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
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© 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. The Mixed Case 351 15.2C. The Nilmanifold Case 353 15.3. Centers 354 15.4. Weakly Symmetric Spaces 357 Bibliography 367 Subject Index 373 Symbol Index 383 Table Index 387 Introduction
Commutative space theory is a common generalization of the theories of com pact topological groups, locally compact abelian groups, riemannian symmetric spaces and multiply transitive transformation groups. This is an elegant meeting ground for group theory, harmonic analysis and differential geometry, and it even has some points of contact with number theory and mathematical physics. It is fascinating to see the interplay between these areas, as illustrated by an abundance of interesting examples. There are two distinct approaches to the theory of commutative spaces: ana lytic and geometric. The geometric approach, which is the theory of weakly sym metric spaces, is quite beautiful, but slightly less general and is still in a state of rapid development. The analytic approach, which is harmonic analysis of commu tative spaces, has reached a certain plateau, so it is an appropriate moment for a monograph with that emphasis. That is what I tried to do here. Commutative pairs (G, K) (or commutative spaces G/K) can be characterized in several ways. One is that the action of G on L2(G/K) is multiplicity-free. Another is that the (convolution) algebra L1(K\G/K) of if-bi-invariant functions on G is commutative. A third, applicable to the case where G is a Lie group, is that the algebra D(G, K) of G-invariant differential operators on G/K is commutative. The common ground and basic tool is the notion of spherical function. In the Lie group case the spherical functions are the (normalized) joint eigenfunctions of the commutative algebra D(G, K). The result is a spherical transform, which reduces to the ordinary Fourier transform when G = Rn and K is trivial, an inversion formula for that transform, and a resulting decomposition of the G-module L2 {G/K) into irreducible representation spaces for G. In many cases this can be made quite explicit. But in many others that has not yet been done. This monograph is divided into four parts. The first two are introductory and should be accessible to most first year graduate students. The third takes a bit of analytic sophistication but, again, should be reasonably accessible. The fourth describes recent results and in intended for mathematicians beginning their research careers as well as mathematicians interested in seeing just how far one can go with this unified view of algebra, geometry and analysis. Part 1, "General Theory of Topological Groups", is meant as an introduction to the subject. It contains a large number of examples, most of which are used in the sequel. These examples include all the standard semisimple linear Lie groups, the Heisenberg groups, and the adele groups. The high point of Part 1, beside
xiii xiv INTRODUCTION the examples, is construction of Haar measure and the invariant integral, and the discussion of convolution product and the Lebesgue spaces. Part 2, "Representation Theory and Compact Groups", also provides back ground, but at a slightly higher level. It contains a discussion of the Mackey Little-Group method and its application to Heisenberg groups, and a proof of the Peter-Weyl Theorem. It also contains a discussion of the Cart an highest weight theory with applications to the Borel-Weil Theorem and to recent results on in variant function algebras. Part 2 ends with a discussion of the action of a locally compact group G on L2(G/T), where Y is a co-compact discrete subgroup. Part 3, "Introduction to Commutative Spaces", is a fairly complete introduc tion, describing the theory up to its resurgence. That resurgence began slowly in the 1980's and became rapid in the 1990's. After the definitions and a num ber of examples, we introduce spherical functions in general and positive definite ones in particular, including the unitary representation associated to a positive definite spherical function. The application to harmonic analysis on G/K consists of a discussion of the spherical transform, Bochner's theorem, the inverse spher ical transform, the Plancherel theorem, and uncertainty principles. Part 3 ends with a treatment of harmonic analysis on locally compact abelian groups from the viewpoint of commutative spaces. Part 4, "Structure and Analysis for Commutative Spaces", starts with rie- mannian symmetric space theory as a sort of role model, and then goes into recent research on commutative spaces oriented toward similar structural and analytical results. The structure and classification theory for commutative pairs (G,K), G reductive, includes the information that (G, K) is commutative if and only if it is weakly symmetric, and this is equivalent to the condition that (GC,KC) is spher ical. Except in special cases the problem of determining the spherical functions, for these reductive commutative spaces, remains open. The structure and classi fication theory for commutative pairs (G, K), where G is the semidirect product of its nilradical N with the compact group K, is also complete, and in most cases here the theory of square integrable representations of nilpotent Lie groups leads to information on the spherical functions. The structure and classification in gen eral depends on the results for the reductive and the nilmanifold cases; it consists of methods for starting with a short list of pairs (G, K) and constructing all the others. Finally there is a discussion of just which commutative pairs are weakly symmetric. At this point I should point out two areas that are not treated here. The first, already mentioned, is the general theory of weakly symmetric spaces, and the closely related areas of geodesic orbit spaces and naturally reductive riemannian homogeneous spaces. That beautiful topic, touched momentarily in Section 13.1C, has an extensive literature. The second area not treated here consists of certain extensions of (at least parts of) the theory of commutative spaces. This includes the extensive but somewhat technical theory of semisimple symmetric spaces, (the pseudo-riemannian analogs of riemannian symmetric spaces of noncompact type), the theory of generalized Gelfand pairs (G,H), and the study of irreducible unitary representations of G that have an iif-fixed distribution vector. It also includes several approaches to NOTATIONAL CONVENTIONS xv infinite dimensional analogs of Gelfand pairs. That elegant area is extremely active but its level of technicality takes it out of the scope of this book.
