Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions: Integral Geometry, Invariant Differential

GROUPS AND GEOMETRIC ANALYSIS Integral Geometry, Invariant Differential Operators, and Spherical Functions

This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks

Editors: SAMUEL EILENBERG AND HYMAN BASS A list of recent titles in this series appears at the end of this volume:

GROUPS AND GEOMETRIC ANALYSIS Integral Geometry, Invaria nt Diff e rent ia I 0perato rs, and Spherical Functions

Sigurdur Helgason Department of Mathematies Massaehusetts Institute of Technology Cambridge, Massachusetts

1984

ACADEMIC PRESS, INC. (Harcourt Brace Jooanovich, Publishers)

Orlando San Diego San Francisco New York London Toronto Montreal Sydney Tokyo Slo Paulo

COPYRIGHT @ 1984, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

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United Kirrgtloni Edition published by ACADEMIC PRESS, INC. (LONDON) LTD.

24/28 Oval Road, London NWI 7DX

Library of Congress Cataloging in Publication Data Helgason, Sigurdur, Date Groups and geometric analysis. (Pure and applied mathematics) Bibliography: v. 1, p. Includes index. Contents: v. 1. Radon transforms, invariant differential operators, and spherical functions. 1. Lie groups. 2. Geometry, Differential. I. Title. 11. Series: Pure and applied mathematics (Academic Press) QA3.P8 [QA387] 510s [512’.55] 83-7137 ISBN 0-12-338301-3 (v. 1) PRINTED IN THE UNITED STATES OF AMERICA

84858687

9 8 1 6 5 4 3 2 1

To Thor and Annie

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CONTENTS PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TO THE READER. . . . . . . . . . . . . . . . . . . . . . . . . SUGGESTIONS TENTATIVE CONTENTS OF THE SEQUEL . . . . . . . . . . . . . . . . . . . . . .

xiii xvii XiX

INTRODUCTION Geometric Fourier Analysis on Spaces of Constant Curvature 1. Harmonic Analysis on Homogeneous Spaces 1. General Problems . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . 2. Notation and Preliminaries . . . . . . . . . . . . . . . . . 2 . The Euclidean Plane R2 . . . . . . . . . . . . . . . . . . . . 3. 4.

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. . . I . Eigenfunctions and Eigenspace Representations . . . . . . . . . . . . 2. A Theorem ef Paley- Wiener Type . . . . . . . . . . . . . . . . . TheSphereS' . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . 2. Proof of Theorem 2.10 . . . . . . . . . . . . . . . . . . . . . . The Hyperbolic Plane H2 . . . . . . . . . . . . . . . . . . . . . . 1. Non-Euclidean Fourier Analysis . Problems and Results

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2. The Spherical Functions and Spherical Transforms . . . . . . . . . . . . 3 . The Non-Euclidean Fourier Transform . Proof of the Main Result . . . . . . 4 . Eigenfunctions and Eigenspace Representations . Proof of Theorems 4.3 and 4.4 . . . . . . . . . . . . . . . . . . . . . . . . 5 . Limit Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 4 4 15 16 16 23 29 29 38

44 58 69 72 78

CHAPTER I Integral Geometry and Radon Transforms 1 . Integration on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 1. Integration of Forms . Riemannian Measure . . . . . . . . . . . . . . . 2. Invariant Measures on Coset Spaces . . . . . . . . . . . . . . . . . . 3 . Haar Measure in Canonical Coordinates . . . . . . . . . . . . . . . . 2 . The Radon Transform on RN. . . . . . . . . . . . . . . . . . . . . . . 1 . Iniroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Radon Transform of the Spaces B(R")and Y ( R " .) The Support Theorem . . . . . . . . . . . . . . . . . . . . . . . .

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81 81 85 96 96 96 91

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3. The Inversion Formulas . . . . . . . . . . . . . . . . . . . . . . . . 4 . The Plancherel Formula . . . . . . . . . . . . . . . . . . . . . . . 5 . The Radon Transform of Disiribuiions . . . . . . . . . . . . . . . . . 6 . Iniegraiion over d-Planes. X-Ray Transforms . . . . . . . . . . . . . . 7. Applicaiions . . . . . . . . . . . . . . . . . . . . . . . . . . . . A . Partial Differential Equations . . . . . . . . . . . . . . . . . . . B . Radiography . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Appendix . Disiribuiions and Riesz Poieniials . . . . . . . . . . . . . . . 3. A Duality in Integral Geometry . Generalized Radon Transforms and Orbital Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . I . A Dualiiy for Homogeneous Spaces . . . . . . . . . . . . . . . . . . 2. The Radon Tramform for the Double Fibraiion . . . . . . . . . . . . . . 3. Orbiial Iniegrals . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . The Radon Transform on Two-Point Homogeneous Spaces. The X-Ray Transform . . . . . . . . . . . . . . . . . . . . . . . . . . I. Spaces of Consiani Curvature . . . . . . . . . . . . . . . . . . . . . A . The Hyperbolic Space . . . . . . . . . . . . . . . . . . . . . . B . The Spheres and the Elliptic Spaces . . . . . . . . . . . . . . . . 2. Compaci Two-Point Homogeneous Spaces . . . . . . . . . . . . . . . . 3. Noncompact Two-Poini Homogeneous Spaces . . . . . . . . . . . . . . 4 . The X - Ray Transform on a Symmeiric Space . . . . . . . . . . . . . . 5. Integral Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Integral Formulas Related to the Iwasawa Decomposiiion . . . . . . . . . 2. Integral Formulas for ihe Carian Decomposiiion . . . . . . . . . . . . . A . The Noncompact Case . . . . . . . . . . . . . . . . . . . . . . B . The Compact Case . . . . . . . . . . . . . . . . . . . . . . . C. The Lie Algebra Case . . . . . . . . . . . . . . . . . . . . . . 3. Iniegral Formulas for ihe Bruhai Decomposiiion . . . . . . . . . . . . . 6 . Orbital Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . Pseudo-Riemannian Manifolds of Consiani Curvaiure . . . . . . . . . . . 2 . Orbital Iniegrab for the Lorentzian Case . . . . . . . . . . . . . . . . 3 . Generalized Riesz Poteniials . . . . . . . . . . . . . . . . . . . . . 4. Deierminaiion of a Function from Its Integrals over Lorenirian Spheres . . . 5. Orbital Iniegrals on SL(2,R) . . . . . . . . . . . . . . . . . . . . . Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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117 122 126 126 130 131

139 139 143 149 150 151 152 161 164 177 178 180 181

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195 196 199 199 203 211 214 218 221 229

CHAPTER II Invariant Differential Operators 1. Differentiable Functions on R" . . . . . . . . . . . . . . . . . . . . . . 2. Differential Operators on Manifolds . . . . . . . . . . . . . . . . . . . . 1. Definiiion. The Spaces O(M) and b(M) . . . . . . . . . . . . . . . . . 2 . Topology of ihe Spaces 9(M)and I ( M ). Disiribuiions . . . . . . . . . . . 3. Effect of Mappings. The Adjoint . . . . . . . . . . . . . . . . . . . . 4 . The Laplace- Belirami Operator . . . . . . . . . . . . . . . . . . . . 3. Geometric Operations on Differential Operators . . . . . . . . . . . . . . 1 . Projections of Differeniial Operators . . . . . . . . . . . . . . . . . . 2. Transversal Paris and Separaiion of Variables.for Differeniial Operators . . .

