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http://dx.doi.org/10.1090/surv/142 Harmonic Analysis on Commutative Spaces Mathematical Surveys and Monographs Volume 142 Harmonic Analysis on Commutative Spaces Joseph A. Wolf American Mathematical Society EDITORIAL COMMITTEE Jerry L. Bona Michael G. Eastwood Ralph L. Cohen Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 20G20, 22D10, 22Exx, 53C30, 53C35. For additional information and updates on this book, visit www.ams.org/bookpages/surv-142 Library of Congress Cataloging-in-Publication Data Wolf, Joseph Albert, 1936- Harmonic analysis on commutative spaces / Joseph A. Wolf. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 142) Includes bibliographical references and indexes. ISBN 978-0-8218-4289-8 (alk. paper) 1. Harmonic analysis. 2. Topological groups. 3. Abelian groups. 4. Algebraic spaces. 5. Geometry, Differential. I. Title. QA403.W648 2007 515'.2433—dc22 2007060807 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2007 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 12 11 10 09 08 07 To Lois Contents Introduction xiii Acknowledgments xv Notational Conventions xv Part 1. GENERAL THEORY OF TOPOLOGICAL GROUPS Chapter 1. Basic Topological Group Theory 3 1.1. Definition and Separation Properties 3 1.2. Subgroups, Quotient Groups, and Quotient Spaces 4 1.3. Connectedness 5 1.4. Covering Groups 7 1.5. Transformation Groups and Homogeneous Spaces 8 1.6. The Locally Compact Case 9 1.7. Product Groups 12 1.8. Invariant Metrics on Topological Groups 15 Chapter 2. Some Examples 19 2.1. General and Special Linear Groups 19 2.2. Linear Lie Groups 20 2.3. Groups Defined by Bilinear Forms 21 2.4. Groups Defined by Hermitian Forms 22 2.5. Degenerate Forms 25 2.6. Automorphism Groups of Algebras 26 2.7. Spheres, Projective Spaces and Grassmannians 28 2.8. Complexification of Real Groups 30 2.9. p-adic Groups 32 2.10. Heisenberg Groups 33 Chapter 3. Integration and Convolution 35 3.1. Definition and Examples 35 3.2. Existence and Uniqueness of Haar Measure 36 3.3. The Modular Function 41 3.4. Integration on Homogeneous Spaces 44 3.5. Convolution and the Lebesgue Spaces 45 viii CONTENTS 3.6. The Group Algebra 48 3.7. The Measure Algebra 50 3.8. Adele Groups 51 Part 2. REPRESENTATION THEORY AND COMPACT GROUPS Chapter 4. Basic Representation Theory 55 4.1. Definitions and Examples 56 4.2. Subrepresentations and Quotient Representations 59 4.3. Operations on Representations 64 4.3A. Dual Space 64 4.3B. Direct Sum 64 4.3C. Tensor Product of Spaces 65 4.3D. Horn 67 4.3E. Bilinear Forms 67 4.3F. Tensor Products of Algebras 68 4.3G. Relation with the Commuting Algebra 69 4.4. Multiplicities and the Commuting Algebra 70 4.5. Completely Continuous Representations 72 4.6. Continuous Direct Sums of Representations 75 4.7. Induced Representations 77 4.8. Vector Bundle Interpretation 81 4.9. Mackey's Little-Group Theorem 82 4.9A. The Normal Subgroup Case 82 4.9B. Cohomology and Projective Representations 84 4.9C. Cocycle Representations and Extensions 85 4.10. Mackey Theory and the Heisenberg Group 87 Chapter 5. Representations of Compact Groups 93 5.1. Finite Dimensionality 93 5.2. Orthogonality Relations 96 5.3. Characters and Projections 97 5.4. The Peter-Weyl Theorem 99 5.5. The Plancherel Formula 101 5.6. Decomposition into Irreducibles 104 5.7. Some Basic Examples 107 5.7A. The Group 17(1) 107 5.7B. The Group SU(2) 107 