Representation theory of semisimple groups : an overview based on examples / Anthony W. Knapp ; with a new preface by the author.
Series Princeton landmarks in mathematics and physicsEditor: Princeton, N.J. : Princeton University Press, 2001Edición: 1st Princeton landmarks in mathematics edDescripción: xix, 773 p. : il. ; 24 cmISBN: 0691090890 (pbk. : alk. paper)Tema(s): Semisimple Lie groups | Representations of groupsOtra clasificación: 22E46 (22-01 22E30) Recursos en línea: Table of contents | Publisher descriptionContents Preface to the Princeton Landmarks in Mathematics Edition xiii Preface xv Acknowledgments xix Chapter I. Scope of the Theory §1. The Classical Groups [3] §2. Cartan Decomposition [7] §3. Representations [10] §4. Concrete Problems in Representation Theory [14] §5. Abstract Theory for Compact Groups [14] §6. Application of the Abstract Theory to Lie Groups [23] §7. Problems [24] Chapter II. Representations of SU(2), SL(2, R), and SL(2, C) §1. The Unitary Trick [28] §2. Irreducible Finite-Dimensional Complex-Linear Representations of sl(2, C) [30] §3. Finite-Dimensional Representations of sl(2, C) [31] §4. Irreducible Unitary Representations of SL(2, C) [33] §5. Irreducible Unitary Representations of SL(2, R) [35] §6. Use of SU(l, 1) [39] §7. Plancherel Formula [41] §8. Problems [42] Chapter III. C∞ Vectors and the Universal Enveloping Algebra §1. Universal Enveloping Algebra [46] §2. Actions on Universal Enveloping Algebra [50] §3. C∞ Vectors [51] §4. Girding Subspace [33] 55. Problems [37] Chapter IV. Representations of Compact LiE Groups §1. Examples of Root Space Decompositions [60] §2. Roots [65] §3. Abstract Root Systems and Positivity [72] §4. Weyl Group, Algebraically [78] §5. Weights and Integral Forms [81] §6. Centalizers of Tori [86] §7. Theorem of the Highest Weight [89] §8. Verma Modules [93] §9. Weyl Group, Analytically [100] §10. Weyl Character Formula [104] §11. Problems [109] Chapter V. Structure Theory for Noncompact Groups §1. Cartan Decomposition and the Unitary Trick [113] §2. Iwasawa Decomposition [116] §3. Regular Elements, Weyl Chambers, and the Weyl Group [121] §4. Other Decompositions [126] §5. Parabolic Subgroups [132] §6. Integral Formulas [137] §7. Borel-Weil Theorem [142] §8. Problems [147] Chapter VI. Holomorphic Discrete Series §1. Holomorphic Discrete Series for SU(1,1) [150] §2. Classical Bounded Symmetric Domains [152] §3. Harish-Chandra Decomposition [153] §4. Holomorphic Discrete Series [158] §5. Finiteness of an Integral [161] §6. Problems [164] Chapter VII. Induced Representations §1. Three Pictures [167] §2. Elementary Properties [169] §3. Bruhat Theory [172] §4. Formal Intertwining Operators [174] §5. Gindikin-Karpelevic Formula [177] §6. Estimates on Intertwining Operators, Part I [181] §7. Analytic Continuation of Intertwining Operators, Part I [183] §8. Spherical Functions [185] §9. Finite-Dimensional Representations and the H function [191] §10. Estimates on Intertwining Operators, Part II [196] §11. Tempered Representations and Langlands Quotients [198] §12. Problems [201] Chapter VIII. Admissible Representations §1. Motivation [203] §2. Admissible Representations [205] §3. Invariant Subspaces [209] §4. Framework for Studying Matrix Coefficients [215] §5. Harish-Chandra Homomorphism [218] §6. Infinitesimal Character [223] §7. Differential Equations Satisfied by Matrix Coefficients [226] §8- Asymptotic Expansions and Leading Exponents [234] §9. First Application: Subrepresentation Theorem [238] §10. Second Application: Analytic Continuation of Interwining Operators, Part II [239] §11. Third Application: Control of K-Finite Z(gc)-Finite Functions [242] §12. Asymptotic Expansions near the Walls [247] §13. Fourth Application: Asymptotic Size of Matrix Coefficients [253] §14. Fifth Application: Identification of Irreducible Tempered Representations [258] §15. Sixth Application: Langlands Classification of Irreducible Admissible Representations [266] §16. Problems [276] Chapter IX. Construction of Discrete Series §1. Infinitesimally Unitary Representations [281] §2. A Third Way of Treating Admissible Representations [282] §3. Equivalent Definitions of Discrete