233 239 239 239 241 242 251 251 253

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3. Radial Parts o f a Differential Operator . General Theory . . . . . . . . . . 259 265 4 . E.vamples of Radial Parts . . . . . . . . . . . . . . . . . . . . . . . Invariant Differential Operators on Lie Groups and Homogeneous Spaces . . . 274 I . Introductory Remarks . Examples . Problems . . . . . . . . . . . . . . . 274 280 2. The Algebra D(G/H) . . . . . . . . . . . . . . . . . . . . . . . . . 3. The Case of Two-Point Homogeneous Space . The Generalized 281 Darboux Equation . . . . . . . . . . . . . . . . . . . . . . . . . . Invariant Differential Operators on Symmetric Spaces . . . . . . . . . . . . 289 1. The Action on Distributions and Commutativity . . . . . . . . . . . . . 289 295 2 . The Connection with Weyl Group Invariants . . . . . . . . . . . . . . . 309 3. The “Polar Coordinate” Form of the Luplacian . . . . . . . . . . . . . 4 . The Luplace-Beltrami Operator for a Symmetric Space of Rank One . . . . 312 5 . The Poisson Equation Generalized . . . . . . . . . . . . . . . . . . . 315 6 . A-sgeirsson’s Mean- Value Theorem Generalized . . . . . . . . . . . . . . 318 323 7. Restriction of the Central Operators in D(G) . . . . . . . . . . . . . . . 8 . Invariant Differenrial Operators.for Complex Semisimple Lie Algebras . . . . 326 9 . Invariant Differential Operators for X = G / K G Complex . . . . . . . . . 329 330 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . 343 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Invariants and Harmonic Polynomials 1 . Decomposition of the Symmetric Algebra . Harmonic Polynomials . 2. Decomposition of the Exterior Algebra . Primitive Forms . . . . . 3. Invariants for the Weyl Group . . . . . . . . . . . . . . . . I . Symmetric Invariants . . . . . . . . . . . . . . . . . . 2 . Harmonic Polynomials . . . . . . . . . . . . . . . . . . 3 . Exterior Invariants . . . . . . . . . . . . . . . . . . . 4 . Eigenfunctions of Weyl Group Invariant Operators . . . . . . 5 . Restriction Properties . . . . . . . . . . . . . . . . . . 4. The Orbit Structure of p . . . . . . . . . . . . . . . . . . . 1. Generalities . . . . . . . . . . . . . . . . . . . . . . 2. Nilpotent Elements . . . . . . . . . . . . . . . . . . . 3. Regular Elements . . . . . . . . . . . . . . . . . . . . 4 . Simisimple Elements . . . . . . . . . . . . . . . . . . . 5 . Algebro-Geometric Results on the Orbits . . . . . . . . . . 5 . Harmonic Polynomials on p . . . . . . . . . . . . . . . . . Exercises and Further Results . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . .

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345 354 356 356 360 363 364 366 368 368 370 373 378 380 380 382 384

CHAPTER IV

Spherical Functions and Spherical Transforms 1 . Representations

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I . Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Compact Groups . . . . . . . . . . . . . . . . . . . . . . . . . .

385 385 390

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2. Spherical Functions: Preliminaries . . . . . . . . . . . . . . . . . . . . 1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Joint Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 3. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Elementary Properties of Spherical Functions . . . . . . . . . . . . . . . 4. Integral Formulas for Spherical Functions . Connections with Representations . 1. The Compact Type . . . . . . . . . . . . . . . . . . . . . . . . . 2 . The Noncompact Type . . . . . . . . . . . . . . . . . . . . . . . . 3. The Euclidean Type . . . . . . . . . . . . . . . . . . . . . . . . . 5. Harish-Chandra's Spherical Function Expansion . . . . . . . . . . . . . I . TheGeneralCase . . . . . . . . . . . . . . . . . . . . . . . . . . 2. The Complex Case . . . . . . . . . . . . . . . . . . . . . . . . . 6 . The c-Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I . The Behavior of q5A at x) . . . . . . . . . . . . . . . . . . . . . . . 2. The Rank-One Case . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Properties of H@) . . . . . . . . . . . . . . . . . . . . . . . . . . 4 . Integrals of Nilpotent Groups . . . . . . . . . . . . . . . . . . . . . 5 . The Weyl Group Acting on the Root System . . . . . . . . . . . . . . 6 . The Rank-One Reduction . The Product Formula of Gindikin-Karpelevit . . 7. The Paley-Wiener Theorem and the Inversion Formula for the Spherical Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Normalization of Measures . . . . . . . . . . . . . . . . . . . . . . 2. The Image of @(G) under the Spherical Transform. The Paley- Wiener Theorem . . . . . . . . . . . . . . . . . . . . . . 3. The Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . 8. The Bounded Spherical Functions . . . . . . . . . . . . . . . . . . . . . 1. Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Convex Hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 . Boundary Components . . . . . . . . . . . . . . . . . . . . . . . . 9 . The Spherical Transform on p, the Euclidean Type . . . . . . . . . . . . 10. Convexity Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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399 399 402 403 407 416 416 417 424 425 425 432 434 434 436 438 439 441 444 448 449

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450 454 458 458 459 461 467 472 481 491

CHAPTER V Analysis on Compact Symmetric Spaces 1. Representations of Compact Lie Groups . . . . . I . The Weights . . . . . . . . . . . . . . . 2 . The Characters . . . . . . . . . . . . . . 2 . Fourier Expansions on Compact Groups . . . . . 1. Introduction . L ' ( Q versus L2(K) . . . . . . . 2. The Circle Group . . . . . . . . . . . . . 3 . Spectrally Continuous Operators . . . . . . . 4 . Absolute Convergence . . . . . . . . . . . 5. Lacunary Fourier Series . . . . . . . . . . 3. Fourier Decomposition of a Representation . . . I . Generalities . . . . . . . . . . . . . . . 2. Applications to Compact Homogeneous Spaces .

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495 496 501 507 507 508 510 519 522 529 529 532

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CONTENTS 4. The Case of a Compact Symmetric Space . . . . . . . I . Finite-Dimensional Spherical Representations . . . . 2. The Eigenfunctions and the Eigenspace . Representations 3 . The Rank-Onecase . . . . . . . . . . . . . . . .

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Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SOLUTIONS TO EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . APPENDIX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . The Finite-Dimensional Representations o f sl(Z.0 . . . . . . . . . . . . . 2 . Representations and Reductive Lie Algebras . . . . . . . . . . . . . . . . 1. Sembimple Representations . . . . . . . . . . . . . . . . . . . . . . 2. Nilpotent and Sembimple Elements . . . . . . . . . . . . . . . . . . . 3 . Reductive Lie Algehras . . . . . . . . . . . . . . . . . . . . . . . . 3 . Some Algebraic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SYMBOLS FREQUENTLY USED . . . . . . . . . . . . . . . . . . . . . . . . . INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

534 534 538 542 543 548 551 597 597 600 600 602 605 607 611 646 650

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This volume is intended as an introduction to group-theoretic methods in analysis on spaces that possess certain amounts of mobility and symmetry. The role of group theory in elementary classical analysis is a rather subdued one; the motion group of R3 enters rather implicitly in standard vector analysis, and the conformal groups of the sphere and of the unit disk become important tools primarily after the Riemann mapping theorem has been established. In contrast, our point of view here is to place a natural transformation group of a given space in the foreground. We use this group as a guide for the principal concepts (like that of an invariant differential operator) and as motivation for the leading problems in analysis on the space. For examples of problems that arise naturally in such a framework, we call attention here to the Eigenfunction Problems (A, B, and C in Introduction $1, No. 1, p. 2), the Radon-inversion and Orbital Integral Problems (A-D in Chapter I, $3, p. 147), and the Invariant Differential Operator Problems (A-E in Chapter 11, $4, p. 277). The present volume is intended as a textbook and a reference work on the three topics in the subtitle. Its length is caused by a rather leisurely style with many applications and digressions, sometimes in the form of exercises (with solutions). The introductory chapter deals with the two-dimensional case, requiring only elementary methods and no Lie group theory. While this example has considerable interest in itself, it serves here to connect our main topics to classical analysis. Its primary purpose, however, is to introduce techniques and theories that generalize to semisimple Lie groups and symmetric spaces, including Harish-Chandra’s c-functions (treated in generality in Chapter IV), the author’s work on harmonic analysis on symmetric spaces (to be treated in generality in another volume), and group-theoretic analysis of the eigenfunctions of the Laplacian, including classical boundary value properties of harmonic functions. In Chapter I we give an exposition of invariant integration on homogeneous spaces and relate this to the structure theory of semisimple Lie groups. We formulate general analytic integral-geometric problems (Radon xiii