Acknowledgment s
Much of the material in Parts 1, 2 and 3 was the subject of courses I taught at the University of California, Berkeley, over a period of years. Questions, comments and suggestions from participants in those courses certainly improved the exposi tion. Some of the material in Part 3 relies on earlier treatments of J. Dieudonne [Di] and J. Faraut [Fa], and much of the material in Part 4 depends on O. Yakimova's doctoral dissertation [Y3]. In addition, a number of mathematicians looked at early versions of this book and made useful suggestions. These include D. Akhiezer (com munications concerning his work with E. B. Vinberg on weakly symmetric spaces), D. Bao (discussions on Finsler manifolds), R. Goodman (advice on how to organize a book), I. A. Latypov and V. M. Gichev (communications concerning their work on invariant function algebras), J. Lauret, H. Nguyen and G. Olafsson (for going over the manuscript), G. Ratcliff and C. Benson (communications concerning their work with J. Jenkins on spherical functions for commutative Heisenberg nilmani- folds), and the three mathematicians who refereed this volume (for some very useful remarks). I especially want to thank O. Yakimova for a number of helpful conversations concerning her work and E. B. Vinberg's work on classification of smooth commu tative spaces.
Notational Conventions
M, C, M and O denote the real, complex, quaternionic and octonionic number systems. If F is one of them, then x H-> X* denotes the conjugation of F over R, Fmxn denotes the space ofmxn matrices over F, and if x G Fmxn then x* eFnXm is its conjugate transpose. We write ReFnxn for the hermitian (x = x*) elements of Fnxn and ReFp Xn for those of trace 0, and we write ImFnXn for the skew-hermitian (x + #* = 0) elements of FnXn; that corresponds to the case n = 1. In general we use upper case roman letters for groups, and when possible we use the corresponding lower case letters for their elements. If G is a Lie group then g denotes its Lie algebra. If I) is a Lie subalgebra of g then (unless it is defined differently) H is the corresponding analytic subgroup of G. Bibliography
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absolutely irreducible representation, 321 Borel subalgebra, 126 adele group, 52 : real, 127 adele ring, 52 Borel subgroup, 126 adjoint representation, 121 : real, 127 affme group, 15 Borel-Weil Theorem, 135 Akhiezer-Vinberg Theorem, 281 Borel-Weil-Bott Theorem, 135 algebra bounded (G, K)—spherical function, 166, : Jordan, 27 184, 242 :: euclidean, 27 bounded symmetric domain :: exceptional, 27 : complex, 232 :: formally real, 27 : quaternionic, 232 :: reduced structure group, 27 : real, 232 :: special, 27 Bourbaki numbering of the simple roots, :: structure group, 27 125 : Lie, 21, 88, 120 Bruhat decomposition, 248 : associative, 26 bundle almost periodic function, 217 : fiber almost-complex manifold, 133 :: associated, 81 almost-complex structure, 133, 229 :: principal, 81 a-representation, 85 : holomorphic line, 88 amenable group, 218 : structure group, 81 antiholomorphic tangent bundle, 133 : vector antiholomorphic tangent space, 133 :: associated, 81, 117 approximate identity, 50, 155 associated vector bundle, 117 c-function, 256 associative algebra, 26 : formula, for symmetric space of automorphism group, 26 noncompact type, 257 Carcano's Theorem, 303 Baker-Campbell-Hausdorff formula, 88 Cartan duality Banach *-algebra, 56 : of orthogonal involutive Lie algebras, Banach algebra 228 : commutative, 179 : of riemannian symmetric spaces, 229 Banach-Steinhaus Theorem, 161 Cartan involution, 20, 21, 171 Bessel function, 244, 307 : conjugacy, 171 bilinear form Cartan subalgebra, 122, 311 : antisymmetric, 68 Cartan subgroup, 122 : degenerate, 25 : compact part, 171 : invariant, 68 : fundamental, 171 : symmetric, 68 : maximally compact, 171 block matrix, 13 : maximally noncompact, 171 Bochner space for G/K, 195 : maximally split, 171 Bochner space for K\G/K, 187 : noncompact (split) part, 171 Bochner Theorem, 187 Cartan's Highest Weight Theorems, 129 Bochner-Godement Theorem, 187 Cartan-Helgason Theorem, 132, 247 Bohr compactification of G, 216 Cartan-Killing form, 121 Borel map, 84 categorical quotient, 238, 242
373 374 SUBJECT INDEX
Cauchy-Kowalevski Theorem, 133 C*-algebra of G, 59 Cauchy-Riemann equations, 133, 134 cyclic representation, 166 Cayley division algebra, 26 cyclic vector, 166 central character of a representation, 129 central reduction of a commutative pair, decomposable Gelfand pair, 345 320 defect of a group action, 317 central reduction of a Gelfand pair, 320 degree (of a vertex in a graph), 350 character (of a representation), 97 differentiability : central, 129 : level, 119 : infinitesimal, 334 differential operator Chebyschev polynomial, 176 : invariant, 160 Chevalley decomposition, 309 direct integral : extremely weak, 312 : of Hilbert spaces, 75 2 : very weak, 310 : L : weak, 310 :: of Hilbert spaces, 75 co—isotropic subspace, 314 : LP co—isotropic symplectic action, 314 :: of Hilbert spaces, 76 co-rank of a group action, 317 :: of linear operators, 76 cocycle representation, 85 :: of representations, 76 coefficient homomorphism, 84 direct product group, 12 coefficient of a representation, 94 discrete series