5.7C. The Group SO(3) 110 5.7D. The Group 50(4) 111 5.7E. The Sphere S2 111 5.7F. The Sphere S3 112 5.8. Real, Complex and Quaternion Representations 113 5.9. The Frobenius Reciprocity Theorem 115 CONTENTS ix Chapter 6. Compact Lie Groups and Homogeneous Spaces 119 6.1. Some Generalities on Lie Groups 119 6.2. Reductive Lie Groups and Lie Algebras 122 6.3. Cartan's Highest Weight Theory 127 6.4. The Peter-Weyl Theorem and the Plancherel Formula 131 6.5. Complex Flag Manifolds and Holomorphic Vector Bundles 133 6.6. Invariant Function Algebras 136 Chapter 7. Discrete Co-Compact Subgroups 141 7.1. Basic Properties of Discrete Subgroups 141 7.2. Regular Representations on Compact Quotients 146 7.3. The First Trace Formula for Compact Quotients 147 7.4. The Lie Group Case 148 Part 3. INTRODUCTION TO COMMUTATIVE SPACES Chapter 8. Basic Theory of Commutative Spaces 153 8.1. Preliminaries 153 8.2. Spherical Measures and Spherical Functions 156 8.3. Alternate Formulation in the Differentiable Setting 160 8.4. Positive Definite Functions 165 8.5. Induced Spherical Functions 168 8.6. Example: Spherical Principal Series Representations 170 8.7. Example: Double Transitivity and Homogeneous Trees 174 8.7A. Doubly Transitive Groups 174 8.7B. Homogeneous Trees 175 8.7C. A Special Case 176 Chapter 9. Spherical Transforms and Plancherel Formulae 179 9.1. Commutative Banach Algebras 179 9.2. The Spherical Transform 184 9.3. Bochner's Theorem 187 9.4. The Inverse Spherical Transform 191 9.5. The Plancherel Formula for K\G/K 192 9.6. The Plancherel Formula for G/K 194 9.7. The Multiplicity Free Criterion 197 9.8. Characterizations of Commutative Spaces 198 9.9. The Uncertainty Principle 199 9.9A. Operator Norm Inequalities for K\G/K 199 9.9B. The Uncertainty Principle for K\G/K 202 9.9C. Operator Norm Inequalities for G/K 203 9.9D. The Uncertainty Principle for G/K 204 9.10. The Compact Case 204 x CONTENTS Chapter 10. Special Case: Commutative Groups 207 10.1. The Character Group 207 10.2. The Fourier Transform and Fourier Inversion Theorems 212 10.3. Pontrjagin Duality 214 10.4. Almost Periodic Functions 216 10.5. Spectral Theorems 218 10.6. The Lie Group Case 219 Part 4. STRUCTURE AND ANALYSIS FOR COMMUTATIVE SPACES Chapter 11. Riemannian Symmetric Spaces 225 11.1. A Fast Tour of Symmetric Space Theory 225 11.1 A. Riemannian Basics 225 11.IB. Lie Theoretic Basics 226 11.1C. Complex and Quaternionic Structures 229 11.2. Classifications of Symmetric Spaces 231 11.3. Euclidean Space 236 11.3A. Construction of Spherical Functions 236 11.3B. General Spherical Functions on Euclidean Space 238 11.3C. Positive Definite Spherical Functions on Euclidean Space 240 11.3D. The Transitive Case 242 11.4. Symmetric Spaces of Compact Type 245 11.4A. Restricted Root Systems 245 11.4B. The Cartan-Helgason Theorem 246 11.4C. Example: Group Manifolds 249 11.4D. Examples: Spheres and Projective Spaces 250 11.5. Symmetric Spaces of Noncompact Type 252 11.5A. Restricted Root Systems 253 11.5B. Harish-Chandra's Parameterization 254 11.5C. Hyperbolic Spaces 255 11.5D. The c-Function and Plancherel Measure 257 11.5E. Example: Groups with Only One Conjugacy Class of Cartan Subgroups 258 11.6. Appendix: Finsler Symmetric Spaces 260 Chapter 12. Weakly Symmetric and Reductive Commutative Spaces 263 12.1. Commutativity Criteria 263 12.2. Geometry of Weakly Symmetric Spaces 264 12.3. Example: Circle Bundles over Hermit