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transforms and orbital integrals) for a double fibration of a homogeneous space and develop their solutions in an elementary fashion for some simple classes of homogeneous spaces. In Chapter I1 we discuss the effect on differential operators of a group action on a manifold. This gives rise to a separation of variables for differential operators, projections, transversal parts, orbital parts, and radial parts of differential operators; all these are useful for problems with built-in invariance conditions. Chapter I11 deals with linear group action on a vector space and with the corresponding invariant polynomials and harmonic polynomials. The results, together with a description of the orbits, find analytic applications in the case of the linear isotropy representation of a symmetric space. In Chapter IV we study the (zonal) spherical functions, that is, the K-invariant eigenfunctions of the G-invariant differential operators on the Riemannian homogeneous space C/K. Their theory, as well as that of the corresponding spherical transform, is worked out in considerable detail for the case of symmetric space (and its tangent space). Here the connection with representation theory is particularly simple and important. In the investigation of the spherical function’s behavior at xi, HarishChandra’s beautiful c-function emerges. Eventually this function is expressed by the formula

the notation being explained at the end of $6. As indicated there, each detail in this formula has its own significance. In particular, the location of the singularities of c(A)is crucial for the proof of the Paley- Wiener-type theorem for the spherical transform, which in turn enters into the proof of the corresponding inversion and Plancherel formula. Chapter V deals with harmonic analysis on compact homogeneous spaces U/K. Because of the intimate connection with finite-dimensional representation theory for U , thechapter begins with a detailed exposition of the weight theory and the character theory for U. Since this book is in part intended as a textbook, we now give some description of its level. The first third of the book is introductory and has on occasion been used as a textbook for first-year graduate students without background in Lie group theory. The remainder of the book requires some standard functional analysis results. The Lie-theoretic tools needed can for example be found in my book “Differential Geometry, Lie Groups, and Symmetric Spaces” (abbreviated [DS]), of which the present book can be considered an “analytic continuation.” Thus, our aim has been to provide complete proofs of all the results in the book. Although this process of

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unification and consolidation has at times led to some simplifications of proofs, we have in the exposition been more concerned with clarity than brevity. Each chapter begins with a short summary and ends with historical notes giving references to source material : an effort has been made to give appropriate credit to authors of individual results. At the same time we have tried to make these notes reflect the fact that the logical order of the exposition often differs drastically from the order of the historical development. Much of the material in this book has been the subject of lectures at the Massachusetts Institute of Technology in recent years; some of the content of my lecture notes [198Oc, 19811 has been incorporated in the Introduction and in Chapter I. Parts of the manuscript have been read and commented on by M. Baum, M. Cowling, M. Flensted-Jensen, F. Gonzales, A. G. Helmink, G . F. Helmink, B. Hoogenboom, A. Koranyi, M. Mazzarello, J. Orloff, F. Richter, H. Schlichtkrull, and G. Travaglini. I am particularly indebted to T. Koornwinder for his numerous useful suggestions.

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SUGGESTIONS TO T H E READER Since this book is intended for readers with varied backgrounds and since it is written with several objectives in mind, we give some suggestions for its use. The primary purpose has been to provide readers having a modest Lietheoretic background with a self-contained account of the three topics in the subtitle. However, the various chapters are largely independent of each other and could individually serve as textbooks for one- or two-semester courses. (i) The introductory chapter is so elementary that it could be used for an advanced undergraduate course. (ii) Chapter I is an independent account of group-invariant integration and analytic integral geometry. A reader wishing only an elementary treatment of the Radon transform with some natural generalizations could read this chapter and skip $1 and $5. (iii) Chapter I 1 (together with $1 and $5 of Chapter I) provides an independent treatment of invariant differential operators. Some of the most natural problems for these operators are discussed in the introduction to & Chapter I, 11. (iv) Chapter 111 deals with invariants, particularly Weyl group invariants, harmonic polynomials, and orbit decomposition for the complex isotropy representation associated with a symmetric space. This chapter is almost entirely independent of the previous ones. (v) Chapter IV, when preceded by Chapter I ($1 and $5) and Chapter 11, provides an independent account of the theory of spherical functions with a certain degree of completeness. (vi) Chapter V (together with Chapter IV, $1, No. 2) gives an independent exposition of analysis on compact homogeneous spaces with emphasis on the symmetric ones. The chapter starts with a brief account of the basic representation theory of compact Lie groups. Exercises and Further Results. Each chapter ends with a few exercises; some of these are routine applications and others develop the general theory

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further. Stating such results with no hint of proof or reference I consider counterproductive; hence solutions are provided at the end of the book. These solutions are often fairly concise and some of them (indicated with a star) rely on other references. Hopefully such exercises can furnish suitable topics for student seminars.

TENTATIVE CONTENTS OF THE SEQUEL Geometric Analysis on Symmetric Spaces I

Duality for Symmetric Spaces The Space of Horocycles. Radon Transform and Its Dual. Spherical and Conical Representations. Conical Distributions and Intertwining Operators.

II The Fourier Transform on a Symmetric Space Plancherel Formula. The Paley- Wiener Theorem. Generalized Spherical Functions. The Case of K-Finite Functions.

111

Differential Equations on Symmetric Spaces Solvability. Eigenfunctions. Integral Representations. Mean-Value Theorems. Wave Equations. Huygens Principle.

IV

Eigenspace Representations Generalities. Irreducibility Criteria for the Compact Type U / K , Euclidean Type, Noncompact Type G / K , the Horocycle Space G / M N , and the Complex Space GIN

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I NTROD UCTlO N

GEOMETRIC FOURIER ANALYSIS ON SPACES OF CONSTANT CURVATURE In this introductory chapter we formulate what might reasonably be called harmonic analysis on homogeneous spaces of Lie groups; the examples of the plane R2. the sphere S2, and the hyperbolic plane H 2 are worked out in detail. For the last case the proofs exemplify those which generalize to symmetric spaces of the noncompact type. Since this chapter is limited to these basic examples, it is elementary and self-contained and does not require familiarity with Lie group theory. After the generalities of 41 we turn to the case R2 as a homogeneous space of its group of isometries. Here the Laplacian generates the algebra of invariant differential operators; we describe its eigenfunctions by means of “entire functionals” and determine the irreducibility of the eigenspace representations. We also state a variation of the classical Paley-Wiener theorem; its proof is postponed to 83, where the necessary tools from the theory of spherical harmonics are developed. The principal section ($4) of this chapter deals with the hyperbolic plane. The nonEuclidean Laplacian is essentially the only invariant differential operator. We give an integral representation of its eigenfunctions and determine the (exceptional) eigenvalues for which the corresponding eigenspace representation is not irreducible. A basic tool is a certain Fourier transform on the hyperbolic plane; when viewed in non-Euclidean terms, the classical Poisson integral formula for bounded harmonic functions becomes a result in this non-Euclidean Fourier analysis. The integral representation mentioned generalizes this formula to all eigenfunctions of the non-Euclidean Laplacian. The radial eigenfunctions are called spherical functions; their behavior at 03 is determined by the so-called c-function of Harish-Chandra which enters in the spectral decomposition of the Laplacian. The knowledge of the singularities of the c-function is also important in the proof of the Paley-Wiener-type theorem for the Fourier transform mentioned; this proof occupies a good part of this section.

81. Harmonic Analysis on Homogeneous Spaces 1. General Problems

Let X be a locally compact Hausdorff space acted on transitively by a locally compact group G. We assume G leaves invariant a positive measure p on X. Let Tx denote the (unitary) representation of G on the Hilbert space L * ( X ) defined by ( T , ( g ) f ) ( x ) = f(g-’ ex) for g E G, 1

2

GEOMETRIC FOURIER ANALYSIS

[Intro.

E L 2 ( X ) , x E X . By harmonic analysis on X is frequently meant the decomposition (in the sense of the so-called direct integral theory) of T, into irreducible representations. The corresponding decomposition of any f’ E L 2 ( X ) will then resemble the classical Fourier integral decomposition. The main tools in the analysis are the measure p and the theory of operator algebras. If X = G/K is a homogeneous space of a Lie group G , K a closed subgroup, the situation changes drastically because the machinery of differential calculus becomes available. To exploit this we consider the algebra D ( G / K ) of all differential operators on G / K which are invuriant under the translations z(g): x K + g x K from G (cf. Chapter 11, $4). A function on G/K which is an eigenfunction of each D E D ( G / K ) will be called a joint eigenfunction of D ( G / K ) . Given a homomorphism

f‘

x: D ( G / K )

+C

the space

E,(X) = { f ’ E Cm(X): Df’ = x(D),f’for all D E D ( G / K ) } is called a joint eigenspace. Let T, denote the natural representation of G on E,(X), that is, (T.,(g)f’)(x) = f’(y-’ ex). These representations are called eigenspace representations. By harmonic analysis on G/K we shall mean “answers” to the following problems: A. Decompose “arbitrary” junctions on X = G / K into joint eigenfirnctions of’ D(G/K). B. Describe the joint eigenspaces E,(X) of’ D(G/K). C. Determine for which x the eigenspace representation 7 is irreducible.