Series [284] §4. Motivation in General and the Construction in SU(1,1) [287] §5. Finite-Dimensional Spherical Representations [300] §6. Duality in the General Case [303] §7. Construction of Discrete Series [309] §8. Limitations on K Types [320] §9- Lemma on Linear Independence [328] §10. Problems [330] Chapter X. Global Characters §1. Existence [333] §2. Character Formulas for SL(2, R) [338] §3. Induced Characters [347] §4. Differential Equations [354] §5 Analyticity on the Regular Set, Overview and Example [355] §6. Analyticity on the Regular Set, General Case [360] §7. Formula on the Regular Set [368] §8. Behavior on the Singular Set [371] §9. Families of Admissible Representations [374] §10. Problems [383] Chapter XI. Introduction to Plancherel Formula §1. Constructive Proof for SU(2) [385] §2. Constructive Proof for SL(2, C) [387] §3. Constructive Proof for SL(2, R) [394] §4. Ingredients of Proof for General Case [401] §5. Scheme of Proof for General Case [404] §6. Properties of Ff [407] §7. Hirai’s Patching Conditions [421] §8. Problems [425] Chapter XII. Exhaustion of Discrete Series §1. Boundedness of Numerators of Characters [426] §2. Use of Patching Conditions [432] §3. Formula for Discrete Series Characters [436] §4. Schwartz Space [447] §5. Exhaustion of Discrete Series [452] §6. Tempered Distributions [456] §7. Limits of Discrete Series [460] §8. Discrete Series of M [467] §9. Schmid’s Identity [473] §10. Problems [476] Chapter XIII. Plancherel Formula §1. Ideas and Ingredients [482] §2. Real-Rank-One Groups, Part I [482] §3. Real-Rank-One Groups, Part II [485] §4. Averaged Discrete Series [494] §5. Sp(2,R) [502] §6. General Case [511] §7. Problems [512] Chapter XIV. Irreducible Tempered Representations §1. SL(2, R) from a More General Point of View [515] §2. Eisenstein Integrals [520] §3. Asymptotics of Eisenstein Integrals [526] §4. The Functions for Intertwining Operators [235] §5. First Irreducibility Results [540] §6. Normalization of Intertwining Operators and Reducibility [543] §7. Connection with Plancherel Formula when dim A = 1 [547] §8. Harish-Chandra’s Completeness Theorem [553] §9. R Group [560] §10. Action by Weyl Group on Representations of M [568] §11. Multiplicity One Theorem [577] §12. Zuckerman Tensoring of Induced Representations [584] §13. Generalized Schmid Identities [587] §14. Inversion of Generalized Schmid Identities [595] §15. Complete Reduction of Induced Representations [599] §16. Classification [606] §17. Revised Langlands Classification [614] §18. Problems [621] Chapter XV. Minimal K Types §1. Definition and Formula [626] §2. Inversion Problem [635] §3. Connection with Intertwining Operators [641] §4. Problems [647] Chapter XVI. Unitary Representations §1. SL(2, R) and SL(2, C) [650] §2. Continuity Arguments and Complementary Series [653] §3. Criterion for Unitary Representations [655] §4. Reduction to Real Infinitesimal Character [660] §5. Problems [665] Appendix A: Elementary Theory of Lie Groups §1. Lie Algebras [667] §2. Structure Theory of Lie Algebras [668] §3. Fundamental Group and Covering Spaces [670] §4. Topological Groups [673] §5. Vector Fields and Submanifolds [674] §6. Lie Groups [679] Appendix B: Regular Singular Points of Partial Differential Equations §1. Summary of Classical One-Variable Theory [685] §2. Uniqueness and Analytic Continuation of Solutions in Several Variables [690] §3. Analog of Fundamental Matrix [693] §4. Regular Singularities [697] §5. Systems of Higher Order [700] §6. Leading Exponents and the Analog of the Indicial Equation [703] §7. Uniqueness of Representation [710] Appendix C: Roots and Restricted Roots for Classical Groups §1. Complex Groups [713] §2. Noncompact Real Groups [713] §3. Roots vs. Restricted Roots in Noncompact Real Groups [715] Notes [719] References [747] Index of Notation [763] Index [767]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode |
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 22 K67r-2 (Browse shelf) | Available | A-9286 |
Originally published: Princeton, N.J. : Princeton University Press, 1986, in series: Princeton mathematical series ; 36.
Incluye referencias bibliográficas (p. 747-762) e índices.
