Preface to the Princeton Landmarks in Mathematics Edition |
| Preface |
| Acknowledgments |
図書
