Representation Theory of Semisimple Groups An Overview Based on Examples 1st Edition by Anthony Knapp 1400883970 9781400883974

Representation Theory of Semisimple Groups An Overview Based on Examples 1st Edition by Anthony Knapp – Ebook PDF Instant Download/Delivery: 1400883970, 9781400883974

Full download Representation Theory of Semisimple Groups An Overview Based on Examples 1st Edition after payment

Product details:

ISBN 10: 1400883970 

ISBN 13: 9781400883974

Author: Anthony W. Knapp

In this classic work, Anthony W. Knapp offers a survey of representation theory of semisimple Lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Although theorems are always stated precisely, many illustrative examples or classes of examples are given. To support this unique approach, the author includes for the reader a useful 300-item bibliography and an extensive section of notes.

Table of contents:

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 sl(2, C)
§3. Finite-Dimensional Representations of sl(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. Gårding 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. Cartan 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-Karpelevič 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
§11. 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(g^C)-Finite Functions
§12. Asymptotic Expansions 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. Analyticity 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 η 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, R)
§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

People also search:

the representation theory of the symmetric group

semisimple representation

group representation theory pdf

representation theory finite groups

group theory representation

Tags: Anthony Knapp, Representation, Theory, Semisimple