Contents --
Preface to the Princeton Landmarks in Mathematics Edition xiii --
Preface xv --
Acknowledgments xix --
Chapter I. Scope of the Theory --
§1. The Classical Groups [3] --
§2. Cartan Decomposition [7] --
§3. Representations [10] --
§4. Concrete Problems in Representation Theory [14] --
§5. Abstract Theory for Compact Groups [14] --
§6. Application of the Abstract Theory to Lie Groups [23] --
§7. Problems [24] --
Chapter II. Representations of SU(2), SL(2, R), and SL(2, C) --
§1. The Unitary Trick [28] --
§2. Irreducible Finite-Dimensional Complex-Linear Representations of sl(2, C) [30] --
§3. Finite-Dimensional Representations of sl(2, C) [31] --
§4. Irreducible Unitary Representations of SL(2, C) [33] --
§5. Irreducible Unitary Representations of SL(2, R) [35] --
§6. Use of SU(l, 1) [39] --
§7. Plancherel Formula [41] --
§8. Problems [42] --
Chapter III. C∞ Vectors and the Universal --
Enveloping Algebra --
§1. Universal Enveloping Algebra [46] --
§2. Actions on Universal Enveloping Algebra [50] --
§3. C∞ Vectors [51] --
§4. Girding Subspace [33] --
55. Problems [37] --
Chapter IV. Representations of Compact LiE Groups --
§1. Examples of Root Space Decompositions [60] --
§2. Roots [65] --
§3. Abstract Root Systems and Positivity [72] --
§4. Weyl Group, Algebraically [78] --
§5. Weights and Integral Forms [81] --
§6. Centalizers of Tori [86] --
§7. Theorem of the Highest Weight [89] --
§8. Verma Modules [93] --
§9. Weyl Group, Analytically [100] --
§10. Weyl Character Formula [104] --
§11. Problems [109] --
Chapter V. Structure Theory for Noncompact Groups --
§1. Cartan Decomposition and the Unitary Trick [113] --
§2. Iwasawa Decomposition [116] --
§3. Regular Elements, Weyl Chambers, and the Weyl Group [121] --
§4. Other Decompositions [126] --
§5. Parabolic Subgroups [132] --
§6. Integral Formulas [137] --
§7. Borel-Weil Theorem [142] --
§8. Problems [147] --
Chapter VI. Holomorphic Discrete Series --
§1. Holomorphic Discrete Series for SU(1,1) [150] --
§2. Classical Bounded Symmetric Domains [152] --
§3. Harish-Chandra Decomposition [153] --
§4. Holomorphic Discrete Series [158] --
§5. Finiteness of an Integral [161] --
§6. Problems [164] --
Chapter VII. Induced Representations --
§1. Three Pictures [167] --
§2. Elementary Properties [169] --
§3. Bruhat Theory [172] --
§4. Formal Intertwining Operators [174] --
§5. Gindikin-Karpelevic Formula [177] --
§6. Estimates on Intertwining Operators, Part I [181] --
§7. Analytic Continuation of Intertwining Operators, Part I [183] --
§8. Spherical Functions [185] --
§9. Finite-Dimensional Representations and the H function [191] --
§10. Estimates on Intertwining Operators, Part II [196] --
§11. Tempered Representations and Langlands Quotients [198] --
§12. Problems [201] --
Chapter VIII. Admissible Representations --
§1. Motivation [203] --
§2. Admissible Representations [205] --
§3. Invariant Subspaces [209] --
§4. Framework for Studying Matrix Coefficients [215] --
§5. Harish-Chandra Homomorphism [218] --
§6. Infinitesimal Character [223] --
§7. Differential Equations Satisfied by Matrix Coefficients [226] --
§8- Asymptotic Expansions and Leading Exponents [234] --
§9. First Application: Subrepresentation Theorem [238] --
§10. Second Application: Analytic Continuation of Interwining Operators, Part II [239] --
§11. Third Application: Control of K-Finite Z(gc)-Finite Functions [242] --
§12. Asymptotic Expansions near the Walls [247] --
§13. Fourth Application: Asymptotic Size of Matrix Coefficients [253] --
§14. Fifth Application: Identification of Irreducible Tempered Representations [258] --
§15. Sixth Application: Langlands Classification of Irreducible Admissible Representations [266] --
§16. Problems [276] --
Chapter IX. Construction of Discrete Series --
§1. Infinitesimally Unitary Representations [281] --
§2. A Third Way of Treating Admissible Representations [282] --
§3. Equivalent Definitions of Discrete Series [284] --
§4. Motivation in General and the Construction in SU(1,1) [287] --
§5. Finite-Dimensional Spherical Representations [300] --