The space P ( X ) has a standard topology (Chapter 11, §2), E , ( X ) is given the induced topology, and irreducibility above means that there are no closed nontrivial invariant subspaces. It may happen that D ( G / K ) has no nontrivial operators, in which case the above considerations have no interest. But we shall see that for the symmetric spaces G / K , Problems A, B, and C are reasonable and interesting. We shall now discuss them in detail for R’, s’, and H’,the three simply connected two-dimensional Riemannian manifolds of constant curvature. 2. Notation and Preliminaries As usual, R and C will denote the fields of real and complex numbers, respectively, and 2 the ring of integers. We write Re c and Im c, respectively, for the real and imaginary parts of a complex number c. Let Z + = Z n Ri. R + = { t E R : t 2 0},

011

1. Harmonic Analysis on Homogeneous Spaces

3

If X is a topological space and A c X , then A denotes the interior of A and A or Cl(A) denotes the closure of A. Also C ( X ) [resp. C,(X)] denotes the space of complex-valued continuous functions (resp. of compact support) on X. Let R" = {x = (x],. . . , x,) : xi E R } and let di denote the partial derivative d/dxi. If (x = (a1,. . . ,a,,) is an n-tuple of integers (xi 2 0, we put = j o ; l . . . a;., = x ; ~. . . ,x: I(xI = a , + ... + a,,. Let Cm(R")denote the space of complex-valued functions with continuous partial derivatives of order Im.Let C"(R") or &R") denote the space of complex-valued C"-functions j ' on R". Given m E Z+ and a compact subset K c R", let (1)

ll.fll,";

=

c

suPl(D:f)(x)I.

121Im .YE

K

The space &(Itn)is topologized by means of the seminorms IIj'll,";.It is then a Frechet space. Let B(R") or C?(R") denote the space C,(R") n Cm(R"). Sometimes this notation C(R"), B(R"), and 9 ( R " ) will be used even when the functions are assumed to be real-valued. If K c R" is compact, let BK(R")denote the space of C"-functions with support contained in K ; this space has a topology given by the norms 11 Ilf, where m = 0, 1, 2 , . . . . A linear functional T on 9(R") is called a distribirtion if its restriction to each QK(Rf')is continuous. Let 9'(R")denote the space of all distributions on R". A locally integrable function F on R" gives rise to a distribution T,: .f -, f(x)F(x) dx on R". A distribution T E ~ ' ( R "is) said to be 0 on an open set U c R" if T ( 4 )= 0 for each 4 ~ 9 ( R " whose ) support is contained in U . If V is the union of all open subsets U , c R" on which T = 0, then a partition of unity argument shows that T = 0 on V. The complement of V is called the sirpport of T. A distribution T of compact support extends to a continuous linear form on &(R")by putting T ( j ' )= T(j0j ' ) , .f' E G'(R"), if .fb is any function in 9 ( R " ) which is identically 1 on a neighborhood of the support of T. The choice of jb is immaterial. In this way the dual space 8'(R") is identified with the space of distributions of compact support. In Chapter I1 we shall define these notions for manifolds M. Here we need them only when M is an open subset of R" and when M is the n-dimensional sphere S". In Chapter I, 52, No. 8 we review some further notions from distribution theory, in particular, differentiation, convolution, and Fourier transforms of distributions.

4

[Intro.

GEOMETRIC FOURIER ANALYSIS

52. The Euclidean Plane R 2 1. Eigenfunctions and Eigenspace Representations

If we view R2 as a homogeneous space of R2 acting on itself by translations, Problems A, B, and C boil down to ordinary Fourier analysis. In fact, the invariant differential operators are just the differential operators with constant coefficients, and the joint eigenfunctions are the constant multiples of the exponential functions. The eigenspace representations are one-dimensional, hence irreducible. However, we can also consider R2 as the homogeneous space R2 = M ( 2 ) / 0 ( 2 )of the group M(2) of all isometries of R Z over the orthogonal group O(2) [leaving (0,O) fixed]. Then it is easy to see that D ( G / K ) consists of the polynomials in the LapIacian L R 2 = a2/ax: + a 2 / a x : . If AEC and o a unit vector, the function x + el"x*W', where ( , ) is the usual inner product, is an eigenfunction of L R 2 with eigenvalue - A 2 . We write the Fourier transform f of a function j ' on R2 in the form

The Plancherel formula now reads

The Fourier inversion formula, valid, for example, if written

1' E 9 ( R z ) , can be

and gives an explicit answer to Problem A. Here r/w denotes the circular measure on S'. This formula suggests that general eigenfunctions should be obtained from the functions eiA('*O)by some kind of superposition. We shall now prove a precise result of this kind. Given a. b 2 0 let E,,b denote the space of holomorphic functions on C - {0} satisfying

(3)

I I j ' I l a , b = SUP(If(Z)le-a'r'-b'Z'-L ) < a. 2

2. The Euclidean Plane R2

823

5

Then Ea.b is a Banach space with the norm 11 Also Ea.bc Ea,,b,if and only if a Ia', b Ib', and in this case l l j ' l l a , b 2 l ~ j ' ~ ~ a so * ~ bthat ~ , the injection of Ea,b into E a ' , b ' is continuous. We can give the union =

u

Ea,b,

a. h

the inductive limit topology. This means that a fundamental system of neighborhoods of 0 in E is given by the convex sets W such that for each (a, h), W n Ea,b is a neighborhood of 0 in We identify the members of E with their restrictions to the unit circle S' and call the members of the dual space E' entire ,firnctionals on S'. Since these generalize measures it is convenient to write T ( f ) =J f ( ~ dT(w), )

f E E,

T E E'.

s1

The following result gives an answer to Problem B. Theorem 2.1. The eigen~irnctions of' the Laplacian on precisely the harmonic functions and the functions

j'(x)

(4)

=

R2 are

/s,ei'(x*u')ilT(o),

where A E C - (0) and T is an entire functional on S'. x

We note first that the right-hand side of (4)is well-defined because if ( x l , x,), the integrand is the restriction to S' of the function

=

2 --+

exp[+(iA)x,(z

+ z - ' ) + +Ax2(z- z-')I,

which indeed does belong to E. Now we need to characterize the members of E in terms of their Laurent series expansions. Lemma 2.2.

Let f be holomorphic in C - (0) and let

j'(z)=

c

an2"

n

he its Lairrent expunsion. Then

.I'E E a + , , b + E j oull r I-: > 0 I / ' iinrl

only if'

,/br euch 6 > 0.

6

GEOMETRIC FOURIER ANALYSIS

[Intro.

Consider the decomposition m

(5)

-1

+ C anzn= f+(z) + jL(z);

f(z) = C anzn 0

-m

since 1; = J' - f - and since 1- is bounded for IzI 2 1, ( 5 ) shows that E , , b = E a , o + E 0 , b . Thus it suffices to show that the function f(z) =

1anzn 0

belongs to

for each

E

tl,, =

> 0 if and only if

o(+), a + 6)''

n 2 0,

for each 6 > 0. Assume first this last condition is satisfied. Then for each 6 > 0 we have a constant C6 > 0 such that

Then oc

I J ( Z )I ~ C Ia,,(IzI"I C6e(a+6)lzl 0

so J ' E E a + 6 . 0 . On the other hand, suppose f E E , + , for all

E

> 0. Put

v = lim sup(n I a, I ' I " ) . n

We may take v > 0, so let 0 < 6 c v. Then for infinitely many n,

(where v - 6 is to be interpreted as an arbitrary large number if v = a). Using If'(n)(O)/n!II (2n1-1

1tl,,1=

J;

~ z ~ - n - ' ~ f ( 1nz1, z)~

zI=r

so

I an I I SUP I .l'(z) I, IzI=r

we have for the sequence r = r, = ne/(v - 6) ( n as in the inequality above)

023

2. The Euclidean Plane R2

It follows that for each

E

so

I!

I ae. Given

E

7

> 0,

> 0 this means

8, =

o(+). a + E)"

This proves the lemma. We can also formulate the result by saying that j ' ( z ) = longs to E,+,,o for all E > 0 if and only if its Laplace transform j'(t)e-" dt

M,Z"

be-

(Re z > a )

has its Laurent series

convergent for IzI > a. Given an entire functional T E E ' we associate with it a Fourier series

(7)

T

-c

a,,eine,

n

where, by definition, a,, = IS,e-in0~ T ( o ) . Proposition 2.3. A series jiinctional #'and only if'

EnaneinOis the Fourier series.for an entire

Proof. Let T E E ' have the Fourier series (7). Then for each a, b > 0 and each E > 0, T has a continuous restriction to E,+,,o and to E O . h + E .

8

GEOMETRIC FOURIER ANALYSIS

The series if

a"z"/n! converges to ea2 in the topology of E,,,,,.

fN(z) =

a"z"

1 1,n .