commutative Banach algebra, 179 : relative, 331 commutative nilmanifold, 319 distributions on manifolds, 161 : irreducible, 320 doubly transitive group, 174 commutative pair, 153 dual homomorphism, 211 : central reduction, 320, 345 dual lattice, 208 : criterion, 345 Dynkin diagram, 125 : decomposable, 345 Engel subalgebra, 311 : indecomposable, 346 e-concent rated, 199 : maximal, 320, 345 equivalence (of representations), 61 : principal, 327, 346 euclidean group, 15 : restricted classification, 327 : proper, 15 : Sp(l)-saturated, 327 exponential map : Sp(l)-saturated Gelfand pair, 346 : riemannian, 225 commutative space, 153 exponential series, 120 commuting algebra, 61, 71 extension compactly embedded subgroup, 279 : of a representation to its stabilizer, 83 compatible root order, 246, 253 extension property, 83 completely reducible representation, 60 extremely weak Chevalley decomposition, complex manifold, 133 312 complex orthogonal group, 21 complex structure, 133 field : integrable, 229 : topological, 32 : invariant, 229 : p-adic, 32 complex symmetric space, 229 Finsler geometry, 260 complex symplectic group, 22 Finsler space, 261 complexification, 30 : Berwald, 262 complexified tangent space, 133 : absolutely homogeneous, 261 complexity of a group action, 317 : distance, 261 concentrated, a measure on a set, 192 : geodesic, 261 convolution : homogeneous, 261 : classical, 47 : isometry, 261 : of (finite Borel) measures, 51 : reversible, 261 : on discrete group, 47 : symmetric, 261 : on locally compact group, 47 Finsler structure, 260 convolution algebra, 49, 153 Finsler symmetric space, 261 convolution of measures, 155 : geodesic symmetry, 261 C*-algebra of (G, K), 188 Fock space, 89 SUBJECT INDEX 375 formal degree, 332 Gelfand transform, 179, 183 Fourier inversion Gelfand-Godement-Helgason Theorem, 160 : compact abelian group, 213 Gelfand-Mazur Theorem, 180 : compact group general linear group, 19 :: scalar, 102 generalized Heisenberg group, 34 : discrete abelian group, 213 geodesic, 225 : for (G, K) geodesic orbit space, 302 :: scalar, 196 geodesic symmetry, 226 :: vector, 196 Gichev-Latypov Theorem, 137 : for a locally compact abelian group, 212 Gindikin-Karpelevic formula, 257 : for symmetric space of noncompact (G,K) spherical functions on En, 236 type, 258 (G, K)-spherical function, 176 : product of abelian groups, 213 (G, K)-spherical function Fourier transform : Harish-Chandra formula, 254 : adjoint, 214 : on symmetric space of noncompact : classical, 208 type, 254 : compact group global inner product, 78 :: operator—valued, 104 Grassmann manifold, 29 : for G/K : complex, 232 :: vector, 196 : oriented real, 232 : for K\G/K, 193 : quaternionic, 232 : for a locally compact abelian group, : real, 232 208, 209 group, 3 : for symmetric space of noncompact : Heisenberg, 33, 87 type, 258 : Heisenberg for Fn, 34 : inverse : Lie, 119 :: classical, 208 : adele, 52 : locally compact abelian group, 214 : affine, 15 Freudenthal multiplicity formula, 131 : algebra, 49 Frobenius Reciprocity Theorem, 116 : automorphism, 26 function : complex orthogonal, 21 : (G, K)-spherical, 176 : complex symplectic, 22 : spherical, 157 : direct product, 12 function algebra, 136 : euclidean, 15 : antisymmetric, 136 : general linear, 19 : rotation-invariant, 136 : generalized Heisenberg for ¥p,q, 34 : self adjoint, 136 : indefinite orthogonal, 22, 23 : skew adjoint, 136 : indefinite special orthogonal, 24 : symmetric, 136 : indefinite special unitary, 24 functional equation (for spherical : indefinite symplectic unitary, 23 functions), 158 : indefinite unitary, 23 fundamental set : linear Lie, 20 : for action of a discrete group, 141 : linear algebraic, 20 : neighbor, 141 : ordinary orthogonal, 22 : normal, 141 : orthogonal, 21, 23 : projective general linear, 26 T function, 238 : proper euclidean, 15 Gelfand pair, 153 : real orthogonal, 22 : central reduction, 320, 345 : real symplectic, 22 : criterion, 345 : semidirect product, 14 : decomposable, 345 : special linear, 20 : indecomposable, 346 : special orthogonal, 22, 24 : maximal, 320, 345 : special unitary, 23, 24 : principal, 327, 346 : symplectic, 21 : restricted classification, 327 : symplectic unitary, 23 : special class, 154 : topological, 3 : Sp(l)-saturated, 327 : topological quotient, 4 : 5p(l)-saturated Gelfand pair, 346 : unitary, 23 376 SUBJECT INDEX
:: projective, 85 infinitesimal character (of a : very generalized Heisenberg for representation), 334 jr5x(t,u)? 34 infinitesimal transvection, 226 : volume preserving, 20 inner product group algebra, 49 : global, 78 integrability condition (C°°), 133 Haar integral, 35 integrability condition (Cw), 133 : left, 35 integrable complex structure, 229 : right, 36 integration on homogeneous space, 45 Haar measure, 35 intertwining operator, 61, 67 : left, 35 invariant integral, 35 : normalized, 93 invariant metric, 15 : right, 36 invariant vector field, 119 Hahn-Banach Theorem, 162 inverse Fourier transform, 208 has square integrable representations, 331 Inverse Function Theorem, 120 Heisenberg commutation relations, 88 inverse spherical transform : uniqueness of, 88 : for (G,K), 191 Heisenberg group, 33, 87, 303 irreducible Heisenberg group over C, 34 : algebraically, 59 Heisenberg group over F, 34 : topologically, 59 Hermite monomial, 89 isometry group, 225 Hermite polynomial, 88 isotropic subspace, 314 hermitian symmetric space, 229 isotropy subgroup, 9, 226 highest weight of a representation, 129 Iwasawa decomposition, 171, 248 Hilbert-Schmidt : inner product, 100, 147 Jacobi function, 256 : norm, 100 Jacobi identity, 120 : operator, 100, 147 Jacobi polynomial, 251, 252 Hirzebruch Proportionality Principle, 141 Jacobson—Morosov Theorem, 127 holomorphic function, 133 joint T>(G, K)-eigenfunctions, 160 holomorphic line bundle, 88 Jordan algebra, 27 holomorphic tangent bundle, 133 : euclidean, 27 holomorphic tangent space, 133 :: table, 28 holomorphic vector bundle, 134 : exceptional, 27 : formally real, 27 Hom(B7ri,B7r2), 67 homogeneous space, 9 : reduced structure group, 27 homogeneous tree, 175 : special, 27 hypergeometric equation, 250, 255 : structure group, 27 hypergeometric function, 251, 255 X-fixed vector, 167 hypergroup, 51, 202 Killing form, 121 Kimmelfeld-Vinberg Theorem, 274 ideal Kirillov construction, 330 : regular, 179 Kostant multiplicity formula, 131 identity component, 6 Kramer classification of spherical spaces, indecomposable 282 : algebraically, 60 : topologically, 60 Laguerre polynomial, 307 indecomposable Gelfand pair, 346 Leray spectral sequence, 135 indefinite orthogonal group, 22, 23 level of a weight, 131 indefinite special orthogonal group, 24 level of differentiability, 119 indefinite special unitary group, 24 Levi component, 309 indefinite symplectic unitary group, 23 Levi decomposition, 300 indefinite unitary group, 23 Levi subalgebra, 300, 309 index (or a vertex of a tree), 175 Lie algebra, 21, 88, 120 indicator (= characteristic) function, 199 : Cartan decomposition, 122 indivisible restricted root, 246, 253 : Cartan subalgebra, 122 induced spherical function, 169 : Dynkin diagram, 125 induction by stages, 79 : Schlafli-Dynkin diagram, 125 SUBJECT INDEX 377
Weyl group, 124 Mackey Little Group Theorem, 83 center, 121 Mackey obstruction, 86 centralizer, 122 Mal'cev splitting of a Lie algebra, 310 commutative, 121 matrix coefficient, 94 direct sum, 120 maximal commutative pair, 320 exponential map, 120 maximal compact subgroup, 171 homomorphism, 120, 121 : conjugacy, 171 ideal, 120 maximal Gelfand pair, 320, 345 nilpotent, 121 maximal ideal space, 179 normalizer, 122 : topology, 182 of a Lie group, 120 : weak * topology, 182 orthogonal involutive, 226 maximal weight of a representation, 129 radical, 121 mean (on a topological group), 218 rank, 124 measurable set, 75 reductive, 121 measure, 75 root : Plancherel :: Weyl chamber, 124 :: for (G,K), 191 :: chain, 124 : Radon, 157 :: hyperplane, 124 : atomic, 104 :: length, 124 : spectral, 218 root decomposition, 122 : spherical, 156 root lattice, 132 measure algebra, 50, 51 root length, 125 measure space, 75 root reflections, 124 complete, 75 root system, 122 finite, 93 positive, 123 product, 93 rank, 124 metaplectic representation, 91 simple, 123 minimal orthogonal involutive Lie algebra, roots, 122 227 semidirect sum, 120 minimal parabolic subalgebra, 171 semisimple, 121 Minkowski norm, 260 simple, 121 modular function, 41 solvable, 121 module (of an automorphism), 41 solvable radical, 121 multiplication of sets, 155 splittable, 310 multiplicative linear functional, 156, 179 subalgebra, 120 multiplicity free, 197 Lie group, 20, 119 multiplicity free vs. "multiplicity free", 307 : Cartan subgroup, 122 multiplicity of a subrepresentation, 70 : Lie subgroup, 120 multiplicity of a weight, 128 : Weyl group, 124 : centralizer, 122 Nelson's Theorem, 162 : exponential map, 120 : Garding's proof, 162 : homomorphism, 121 Newlander-Nirenberg Theorem, 133 : normalizer, 122 nilradical or nilpotent radical : rank, 124 : of a Lie algebra, 300 Lindelof, 142 : of a Lie group, 300 linear algebraic group, 20, 272 norm complex, 272 : global real, 272 :: L°°, 78 reductive, 272 :: LP, 78 linear algebraic groups, 82 normalized character (of a representation), linear functional 97 : multiplicative, 156, 179 linear Lie group, 20 octonion division algebra, 26 LP S-bandlimited, 202 automorphism group, 27 Lp e-concentrated, 202 multiplication diagram, 27 Lp-induced spherical function, 169 multiplication table, 26 Lp 5-concentrated, 202 octonion hyperbolic plane, 233 378 SUBJECT INDEX octonion projective plane, 233 Pontrjagin Duality Theorem, 214 1-parameter subgroup, 119 positive definite (G, K)-spherical function, one parameter subgroup, 119 167, 184 operator positive definite function, 165 : Hilbert-Schmidt, 147 : spherical, 167 : compact, 72 positive restricted root system, 171 : completely continuous, 72 positive Weyl chamber, 254 : trace class, 147 primary constituents of a representation, 71 orbit, 9 primary decomposition of a representation, ordinary orthogonal group, 22 71 orthogonal group, 21, 23 primary representation, 71 orthogonal involutive Lie algebra, 226 primary subrepresentation, 71 : Cartan duality, 228 primary subspace (of a representation : compact group, 228 space), 71 : compact type, 228 principal fiber bundle, 81 : direct sum, 227 principal Gelfand pair, 327, 346 : euclidean, 227 principal series representation, 174 : four classes of irreducible, 228 principal triple (F, F, V), 351 : irreducible, 227 product : isomorphism, 227 : direct, 12 : minimal, 227 : semidirect, 14 : noncompact type, 228 projective general linear group, 26 oscillator representation, 91 projective kernel, 140 projective space, 28 p-adic integers, 51 projective unitary group, 85 p-adic number field, 32 projective unitary representation, 85 Panyushev Theorems, 276 parabolic subalgebra, 126 quasi-character, 207 : minimal, 171 quaternion : real, 127 : structure parabolic subgroup, 36, 126 :: invariant, 113 : real, 127 : structure on a vector space, 113 parallel tensor field, 229 quaternionic structure, 230 Peter-Weyl Theorem, 99 quotient representation, 60 PfafRan : of an antisymmetric bilinear form, 333 radial part of the Laplace—Beltrami : polynomial, 333 operator, 254 Plancherel density radical : for symmetric space of noncompact : nilpotent, 300 type, 258 : solvable, 300 Plancherel formula : unipotent, 309 : compact group Radon measure, 35, 157 :: Hilbert-Schmidt, 101 rank :: operator-valued, 104 : of a riemannian symmetric space, 229 :: trace form, 102 : real, of a semisimple Lie group, 229 : for G/K, 196 rank of a group action, 317 : for K\G/K, 193 real form, 30 : locally compact abelian group, 214 : of a complex representation, 113 Plancherel measure : of a complex vector space, 113 : for (G,K), 191 real orthogonal group, 22 point mass, 155 real polarization, 329 Poisson algebra, 314 real rank, 229 Poisson bracket, 314 real symplectic group, 22 polarization reductive component, 309 : real, 329 reductive subalgebra, 309 polonais regular ideal, 179 : group, 84 regular set, 130 : space, 84 relative discrete series representation, 331 SUBJECT INDEX 379 representation : topologically completely irreducible, 62 : Banach algebra on a Banach space, 57 : unitary, 56 :: bounded, 57 :: projective, 85 : Segal-Shale-Weil, 91 : unitary equivalence, 61 : Weil, 91 : unitary principal series, 174 : absolutely irreducible, 321 : weight, 128 : adjoint, 121 : weight lattice, 131 : admissible, 148 : weight space decomposition, 128 : algebraic direct sum, 65 : L°° discrete direct sum, 65 : basic weight, 131 : Lp direct sum, 64 : character, 148 : Lp discrete direct sum, 65 : cocycle, 85 representation space, 56 : compact, 72 representative function, 140 : completely continuous, 72 restricted root space decomposition, 246, : completely reducible, 60 253 : complexification of real, 113 restricted root system, 171, 253 : contragredient, 64, 329 restricted Weyl group, 254 : distribution character, 148 Riemann-Lebesgue Lemma, 183, 186 : dual, 64, 329 riemannian covering, 229 : equivalence, 61 riemannian homogeneous space, 226 : finite dimensional riemannian nilmanifold, 301 :: character, 97 riemannian symmetric space, 226 :: normalized character, 97 : for an orthogonal involutive Lie : finite multiplicity, 71 algebra, 227 : fundamental weight, 131 : rank, 229 : global character, 148 Riesz-Thorin interpolation, 200 : group on Banach space, 56 root lattice, 132 :: bounded, 56 : induced scalar Fourier inversion
:: by stages, 79 : for (G,K)t 196 :: LP, 78 scalar Fourier transform, 193 unitary, 78 scalar part, 230 : infinitesimal character, 334 Schlafli-Dynkin diagram, 125 : left regular Schur Orthogonality Relation, 96 :: of group, 56 Schur's Lemma, 61 :: of group algebra, 57 Segal-Shale-Weil representation, 91 :: of measure algebra, 57 semidirect product group, 14 :: on LP{G), 56 seminorm, 161 : linear, 85 semisimple representation, 60 : metaplectic, 91 :: complex, 272 symmetry :: real, 272 : geodesic, 226 : pair symplectic action, 314 :: complex, 272 : co—isotropic, 314 :: real, 272 symplectic group, 21 : subgroup, 272 symplectic manifold, 314 (G, K)-spherical functions, 184, 242 symplectic unitary group, 23 spherical function, 157 : bounded, 166 tangent space, 225 : of type <5A, 270 tensor power : positive definite, 167 : antisymmetric spherical function for (G, K) :: of Banach representations, 67 : positive definite, 184 :: of Banach spaces, 67 spherical functions for (G,K), 184, 242 : of Banach representations : bounded, 184 :: projective, 67 spherical harmonics, 137 : of Banach spaces spherical inversion :: projective, 67 : for symmetric space of noncompact : symmetric type, 258 :: of Banach representations, 67 spherical measure, 156 :: of Banach spaces, 67 spherical principal series representation, tensor product 174 : of Banach algebras spherical transform, 184 :: projective, 68 : for K\G/K, 193 : of Banach representations : inverse :: exterior projective, 66 :: for (G,K), 191 : of Banach spaces splittable Lie algebra, 310 :: algebraic, 65 stabilizer :: projective, 66 : of a representation of a subgroup, 83 : of Hilbert spaces Stone's Theorem, 219 :: projective, 67 Stone-Weierstrass Theorem, 136 : of unitary representations subalgebra :: exterior projective, 67 : Borel, 126 :: interior projective, 67 :: real, 127 Thomas' Theorem, 160 : Cartan, 122, 311 Titchmarch Inequality, 200 : Engel, 311 topological action, 8 : parabolic, 126 topological field, 32 :: real, 127 topological group isomorphism, 11 : reductive, 309 topological space : reductive in the large algebra, 309 : regular, 3 subgroup topological transformation group, 8 : Borel, 126 topology :: real, 127 : quotient, 4 : Cartan, 122 : subspace, 4 : isotropy, 226 totally real submanifold, 