§6. Duality in the General Case [303] --
§7. Construction of Discrete Series [309] --
§8. Limitations on K Types [320] --
§9- Lemma on Linear Independence [328] --
§10. Problems [330] --
Chapter X. Global Characters --
§1. Existence [333] --
§2. Character Formulas for SL(2, R) [338] --
§3. Induced Characters [347] --
§4. Differential Equations [354] --
§5 Analyticity on the Regular Set, Overview and Example [355] --
§6. Analyticity on the Regular Set, General Case [360] --
§7. Formula on the Regular Set [368] --
§8. Behavior on the Singular Set [371] --
§9. Families of Admissible Representations [374] --
§10. Problems [383] --
Chapter XI. Introduction to Plancherel Formula --
§1. Constructive Proof for SU(2) [385] --
§2. Constructive Proof for SL(2, C) [387] --
§3. Constructive Proof for SL(2, R) [394] --
§4. Ingredients of Proof for General Case [401] --
§5. Scheme of Proof for General Case [404] --
§6. Properties of Ff [407] --
§7. Hirai’s Patching Conditions [421] --
§8. Problems [425] --
Chapter XII. Exhaustion of Discrete Series --
§1. Boundedness of Numerators of Characters [426] --
§2. Use of Patching Conditions [432] --
§3. Formula for Discrete Series Characters [436] --
§4. Schwartz Space [447] --
§5. Exhaustion of Discrete Series [452] --
§6. Tempered Distributions [456] --
§7. Limits of Discrete Series [460] --
§8. Discrete Series of M [467] --
§9. Schmid’s Identity [473] --
§10. Problems [476] --
Chapter XIII. Plancherel Formula --
§1. Ideas and Ingredients [482] --
§2. Real-Rank-One Groups, Part I [482] --
§3. Real-Rank-One Groups, Part II [485] --
§4. Averaged Discrete Series [494] --
§5. Sp(2,R) [502] --
§6. General Case [511] --
§7. Problems [512] --
Chapter XIV. Irreducible Tempered Representations --
§1. SL(2, R) from a More General Point of View [515] --
§2. Eisenstein Integrals [520] --
§3. Asymptotics of Eisenstein Integrals [526] --
§4. The Functions for Intertwining Operators [235] --
§5. First Irreducibility Results [540] --
§6. Normalization of Intertwining Operators and Reducibility [543] --
§7. Connection with Plancherel Formula when dim A = 1 [547] --
§8. Harish-Chandra’s Completeness Theorem [553] --
§9. R Group [560] --
§10. Action by Weyl Group on Representations of M [568] --
§11. Multiplicity One Theorem [577] --
§12. Zuckerman Tensoring of Induced Representations [584] --
§13. Generalized Schmid Identities [587] --
§14. Inversion of Generalized Schmid Identities [595] --
§15. Complete Reduction of Induced Representations [599] --
§16. Classification [606] --
§17. Revised Langlands Classification [614] --
§18. Problems [621] --
Chapter XV. Minimal K Types --
§1. Definition and Formula [626] --
§2. Inversion Problem [635] --
§3. Connection with Intertwining Operators [641] --
§4. Problems [647] --
Chapter XVI. Unitary Representations --
§1. SL(2, R) and SL(2, C) [650] --
§2. Continuity Arguments and Complementary Series [653] --
§3. Criterion for Unitary Representations [655] --
§4. Reduction to Real Infinitesimal Character [660] --
§5. Problems [665] --
Appendix A: Elementary Theory of Lie Groups --
§1. Lie Algebras [667] --
§2. Structure Theory of Lie Algebras [668] --
§3. Fundamental Group and Covering Spaces [670] --
§4. Topological Groups [673] --
§5. Vector Fields and Submanifolds [674] --
§6. Lie Groups [679] --
Appendix B: Regular Singular Points of Partial Differential Equations --
§1. Summary of Classical One-Variable Theory [685] --
§2. Uniqueness and Analytic Continuation of Solutions in Several Variables [690] --
§3. Analog of Fundamental Matrix [693] --
§4. Regular Singularities [697] --
§5. Systems of Higher Order [700] --
§6. Leading Exponents and the Analog of the Indicial Equation [703] --
§7. Uniqueness of Representation [710] --
Appendix C: Roots and Restricted Roots for Classical Groups --
§1. Complex Groups [713] --
§2. Noncompact Real Groups [713] --
§3. Roots vs. Restricted Roots in Noncompact Real Groups [715] --
Notes [719] --
References [747] --
Index of Notation [763] --
Index [767] --
MR, MR1880691
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