[Intro. In fact,

gN(z) = fh(z)e-(a+E)IZI,

N+l

then IgN(z)l Ie-'lz1, so for R > 0 sup Ig N ( Z ) I

lzlsR

2

5

1 g d Z )I + I

SUP ~ N ( z ) I sR

I4 > R

Ig N ( ' ) I

+ SUP e-&l2l. I4 > R

121

By the uniform convergence of f N to 0 on compact sets this implies -, 0 in as claimed. Consequently,

fN

c +. N

ear''

dT(e) = Iim N-0

Jsl

an a

0

Using a similar argument for E O , b + r ,the convergence (8) follows. On the other hand, suppose (8) holds. Then we can define a linear form T on E by T ( f )=

1ana-n n

1.

iff = a,,z". By the definition of the inductive limit, T is continuous provided that for each a, b 2 0 the restriction TIE,,, is continuous. Now it is easily seen by means of the closed graph theorem that the map f E E a , b + h E E a . O ,

with the notation of (9,is continuous. Thus it suffices to prove that for each a > 0 the restriction TI E a m 0is continuous. For 1'E Ea,O let j ( z ) = a,zn denote its Taylor series and F as in ( 6 ) its Laplace transform. Let ( X ) c Ea,o be a sequence converging to 0. Then

1;

IX ( Z ) I I

(9)

where Ak (10)

-+

A , ealzl,

0. The Laplace transforms F k ( z ) then satisfy for x > a IFk(X)I

5 A,

/omf2'a-x)'

dr = Ak(x - a ) - ' .

But since the right-hand side of (9) is rotation-invariant and since the function z --t J(eioz) has Laplace transform z -+ e-"F(e-"z) [because of (6)], the estimate in (10) implies (11)

IFk(Z)I 5 Ak(lzl - a)-'

for IzI > a.

2. The Euclidean Plane R2

Q21

9

If aik)are the Taylor coefficients of j i ( z ) , we have

with the integration taken, for example, over the circle zI = a then by (1 1) n!laLk'lI Ak(a

+ 1. But

+ l)n+l,

so that la-,,l(a

+ I)"+'(

!!)- 1.

Using our assumption (8), we deduce T ( j J -,0, proving the continuity.

+

Lemma 2.4. Let A 0, n E Z. The solutions f to LR2.f = -A2f sat= ei"ef(z)are the constant multiples of the jirnction isfying ,f'(eiez) 4l.,,(x) =

&

j ' , e i ~ ( x ~ ~ ) z , ,dw, (w)

where Xn(eie)= e'"'. Proof. It is clear that two such functions f are proportional on the unit circle; hence by Theorem 2.7 in Chapter I1 they are proportional on R2. A more direct proof is obtained by noting that in polar coordinates, a2 I a 1 a2 LR2=-+--+-ar2 r ar r2 do2' so

Because of the ellipticity of L R 2 ,the eigenfunction f is analytic. Expanding it in a power series in r, the differential equation gives a recursion formula for the coefficients which readily shows that all the solutions are pro port ional. To conclude the proof of the lemma it just remains to remark that 4,,nf 0. This is a special case of the elementary Lemma 2.7. Proposition 2.5. A function f satisfying LR2f = -1zf satisjes the functional equation I

r2n

2J f(z 2n 0

+ eiew)do = f ( z ) ~ $ ~ , ~ ( w z,) , w E C.

Conversely a continuous function f satisjying (12) is automatically of class C" and satisjes LRlf = - A 2 j .

10

[Intro.

GEOMETRIC FOURIER ANALYSIS

Proof.

If .f' is an eigenfunction, so is the function

F:w

+ eiew) (10.

+ Jo2n/,(i

In addition, F is radial, so by Lemma 2.4 it equals the function 4a,o(w) multiplied by some factor. This factor is obtained by putting w = 0; (12) follows. On the other hand, suppose J' is a continuous function satisfying (12). We multiply (12) by a function h(w) E 9 ( R 2 ) and integrate with respect to MI.By the invariance of dw under M(2) we get

;

j R j ( d h ( f ? - ' e ( s - 2 ) ) rlu

Jo2,.

=

.f'(z)

4A*O(W)h(M')

rlw

JR2

and this shows that j ' is of class C". Now apply L = LRz to (12) as a function of w. By its invariance under M(2) we get 1

r2n

Putting w = 0 we get Lf = -1% so the lemma is proved. We can now prove Theorem 2.1. Let f satisfy LR2.J'= -LZf, where Iz E C - (0). We expand the function 8 + f ( e i o z ) in an absolutely convergent Fourier series (13)

f'(eiez)= C cn(z)ei"', n

where

But this is an eigenfunction of the Laplacian satisfying the homogeneity assumption of Lemma 2.4. Hence cn(Z) = an 4A,n(z),

an E C.

Also, substituting tJ = (7r/2) - 4, we get

JO2=

4a,n(r) = if1(2~)1

eidrsin+

r -in+ dfp.

This shows incidentally that -n(r)

= (-1Y'4A,n(-r)*

2. The Euclidean Plane RZ

621

11

Consider now the Bessel function a '

J,(z) =

1 (-l)k(tz)zk++n[k!r(n+ k + 1)]-l, k=O

defined for all n, Z E C .(For - ~ E Z +the first coefficients vanish.) If in the integral for +A.,l we expand eiArsin@ in a power series and integrate term by term, we obtain (134

$i.,n(r)

=

iflJn(~r)*

O n the other hand, we have for n E Z +

r(n + i)(;z)-v,,(z)

+ h(z),

= 1

where

so lim n!(tz)-"J,(z)

=

1.

n++m

Hence the relation

c

<

I%Jn(JI.>I

a

7

which follows from (13) and the formulas

J-,,(Ar)

=

J,( - h)= ( - l)"J,,(Ar),

which follow from (1 3a), imply

1 lafll(r'n'/ln[!) < m. n

Thus by Prop. 2.3 there exists a T E E ' such that

-1 The formula for 4A,fl shows that c4A. T

eil(x.r'e)

u,eine.

=

,we-

ine.

n

which, if x

= ( x ~xz), ,

gives the Laurent expansion

12

[Intro.

GEOMETRIC FOURIER ANALYSIS

As we saw for the Taylor series for eaz, the series (14) converges in the topology of E . Thus we can apply T to it term by term. This gives

L,~~A(X*C~J) c IT(^))

4,.n(x)an= !(XI,

=

n

which is the desired representation of ,/'. On the other hand, let T E E' and consider the "integral" j ' ( x ) = Js,ei~lx~C~) rlT(w)

as well as the Fourier expansion Applying T to the expansion (14), we get

by (8) and the above estimates for J,(Ar) this series converges uniformly on compact sets, so ,/' at least is continuous. The functions q5,.,, all satisfy (12), and so does ,/' because of the uniform convergence. Hence it is an eigenfunction of L R 2 and Theorem 2.1 is proved. Problem C in 51 will now be answered with the following result. For 1E C let &,(R2) denote the eigenspace 8',(R2) =

1.1' E 6 ( R 2 ): L R 2 j '

=

-If/']

and let T, denote the natural representation of M(2) on &,(R2); that is, ( T , ( g ) f ) ( x )= f ( g - ' . x ) . The topology of 8 ( R 2 ) was explained in $1. No. 2; b , ( R 2 ) is a closed subspace. Theorem 2.6. only i f A # 0.

The eigenspuce representation T, is irretlircible ifand

We first have to prove a couple of lemmas. Given F e L 2 ( S 1 )consider , the function

Lemma 2.7. is one-to-one.

Let

A

# 0. Then the rnapping F

f defined by (16)

Proof. Let p([) = p(il, i2) be a polynomial and D the corresponding constant coefficient differential operator on R2 such that

D

( e i l x . i ) ) = p(i)ei(x.i) ri

$21 for

2. The Euclidean Plane R2 E C2. If

J'

13

= 0 in (16), we deduce from the above equation that P ( , ? w ~,?o~)F(o) , d o = 0.

Since p is arbitrary and ,? # 0, this implies F = 0. For 1 # 0 let X I denote the space of functions J' as defined by (16); is a Hilbert space if the norm of j ' is defined as the L2-norm of F on S'. Because of Lemma 2.7 this is well-defined. Lemma 2.8.

Let 3, # 0. Then the space X i is dense in &a(R2).