239 : one-parameter, 119 trace class : parabolic, 126 : operator, 100, 147 :: real, 127 translation, 3 submanifold : left, 3 : regularly embedded, 120 : on quotient space, 5 subquotient representation, 60 : right, 3 subrepresentation, 60 translation—invariant vector field, 119 symmetric algebra, 334 transvection, 226 symmetric space tree, 175, 350 : grassmannian, 232 : homogeneous, 175 : hermitian, 229 : rooted, 350 : quaternionic, 230 :: root, 350 : riemannian, 226 trigonometric polynomial, 217 SUBJECT INDEX 381 2-step Nilpotent Theorem, 299 : structure of commutative spaces, 264 Two-Step Nilpotent Theorem, 299 Type I, 82 uncertainty principle : classical, 199 : for G/K, 204 : for K\G/K, 202 : for locally compact abelian groups, 199 unimodular (group), 41 unipotent radical, 309 unitary dual, 70 unitary dual group, 208 unitary equivalence (of representations), 61 unitary group, 23 unitary principal series representation, 174 universal covering group, 8 universal covering space, 7 universal enveloping algebra, 334 vector field : invariant, 119 vector Fourier inversion : for (G,K), 196 very generalized Heisenberg group, 34 Vinberg's Decomposition Theorem, 318 volume preserving group, 20 weak containment, 55 weak symmetry : conditions on, 267 : differential—geometric, 266 : group-theoretic, 265 weakly commutative coset space, 314 weakly commutative pair, 314 weakly symmetric : complex pair (GC,HC), 276 :: compact real form of, 276 :: real form of, 276 :: weak symmetry of, 276 : coset space G/K, 265 : pair (G,K), 265 : riemannian manifold, 264 : weak symmetry, 265 weight : highest, 129 : maximal, 129 : multiplicity, 128 : of a representation, 128 : space decomposition of representation space, 128 weight lattice, 131 : positive, 132 Weil representation, 91 Weyl character formula, 130 Weyl degree formula, 131 Weyl group, 124 Weyl involution, 277, 358 working assumptions Symbol Index A = expG(a), 245 C, complex number field, xv A(R), range of the spherical transform, 191 CmXn, m x n complex matrices, xv A : G = NAK -+ a by C(K\G/K), if-bi-invariant continuous g = v(g)expA(g)n(g), 254 functions that vanish at 00 on G, 153 .A(7r), commuting algebra of 7r, 71 Coo(X), continuous functions that vanish at a, maximal abelian subspace of s in Cartan infinity on X, 183 decomposition gg = t + s, 245 Coo(MA), 183 Ad, adjoint representation of Lie group, 121 Xn, character of finite dimensional ad = dAd, adjoint representation of Lie representation 7r, 97 algebra, 121 Xn(g) = trace n(g), 97 n ag, conjugation, 3 C //Kc, categorical quotient, 238 AP(G), almost periodic functions on G, C*(G), G*-algebra of G, 59 217 C*(G, K), the G*-algebra of (G, K), 188 Aut(Hn), automorphism group of Hn, 87, 303 P(G), left-invariant differential operators on G, 160 B(G/K), Bochner space for G/K, 195 V(G,K), G-invariant differential operators B(K\G/K), Bochner space for K\G/K, on G/K, 160 187 deg(7r) = dimi^Tr, degree of the Bq(G; A), Borel g-coboundaries, 84 representation 7r, 96 Bn, representation space of 7r, 56 deg(7r), formal degree of relative discrete B(B), algebra of bounded operators on series representation 7r, 332 Banach space B, 56 AG, modular function of G, 41 &/(€,»?) = /(K, 17]), 329 AG/H(h) = AG(h)/AH(h),U BS = BS(G,K), bounded (G, K)-spherical Der(Q, Lie algebra of derivations of [, 120 functions, 184 dist, distance function on V(T), 175 C(K\G/K), K-bi-invariant continuous E(T), edges of tree T, 175, 350 functions on G, 153 ^6,F4, collineation group of octonion 0 Ct{G) = {fe CC{G) I f(G) C M= }, 37 projective plane, 233 w C , real analytic, 119 exp^ : TX(M) —+ M, riemannian C-°°(G), distributions on G, 161 exponential map, 225 C-°°IG/K), distributions on G/K, 161 C-°°(K\G/K), distributions on K\G/K, [^r(/)](a;) = ^uj(f)uuj G HUJ, vector Fourier 161 transform, 196 p,q CC(G), continuous compactly supported ¥ , p x q matrices over F with hermitian functions G -+ C, 37 form of signature (p, q), 33 UJ 1 CC(K\G/K), K-bi-invariant continuous /M = fGf(9) (9~ )diJ,G(g) =mw(/), compactly supported functions on G, spherical transform, 184 153 f(n) = trace 7r(/), scalar-valued Fourier Cc~°°(G), compactly supported transform, 102 distributions on G, 161 / H^ JF(/) = (7r(/))[7r]Gg, operator-valued C^°°(G/K), compactly supported Fourier transform, 104 distributions on G/K, 161 /°(P) = /(P"1),154 C^~°°(K\G/K), compactly supported k fHg) = IK IK f( i9k2) dp,K {k1)dfiK (fc2), distributions on K\G/K, 161 153 384 SYMBOL INDEX 2 2 f(g) = 1(0(9)), 154 7i = fY Hy dr(y), L direct integral of fu,v, matrix coefficient, 94 Hilbert spaces, 75 p p 2F1, hypergeometric function, 251, 255 H = JY Hy dr(y), L direct integral of F4/Spin(9), octonion projective plane, 233 Hilbert spaces, 76 ^4^4/Spin(9), octonion hyperbolic plane, \) := t+ a, maximally split Cartan 233 subalgebra of g, 245, 253 Hom(B7ri,B7r2), 67 G —* G/H, projection to quotient, 121 Homer(Bni, BTT2), intertwining operators, G°, topological component of 1 EG, 6 67 G^), Borel g-cochains, 84 G*A, adele group of GJK, 52 I(TV,TT'), intertwining operators Bn —> Bnf, 61 Gc, complexification of G, 30 jT(7ri,7T2), intertwining operators, 67 Gfc n(C), complex Grassmann manifold, 29 ImCnXn, complex skew-hermitian n x n, Gfc n(H), quaternion Grassmann manifold, ' 29 xv ImHnXn, quaternion skew-hermit ian Gfc>n(R), real Grassmann manifold, 29 n x n, xv GpjqjF = #p,q;F X ^(p,^), 34 nXn (25, universal enveloping algebra of Q, 164 ImO , octonion skew-hermitian n x n, xv G = G^, Bohr compactification of G, 216 ImIRnXn, real skew-hermitian n x n, xv G, unitary dual of G, 70 IndS(C), spherical function L2-induced G, universal covering group of G, 8 from C, 169 Q, Gelfand transform, 183 IndQ'p(C), spherical function Lp—induced (g, 7J^, if-fixed vectors in H^, 167 lg, left translation, 3 ifn = Hn.