ian Symmetric Spaces 268 12.4. Structure of Spherical Spaces 272 12.5. Complex Weakly Symmetric Spaces 275 12.6. Spherical Spaces are Weakly Symmetric 277 12.7. Kramer Classification and the Akhiezer-Vinberg Theorem 282 12.8. Semisimple Commutative Spaces 287 CONTENTS xi 12.9. Examples of Passage from the Semisimple Case 290 12.10. Reductive Commutative Spaces 293 Chapter 13. Structure of Commutative Nilmanifolds 299 13.1. The "2-step Nilpotent" Theorem 299 13.1 A. Solvable and Nilpotent Radicals 299 13.1B. Group Theory Proof 300 13.1C. Digression: Riemannian Geometry Proof 301 13.2. The Case Where N is a Heisenberg Group 303 13.3. The Chevalley-Vinberg Decomposition 309 13.3A. Digression: Chevalley Decompositions 309 13.3B. Weakly Commutative Spaces 314 13.3C. Weakly Commutative Nilmanifolds 317 13.3D. Vinberg's Decomposition 318 13.4. Irreducible Commutative Nilmanifolds 319 13.4A. The Irreducible Case — Classification 320 13.4B. The Irreducible Case — Structure 321 13.4C. Decomposition into Irreducible Factors 326 13.4D. A Restricted Classification 327 Chapter 14. Analysis on Commutative Nilmanifolds 329 14.1. Kirillov Theory 329 14.2. Moore-Wolf Theory 330 14.3. The Case where N is a (very) Generalized Heisenberg Group 335 14.4. Specialization to Commutative Nilmanifolds 338 14.5. Spherical Functions 341 14.5A. General Setting for Semidirect Products N x K 342 14.5B. The Commutative Nilmanifold Case 342 Chapter 15. Classification of Commutative Spaces 345 15.1. The Classification Criterion 345 15.2. Trees and Forests 350 15.2A. Trees and Triples 350 15.2B.
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  • Nilpotent Orbits and Theta-Stable Parabolic Subalgebras

    Nilpotent Orbits and Theta-Stable Parabolic Subalgebras

    REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 2, Pages 1{32 (February 3, 1998) S 1088-4165(98)00038-7 NILPOTENT ORBITS AND THETA-STABLE PARABOLIC SUBALGEBRAS ALFRED G. NOEL¨ Abstract. In this work, we present a new classification of nilpotent orbits in a real reductive Lie algebra g under the action of its adjoint group. Our classification generalizes the Bala-Carter classification of the nilpotent orbits of complex semisimple Lie algebras. Our theory takes full advantage of the work of Kostant and Rallis on p , the “complex symmetric space associated C with g”. The Kostant-Sekiguchi correspondence, a bijection between nilpotent orbits in g and nilpotent orbits in p , is also used. We identify a fundamental C set of noticed nilpotents in p and show that they allow us to recover all other C nilpotents. Finally, we study the behaviour of a principal orbit, that is an orbit of maximal dimension, under our classification. This is not done in the other classification schemes currently available in the literature. Introduction Let g be a semisimple Lie algebra and G its adjoint group. We say that an C C element x of g is nilpotent if and only if, ad : y [x; y] for all y g ,isa C x C nilpotent endomorphism of g . Kostant (see also Dynkin→ [Dy]) has shown,∈ in his C fundamental 1959 paper [Ko], that the number of nilpotent orbits of G in g is C C finite. The Bala-Carter classification can be expressed as follows: There is a one-to-one correspondence between nilpotent orbits of g and con- C jugacy classes of pairs (m; p ), where m is a Levi subalgebra of g and p is a m C m distinguished parabolic subalgebra of the semisimple algebra [m; m].