Proof. Let J ' € g A ( R 2and ) let J"((z) = ,f'(e''z). Consider the mapping @: 8 -,J'e of s' into the Frechet space &,(R2). Using the seminorms ( 1 ) in $1 on &(R2), it is clear what is meant by a differentiable mapping of a manifold into &(R2). In that sense the map @ is differentiable. The series (13) can be written in the form

@(O)

=

1a , ~ , ~ n e i n ' , n

so u n 4 A , nis the Fourier coefficient of the vector-valued function 0.Since the series (13) can be differentiated at will with respect to 0, we obtain for the kth derivative O(k)of 0.

s:'

an4,," = ( i t 1 - ~ ( 2 x ) - '

@(k)(8)e-ine no.

This gives an estimate Ila,4j.,nll I(const.)Kk for each of the indicated seminorms 11 11 of &(RZ),so by the completeness the series

C a,,4 a ,

n ein'

n

converges absolutely in the topology of & ( R 2 )to some limit (Do(@. But then @ and will have the same Fourier coefficients, so Q0 = @ and N

lim

c

N-+m - N

4 4 A . n

=

0) = .I

in the topology of &(R2).This proves the lemma. We can now prove Theorem 2.6. If ,? = 0, &. is obviously not irreducible since the constants form a nontrivial invariant subspace. Assuming now ,? # 0, we first prove that M(2) acts irreducibly on .PA.Let V # 0 be a closed invariant subspace of the Hilbert space X i . Then there exists an h E V such that h(0) = 1. We write

14

GEOMETRIC FOURIER ANALYSIS

[Intro.

Because of Lemma 2.4 the average h"Z) = 1 lo2nh(ei%) (10 271

is then given by

If ,/' in (16) lies in the annihilator V o of V in X i , the functions F and H are orthogonal on S'. Since V o is O(2)-invariant, this remains true for H replaced by its average over S'; in other words, 4A,obelongs to the double annihilator (VO)', which, by Hilbert space theory, equals I/. But then, since V is invariant under translations, it follows that for each t E R2 the function -, /s,eiilx.wleiJ.li.wi

belongs to V. Lemma 2.7 then implies that V o = {O}, whence the irreducibility of M(2) on X A . Passing now to bA,let W c bi be a closed invariant subspace. Then W n X i is an invariant subspace of H A . Let (jn) be a sequence in Wn converging to j ' € X A(in the topology of HA).From (16) for each jk we see, using Schwarz's inequality, that 1; -, j ' in the topology of b,. Thus j ' W, ~ so W n S Ais closed in Xi. Hence, by the above, W n X Ais X i or {O}. In the first case, W = 8, because of Lemma 2.8. In the second case, consider for each ,f E W the expansion (15). Each term L I , , ~ ~ , ,is, given by

This belongs to W n X i because the left-hand side is in X i , and the right-hand side belongs to W as a limit [in &(R')] of translates of members of W . It follows that j ' = 0 and the irreducibility is proved. Remarks. (i) Theorem 2.6 generalizes to arbitrary dimensions (cf. Exercise A2). (ii) For the exceptional value A = 0 the solution space B'o(R2) consists of the space of harmonic functions. In this case there is a bigger group acting on the solution space, namely, the conformal group (or rather its Lie algebra), and again a certain irreducibility holds (cf. Exercise A4).

2. The Euclidean Plane R2

621

15

2. A Theorem of Paley-Wiener Type A holomorphic function F on C" is said to have exponential type A > 0 if for each N E 2' there exists a constant C N such that

I F ( [ ) [ I C,(1

+ l[l)-NeA1'mcl,

(EC".

Here l i l ' = and Im i denotes the imaginary part of 1' + .. . + the vector ( = (i,, . . . , (J. Let ,XA(C")denote the space of holomorphic functions F satisfying the condition above and let X ( P )=

IJ H"A(C"). A>O

For simplicity we write g A ( R n )for the space 9 E A ( o j ( R "of) C"-functions on R" with support in the ball 1x1 I A. Then we have the following classical result. Theorem 2.9. ,f, where

,f

+

is

LI

(Paley-Wiener)

The

Fourier-Loplace

transjbrm

bijection of g A ( R " )onto .#'(C").

O u r viewpoint of harmonic analysis on the homogeneous space R2 = M ( 2 ) / 0 ( 2 ) , and more generally on R" = M(n)/O(n), relates functions /' on R" to functions 4 on R x s"- ' via the Fourier transform (18)

dX.

(We pass from n = 2 to general n because the proof of the result below requires a kind of an induction on the dimension). While Theorem 2.9 characterizes the holomorphic functions ,f(() it is more appropriate in the present context to ask for an intrinsic characterization of the functions 4(1,(0)in (18) as ./' runs through g(R"). (The similar problem for j ' E L' offers no difficulties.) We shall now obtain such a characterization. Let Im c denote the imaginary part of a complex number c. A vec. . . , u,J E C" is called isotropic if a: + . . . + a,' = 0. tor ( I = Theorem 2.10. The mapping j ' + f m p s 9 ( R " ) onto the set yf'jirnctions . f ( ~ w=) +(A, w ) E C ~ ' ( Rx s"- ') sutisfjing:

16

A

GEOMETRIC FOURIER ANALYSIS

[Intro.

(i) There exists u constant A > 0 such that jbr euch o the jirnction + $(A, w ) extends to u holomorphic firnction on C with the property

(19)

supl4(A, o)(l + I A I ) ~ ~ - ~ I ' "I *
for euch N E Z . (ii) For euch k E Z f und euch isotropic vector u E C" the jirnction

A .+ A - k

s,.-

,$(A, O)(N,o)A t l o

is even and holomorphic on C .

Since the proof of this result uses several facts concerning spherical harmonics, we postpone it until the next section. In the process we shall also give a proof of Theorem 2.9.

83. The Sphere S2 1. Spherical Harmonics

In this section we consider Problems A, B, and C in 41 for the sphere S 2 viewed as a homogeneous space 0(3)/0(2). O(n) denoting the orthogonal group in R". Using the spherical polar coordinates x 1 = r cos I,/I sin 8, x 2 = r sin I,/I sin 0, x 3 = r cos 8

on R3, the Laplacian takes the form

a2

2 a

I

L,., = - + - - + - L , ar2 r ar r2

where

a + sin-2 d-.a2 ae w2

L = - + cot 8a2

a02

More generally, the Laplacian LRn(n > 1) has the form

(3) where L is the Laplacian on S"-' (cf., Chapter 11, 55). The operator L generates the algebra of O(n)-invariant differential operators on S"= O(n)/O(n- I ) (cf. Chapter 11, Cor. 4.1 1). Let ( s l , .. . ,s,) denote the Cartesian coordinates of points on S"-'.

531

3. The Sphere S2

17

Theorem 3.1.

(i)

' are of the form Ek = span of {f';.k(s)= ( a l s l + . . . + a,s,)k, s ~ S " - l } .

The eigertspaces of' L on S"-

Here a = ( u l r. . . , u,) E C" is an isotropic vector and k E 2 ' . The eigenvalire is - k(k + 11 - 2). (ii) Each eigenspuce representation is irreducible. (iii) L2(S"- ') = Ek (orthogonal Hilhert space decomposition).

1;

Proof. Let Pk denote the space of homogeneous polynomial functions p(xl, . . . ,x,) of degree k on R" and Hk c Pk the subspace of harmonic polynomials. Consider the bilinear form ( , ) on F' = p,, given by

A simple computation shows that ( , ) is strictly positive definite and that ( p , q ) = ( 4 , p). Also, if p, q, r E P and if we put

. . . ,a~,,)~

d(p) = p -,

(a:,

then

(P,q r )

= (a(qr)P)(o)=

( W M d P ) ( O ) = (a(q)P, r ) ,

so that the operators r + q r and r -,d(q)r are adjoint operators. In particular, if p~ P k - , , q = x: + ... x,' = [XI,,

+

(qp, h ) = ( P , d(q)h),

so that H,, is the orthogonal complement of qPk-2 in Pk; hence

P,

(4)

=

qPk-2

+ Hk

and by iteration (5)

Pk = Hk

+ 1xI2Hk-2 + + IXIZrnHk_2,,,, m = [ik]. .*'

On the other hand, if c = (cl, . . . ,c,) E C" satisfies

+ c,' = 0, + c,x,)& belongs h,(x) = (clxl + (c, c ) = c:

+

* *

to H k . Let then the polynomial H t be the span of the h, for (c, c) = 0 and suppose h E H k is orthogonal to H t . A simple computation shows for p € P k ,

+ +

d(p)(c,x, * * * C,XJrn = m(m - l ) . . . (m - k

+ l)p(c)(c,x, + ... + c,x,)"-~,

18

[Intro.

GEOMETRIC FOURIER ANALYSIS

so that, in particular, h(c) = 0

if (c, c) = 0.