£, usual Heisenberg group, 34 Lm , generalized Laguerre polynomial of iin, Heisenberg group, 87, 303 order n - 1, 307 k th H(i/>)i ^-primary subspace of Hn, 71 A (B), k antisymmetric power of Banach ^n;F> generalized Heisenberg group based space B, 67 n ' onF , 34 Afc(7r), kth antisymmetric power of Banach ^p,q;F, generalized Heisenberg group based representation 7r, 67 ' onF^, 34, 303 Art, root lattice of a Lie algebra, 132 ,t,tt;F) very generalized Heisenberg group Awt,G, weight lattice of a Lie group, 132 based on Fsx^'u), 34 ^wt G~> Positive cone in weight lattice of G, H, quaternion division algebra, xv '132 mXn H , m x n quaternion matrices, xv Awt, weight lattice of a representation, 131 SYMBOL INDEX 385 A Jt, positive cone in weight lattice of g, PGL(-), projective general linear group, 26 132 ^>a, simple restricted root system, 171 A, left regular representation, 56 4% , Jacobi function, 256 Xf, eigenvalue on / of convolution by CJ, iX+l )(M )) Vxti) = fKe< ' '"' dnK(k), 158 Harish—Chandra formula for spherical function, 254 M = Z (A), centralizer of A in K, 245 K n = SY ny ^T(^)' Lp direct integral of M(R), finite complex-valued Radon representations, 76 measures on R, 189 [iTf] G A/", representation class defined by M+(P), non-negative finite Radon / G n*, 329 measures on P, 189 Pn,V(z) = Fi(-n,iz + v + n + l,w + l; ^), A4^4, multiplicative linear functionals on 2 Jacobi polynomial, 251 commutative Banach algebra A, ProjRep(G), projective unitary maximal ideal space of A, 179 representations of G, 85 m = 3e(a), centralizer of a in t, 245 ^, simple subsystem of root system S, 123 m 1 157 (/) = JG/^M^" )^^)' PU(H), projective unitary group, 85 Meas(G), measure algebra of group G, 50 \±G, left Haar measure on G, 35 Qp, p—adic number field, 32 J^P, Plancherel measure for (G,K), 191 TO(^,7T), multiplicity of ^ in TT, 70 R = R(G, K), maximal ideal space of C*(G,K), 188 AT = exp(n), nilradical of minimal parabolic R, real number field, xv subgroup, 254 MmXn, m x n real matrices, xv NG{H), normalizer of H in G, 122 rg, right translation, 3 7 nXn n = $^7Gl]+(a a) 0~ > nilradical of minimal ReC , complex hermitian n x n, x.v parabolic subalgebra, 254 ReFjXn, trace 0 elements of FnXr\ xv rig(rj), normalizer of f) in g, 122 ReHnXn, quaternion hermitian n x n, x.v ||0||oo, global L°° norm, 78 ReOnXn, octonion hermitian n x n, xv nXn \\(J)\\P, global L^ norm, 78 ReM , real hermitian n x n, x.v IMIspec» spectral radius of a, 180 Reprai(G), a-representations of G, 85 ||/1|00, Lebesgue sup norm, 37 p, half the sum of the positive roots, 130 v:G = NAK -• AT by p, right regular representation, 57 fir = i/(flf)expi4(^)«(^), 254 Rn//K, categorical quotient, 242 X, semidirect product, 14 o(y,b),o(y), 21 0(n) = £/(n;M), orthogonal group, 23 S = S(G, K), (G, AT)-spherical functions, 0(n;C), complex orthogonal group, 21 184 k th 0(Pi q) — U(p, q\ M), indefinite real S (B), k symmetric power of Banach orthogonal group, 23 space P, 67 k th 0(Pj P(f) = Pf(uf), Pfaffian polynomial, 333 123 2 P (©), octonion (or Cayley) projective £a, restricted root system, 171 plane, 29 £a , positive restricted root system, 171 Pn(C), complex projective n-space, 29 SO(p,q), indefinite special orthogonal U(p,q) = U(p,q;C), indefinite complex group, 24 unitary group, 23 SO(r + s)/[SO(r) x SO(s)], oriented real U(p, g;C), indefinite complex unitary Grassmann manifold, 232 group, 23 SO(r,s)/[SO(r) x SO(s)}, real bounded domain, 232 V(T), vertices of tree T, 175, 350 SO*(V,s),SO*(2n), defined by quaternion W(g, a), restricted Weyl group, 254 skew-hermitian form, 24 Wt(r), weight system of representation r, Sp(V,b),Sp(V), 21 128 5p(m;C), complex symplectic group, 22 Sp(m;R), real symplectic group, 22 X/A, quotient of space X by algebra A, Sp(n) = £/(n; H), symplectic unitary group, 136 23 £i, basic (fundamental) weights, 131 Sp(p,q) = U(p,q;M), indefinite symplectic group, 23 Zi(G;A), Borel g-cocycles, 84 Sp(r + s)/[Sp(r) x Sp(s)\, quaternionic ZG(H), centralizer of H in G, 122 Grassmann manifold, 232 Zf, ring of integers in algebraic number Sp(r,s)/[Sp(r) x 5p(s)], quaternionic field F, 51 bounded domain, 232 Zp, ring of p—adic integers, 51 SU(n), special unitary group, 24 3, universal enveloping algebra of 3, 334 SU(n;F), special unitary group, 23 3g(rj), centralizer of J) in 0, 122 SU(p,q), indefinite special unitary group, 24 SU(r + s)/S(U(r) x U(s)), complex Grassmann manifold, 232 SU(r,s)/S(U(r) x U(s)), complex bounded domain, 232 Supp(/), support of a function /, 37 sx : M —>• M, geodesic symmetry to M at x, 226 SymCmXm, 234 T = fYpTy dr(y)7 LP direct integral of linear operators, 76 T, (rooted) tree, 350 T0,1(X), antiholomorphic tangent bundle, 133 T1'°(X), holomorphic tangent bundle, 133 TX(M), real tangent space to M at x, 225 TX(X)C, complexified tangent space, 133 Tx' (X), antiholomorphic tangent space, 133 2V (X), holomorphic tangent space, 133 t, Cartan subalgebra of m, 245 (T,S)HS = trace TS*, 100 TV , character of finite dimensional representation 7r, 97 TTT(#) = (deg?r)trace n(g), 97 Tgi translation by g, 5, 8 TCI, topologically completely irreducible, 62 Trig(G), trigonometric polynomials on G, 217 tu(^y(t)) = 7(t + w), transvection along geodesic 7, 226 C/(n) = C/(n;C), unitary group, 23 U(n;¥), unitary group over F, 20, 23 Table Index Adjoint Representations, Classical Groups, : Hyperbolic Spaces, 255 130 : Noncompact, Real Rank 1, 255 Adjoint Representations, Exceptional : Spheres and Projective Spaces, 250 Groups, 130 Riemannian Symmetric Spaces : G/K with G Classical Simple, 232 Benson-Jenkins-Ratcliff Classification, 305, : G/K with G Exceptional Simple, 232 306 : Complex Cases, 233 Brion-Mikityuk-Yakimova Classification : Group Manifolds and their : Compact, Semisimple, Non—Simple Noncompact Duals, 231 Weakly Symmetric Spaces, 289 : Quaternionic Cases, 235 : Complex Semisimple Non-Simple : Spaces of Rank 1, 233 Weakly Symmetric Spaces, 288 Simple Formally Real Jordan Algebras, 28 Classical Simple Lie Groups, 126 Commutative Principal Pairs, 355 Vector Representations, 129 Comparison of Tables 13.4.1 and 13.2.5, 326 Vinberg Classification of Maximal Connected Dynkin Diagrams, 125, 126 Irreducible Nilpotent Gelfand Pairs, 320 Dynkin Diagrams, 125 Weakly Symmetric Spaces Hermitian Symmetric Spaces, 233 : G/K with G Compact Simple, 282 : Real Forms, 235 : Branching Compact Non-Semisimple Circle Bundles, 292 Kac Classification, 305, 306 : Branching Noncompact Reductive Kramer Classification Non-Semisimple Circle Bundles, 293 : of Compact Spherical Spaces, 282 : Circle Bundles over Hermitian : of Complex Spherical Spaces, 283 Symmetric Spaces, 268 : of Noncompact Real Spherical Spaces, : Compact Non—Semisimple Circle 286 Bundles, 292 Maximal Indecomposable Principal : Compact, Semisimple, Not Simple, 289 Saturated Pairs, 328, 347 : Complex Simple Cases, 283 Maximal Irreducible Nilpotent Gelfand : Complex, Semisimple, Not Simple, 288 Pairs, 320 : Noncompact Real Simple Cases, 286 Maximal Principal Indecomposable : Noncompact Reductive Non-Reductive Non-Nilmanifold Non-Semisimple Circle Bundles, 292 Non-5p(l)-Saturated Gelfand Pairs, Yakimova Classification 352, 353 : Maximal Indecomposable Principal "Multiplicity Free" Irreducible 5p(l)-Saturated Gelfand Pairs, 348 Representations, 305, 306 : Maximal Indecomposable Principal Octonion Division Algebra, 27 Saturated Nilpotent Gelfand Pairs, 328 Quaternionic Symmetric Spaces, 235 : Maximal Principal Indecomposable : Complex Forms, 236 Non-Reductive Non-Nilmanifold Non-Sp(l)-Saturated Gelfand Pairs, Rank 2 Root Systems, 124 352, 353 Restricted Root System : Compact, Real Rank 1, 250 387 Titles in This Series 142 Joseph A. Wolf, Harmonic analysis on commutative spaces, 2007 141 Vladimir Maz'ya and Gunther Schmidt, Approximate approximations, 2007 140 Elisabetta Barletta, Sorin Dragomir, and Krishan L. Duggal, Foliations in Cauchy-Riemann geometry, 2007 139 Michael Tsfasman, Serge Vladu^, and Dmitry Nogin, Algebraic geometric codes: Basic notions, 2007 138 Kehe Zhu, Operator theory in function spaces, 2007 137 Mikhail G. Katz, Systolic geometry and topology, 2007 136 Jean-Michel Coron, Control and nonlinearity, 2007 135 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: Techniques and applications, Part I: Geometric aspects, 2007 134 Dana P. Williams, Crossed products of C*-algebras, 2007 133 Andrew Knightly and Charles Li, Traces of Hecke operators, 2006 132 J. P. May and J. Sigurdsson, Parametrized homotopy theory, 2006 131 Jin Feng and Thomas G. 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Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, second edition, 2003 104 Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas 'Ward, Recurrence sequences, 2003 103 Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanre, Lusternik-Schnirelmann category, 2003 102 Linda Rass and John Radcliffe, Spatial deterministic epidemics, 2003 101 Eli Glasner, Ergodic theory via joinings, 2003 100 Peter Duren and Alexander Schuster, Bergman spaces, 2004 99 Philip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, 2002 97 V. A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Martin Markl, Steve Shnider, and Jim Stasheff, Operads in algebra, topology and physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D. Neusel and Larry Smith, Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2: Model operators and systems, 2002 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1: Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery, A tour of subriemannian geometries, their geodesies and applications, 2002 90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Prenkel and David Ben-Zvi, Vertex algebras and algebraic curves, second edition, 2004 87 Bruno Poizat, Stable groups, 2001 86 Stanley N. Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential operators, and layer potentials, 2000 80 Lindsay N. Childs, Taming wild extensions: Hopf algebras and local Galois module theory, 2000 For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/.