In other words. h vanishes identically on the variety {c E C": (c,c) = O}. By Hilbert's Nullstellensarz some power h" belongs to the ideal (x:

+ ... + x p . + + x,'

If n 2 3, a simple computation shows that x: the ideal is a prime ideal, so h itself is divisible by x: by (4), h = 0. Consequently, if n 2 3 (6)

H , = span of {(clxl

is irreducible, whence

+ ... + x,',

+ . . + c,,x,,)~:(c, c) = 0).

This holds also for n = 2 ; in fact, we can represent a real harmonic polynomial u(xl, x2) as the real part of a holomorphic function

f(xl

+ ix,) = 4 x I ,x2) + iu(x,, x2).

By the Cauchy-Riemann equations, u is a polynomial in xl, x 2 . The function J; being holomorphic, is therefore a polynomial in z. Hence u = gf + f )is a linear combinations of powers (xl i x J k as claimed. By definition, and by (6), Ek consists of the restrictions H k l S " - ' . It follows from (3) that each h E Ek satisfies Lh = - k ( k + n - 2)h. The eigenvalues -k(k + n - 2 ) being different for different k, the symmetry of L (Chapter 11, Prop. 2.3) shows that the spaces Ek are mutually orthogonal. By the Stone-Weierstrass theorem the algebraic direct sum m

which by ( 5 ) equals E k , is dense in C(Sn-l) in the uniform norm; this implies (iii). Now consider the normalized measure d'o = Q-'dw on S"-' (0, being the area of S"-l) and let ( , ) denote the inner product on L'(S"-') corresponding to the measure d'o. Let Skm [I I m 5 d(k)] be an orthonormal basis of E k , and let F be any eigenfunction of L, L F = cF (cEC). We have the expansion

(7)

1akmSkmnr

akm

= ( F , skm>*

k,m

But then, using the symmetry of L,

so c is one of the eigenvalues -ko(ko responding eigenspace. This proves (i).

+ n - 2)

and

Eko

is the cor-

031

3. The Sphere S 2

19

For (ii) suppose E , = E' Q E" is an orthogonal decomposition of E, into two nonzero invariant subspaces. By the O(n) invariance each of the spaces E', E" contains a function which equals 1 at the point u = (0,. . . , 0 , 1). By averaging over O(n - l), the isotropy group at o, we would obtain functions 4' E E', 4" E E", with #(o) = 4"(u) = 1, depending only on the distance 8 from u. Using Chapter 11, Theorem 2.7, we deduce that 4' = 4". A more direct proof of the identity 4' = 4'' can be obtained by observing that on functions on S"-' depending only on 8, L has the form d2/dB2 + (n - 2) cot 8 d/de (Chapter 11, Proposition 5.26). Thus 6 and 4" both satisfy the singular differential equation

Using a power series expansion obtains the formulas a , = 0,

(m

4(e) =

a, sinme for small

8, one

+ 2)(m + n - l)c~,,,+~ = [m(m + n - 2) - k(k + n - 2)3a,.

Thus 4 is determined up to a constant factor. Hence Theorem 3.1 is proved. Note that 4' must be real-valued.

4' = 4" and

Remark. The eigenfunctions of L on S"-' are called spherical harmonics. The name is derived from the fact that they coincide with the set It is useful to remark that the restriction of restrictions u k l O ( HIf?""). k mapping p + plS"-' of Hk onto Ek is one-to-one.

According to Theorem 3.1 each j'E L2(S"-l ) can be expanded in a series of spherical harmonics. We shall now need some refinements of this expansion. Classically this is done by a direct investigation of the functions in E k . A simpler method (which is also closer to the spirit of this book) is to represent the members of E , as representation coefficients (Chapter IV, 01) of the orthogonal group O(n). For example, with this approach, the formula from Prop. 3.2 below,

P(u * O)

= d(6)( p , & U ) 4 k ) ,

replaces the classical formula &(s)

=c

s,"-

(0.s" Us') do(s')

, P p - 2 )

(Heine [1878]), where & E E k , c = $(in

-

l)()n

+ k - 1)n-(li2)",

20

[Intro.

GEOMETRIC FOURIER ANALYSIS

and P:1'2)(n2, is the ultraspherical polynomial defined by (1

(84

- 2zt

1 00

+

z2)1 - ( 1 / 2 ) n

=

Zk/A1/2)(n-2) (t). k

0

The group-theoretic method is still quite elementary because the only required tools are the Haar measure on O(n) and the Schur orthogonality relations (Chapter IV. 41, No. 2). If s ES"-I,UE O(n), let us denote the image of s under u and let the Haar measure du on O(n) be normalized by F(u

(9)

0)du

= Jsn- F(s) d'w(s),

Job,

(cf. Chapter I, $1). Proposition 3.2. For a jixed k E Z+ let 6 be the nutural representation of' O(n) on Ek and d(6) = d(k) its degree. Let +k E Ek be the unique element which is invariant under the isotropy group O(n - 1) at 0, normalized by &(o) = 1. Then

(i) (ii)

dZ4O ) ( 4 k r *

(4k9

4k)

4 k ) = ( p , 6 ( u ) 4 k ) .fbr p E k ; = l / W , M u - * 0) = 4kb 01,

-

5 1.

I4k(S)l

Proof. (i) Let M = O(n - 1) and dm its normalized Haar measure. Since 4 = 4 k is M-invariant. we have for m E M,

i,.-Ip(s)4(s)

d'&)

=

=

=

so that, since

I,"I,.-14(s)

Ip(m * s)4(s) Jw(s)

1

( I , p ( m . s) h d ' W

,4(s)p(oW(s)d ' d s ) ,

4 is real, (P,

4)

= P(0)(4, 4)s

Replacing p(s) by (S(u)p)(s)= p(u- Is), relation (i) follows. Using Schur's orthogonality relations (Chapter IV, 61, No. 2) on relation (i), we obtain

21

3. The Sphere S 2

031

which proves (4, 4) = d(d)-'. Since ness) we have by (i)

4kU O ) = d(6)( 4 k '

and the relation

l4kl

9

d(u)4k)

4 k

is real-valued (by the unique-

= d(s) ( 6 ( u ) 4 k

7

4k)

=

4k(u-

O)

I 1 follows from Schwarz's inequality.

Corollary 3.3. For each so E S"-'

and each p E E k r F E C(S"-'),

the junction 9:s + S,,.F(u . s)p(u. so) du,

belongs fo

s E sn-1,

Ek.

For this it is sufficient to consider the case so = 0. Writing s = u . 0 , the integral is g(u ' O ) = d(d)

[o(nF(w ( p ? d(wuF(w

*

O)

'

0)(d(w-

')4k)

dw

' ) p , 6(u- ' ) 4 k ) dw = q ( V -

0)

if

q

=

j0,.,F(w. o)d(w-')p dw.

Clearly q E E k . Taking u as a rotation in the two-plane through 0, s, and 0, we see that g(s) = q(a.s), where a is the symmetry of S"-' with respect to 0. Hence g E E k r as claimed. Theorem 3.4. For k € Z + let

ma/ basis

oJ'E~.

S k m [l I m I d(k)] be an orthonorLet FE&(S"-') and akm= (f-,skm). Then

( i ) F = 1 k . m a k m S k m , the series being absoliitely and uniformly convergent. (ii) The mapping F + {a,,,,} maps &(Sn- ') onto the set of' a / / sequences a k m [ k E Z + , 1 I m I d(k)] satisjjing

for each q E Zt Proof. By Hilbert space theory,

22

[Intro.

GEOMETRIC FOURIER ANALYSIS

Replacing here F by LqF, inequality (10) follows. Putting p = Skm in Prop. 3.2, we obtain by Schwarz's inequality

I skm(s) I 5 d(@1'2,

(1 1)

which by (4) is bounded by a fixed power of k . This proves (i). It remains to prove that a sequence { a k m }satisfying (10) gives a sum

which is necessarily smooth on S"-'. For this we define for F , , F , E C(S"-l) the convolution F , * F , by

( F , * F 2 ) ( s ) = /o(nFl(id

(12)

- o)F,(u-

. s) du.

Now, as well as later, we need the following simple lemma. Lemma 3.5.

(13)

p

I f p E E k , we have

*4 k

=

d(k)- ' p ,

4k *p

= p(o)cj(k)-

4k.

I n the notution oj' Theorem 3.4,

so that (i) reads

(15) Proof. We have by Prop. 3.2, ( P * 4 k ) ( v ' o> =

d ( k ) 2 ( p ?6 ( l l ) $ k )

(4kY

d(u-

'U)+k)

dLr,

L n )

which by the orthogonality relations equals ( p ? 6 ( v ) 4 k ) = d(k)- ' d v' O)*

Second, * p is O(n - 1)-invariant, hence proportional to 4 k , so evaluating at o and using Proposition 3.2, we obtain (13). If F , is orthogonal to E k , the definition (12) shows F , * 4 k = 0. Hence ( 1 3) implies (14) and (15).

331

23

3. The Sphere S 2

Finally, to prove that F in (1 la) is smooth, we consider for y E Z + the functions akni(k

+ 1)4skn1(S)9

1( k +

lj(k)4k(s),

F,(s) = k. m $q(S)

=

k

which by (13) satisfy F

=

F, * $,.

Let I I = 3. Using part (i) of Theorem 3.1 and the invariance of 4k under O(n - I), we conclude that +k is the restriction to S 2 of the,polynomial

i 271

-n

( i x , cos 4

+ ix, sin 4 + x3)k d+.

Hence $k(S)

=

'J

-

= o

(cos O

+ i sin O cos 4)kd4,

SE

s2,

with s3 = cos 0. For n arbitrary we have similarly [for example, by verifying (8)]

if s,, = cos 8. Given N E Z t we can therefore choose q so large that the series for t j , can be differentiated term by term N times, so $ q has continuous derivatives up to order N . The function F , is continuous [by (10) and (1 l)], so, using F = F , * $, we conclude that F is N times differentiable. Thus F E &(S"- l), so Theorem 3.4 is proved. 2. Proof of Theorem 2.10

We first prove that conditions (i) and (ii) are satisfied for each ( 1 8) in $2 can be written

.f if .I'E 9 ( R " ) . First note that

+(A, o)= , j ( ~ w=)

s.

f(o, p)e-i'p tip,

(16) where .f((u. p ) is the integral of .f over the hyperplane (x. w ) = p . Since the function p --* ,f(w, p ) belongs to g ( R ) , property (i) follows immediately. Also (18) in 42 implies that dY/dAk at A = 0 is a homogeneous kth-degree polynomial in (o,, . . . , on),which according to (4) can be

24

GEOMETRIC FOURIER ANALYSIS

[Intro.

expressed as a harmonic polynomial of degree 5 k. Expanding the function 1 -, +(A, w ) = f(1o) in a convergent Taylor series around 1 = 0 and integrating against ( a , ~ )property ~, (ii) follows since Ek and El are orthogonal for k > 1. Conversely, suppose 4 satisfies (i) and (ii) and define .f'eC(R")by (17)

f(x) = (2n)-"

s

Rt

4(1,w)ei"'X*m)l"-d 1 dw. X p - 1

We shall show that f(x) = 0 for 1x1 > A and then the inversion formula for the Fourier transform implies f= 4. Since the assumptions (i) and (ii) are invariant under rotations around 0, it suffices to prove f(x) = 0 foi x = re,,, where e,, = (0,. . . ,0, 1) and r > A . Let U = O(n) and A4 [=O(n - l)] be the subgroup leaving e, (i.e., the north pole 0) fixed. We can apply (15) to the function F(u . o ) = f ( u . x). Evaluating at u = e, we obtain m

c

931

3. The Sphere S 2

25

where

(20)

fim(X)

=(

4km(1)Skm(o)eiA(x’O)jlnd l do.

2~)~“

R + xSn-1

Because of our assumptions about holomorphic and

I&,,(A)l

(21)

4, the

ICN(I

function

f#&(n)n-k

is even and

+ JAJ)-NeAl’mAl.

In order to deal with (20) we shall use the following classical lemma (cf., Bochner [1955]. p. 37. Erdelyi et a/. [1953], Vol. 11, p. 247, or Vilenkin [1968], p. 554). We shall indicate a group-theoretic proof later. Lemma 3.6.

For each p E E k ,q E s”- I ,

where c,,k = ( 2 ~ ) ‘ ” ~ ’ ” i ~ .

Because of this lemma and (19), (20), our problem is reduced to proving

for r > A . First let us assume &A, o)= &A) is independent of o.We “extend” invariant holomorphic function
4 to an

@(O = . l n ) = 441) if 1’ = [:+ ... + [i.Since 4 is even, this is possible. Writing Q(C19

1=p

+ iv,

*

. 7

[=5

+ iq,

5, q E R”,

we have

(23) whence

p2

- v2

= 1512 - l q 1 2 ,

11t4= (iti2-

(24)

p2v2

=

( 5 .q)2,

I ~ I +~w )q ~) 2

and

2 1 1 m1 1 =~ IVI~ 1t12+ [(lt12In particular, since (25)

Ir11212 + 4 ( 5 . ~ l ) ~ I ” ~ .

I (t . q) I I15 I I q I, we have IIm 1 15

Ivl.

26

[Intro.

GEOMETRIC FOURIER ANALYSIS

Now

4 satisfies for each

N

I#(A)l

(26)

E

Z+

I CN(1

+ IAI)-NeA"ml'

and (17) can be written f ( x ) = (27r-"

/R(D(

(I(.

By a suitable variation of the classical Paley-Wiener theorem argument we shall prove .f' E 9 ( R " ) . If qeR" is any fixed vector, we can by Cauchy's theorem shift the integration into the complex domain, giving f(x) = (27c)-"

(27)

+

JR?(<

iq)ei(x.c+iq)

d<.

In fact, (24) and (26) show that, q being fixed, a(<+ i t q ) is rapidly decreasing in <,uniformly for 0 Ir I 1. Also, by (25), and (24), we have

(@(t + iq)l

_< C,(1

+ IAI)-NeAl'll

I C,eAI'lI(l

Taking N = n

+ 11<12

- 1q1211'2)-N.

+ 1 we have

JRfl+ 1 1 < 1 2 (1

- ( q I Z ( 1 ' 2 ) - N d
+ b.

where a and b are constants, as we see by breaking the integral up into the regions 151 5 21471 and > 2 ) ~ )Thus . (27) implies

If'(x)l

(28)

s (27c)-"C,+

l(alql"

+ b)eAlql-(x*q),

valid for all x, V E R". Now fix x with 1x1 > A and put q = t x in (28). Since (a I x ("t" + b)erlxl(A-Ixl)+ 0

as t (29)

+

+a we obtain f(x)=O

for 1x1 > A.

Consider now the general 4(A,w). We recall that & , , ( A ) n - ' is even and holomorphic and &,, satisfies (21). We now write the expression in (22) as # times

$31

3 . The Sphere S 2

27

But as a special case of Lemma 3.6

where r = 1x1 and c is a constant. Consequently, the expression (30) is up to a constant factor the Fourier transform in Rn+2k of the radial function @km(
Letting

4km(A)A-k

<ER"+2k. above, the function


play the role of

@(c

19

$(A)

... i n +2 k ) 9

restricts to @knr on R"+Zk.By the conclusion (29) we therefore find that the integral (30) vanishes for r > A, as desired. Finally, we give a proof of Lemma 3.6. Consider the function (31) u denoting

@n,s(x) =

J-"

( u , 6(u)u) du,

eiA(x. ut:,)

X.E

R",

the unit vector d(6)1'24kE E k .

Lemma 3.7. Let w E Ek und

x = lare,,,

IEU, rER.

Then

Proof. Define the map F : R" ~ ( x= )

--+

Ek by

1"

e - ; ~ ( x ~ ~ ~ n lnu. d(~,)u

Then the left-hand side of (32) equals ( w , F ( x ) ) . Also, F(u * X) = d(u)F(x), so in particular, the vector F(ren) is M-invariant and hence proportional to u. Thus

F(ren) = (@A, a(re,,))-0, so that

(w,F ( I *re,,)) = (w,a(OF(re,,)) = ( w , d(O~)@A,a(~e,,), proving the lemma.

28

[Intro.

GEOMETRIC FOURIER ANALYSIS

It remains to compute

aA.&(x). We have of course

(33) is M and we shall now 'use (3) to separate the variables. Since invariant, it is not hard to see that @A,6(1.e,,)= @A.d(l-le,).Hence

@A,6(rle,,)= @A,d(rl-' e n )=

1"

ei"""n~'u'~)

(0, d(u)u)

du,

so that @A, d(r/e,,)= J;i~r(l,n.uen)

(d(l)u, d(u)u)

mi.

Thus for a fixed r, the function IM -, @A,d(rle,)on S"-' is an eigenfunction of L with eigenvalue - k ( k + n - 2). Combining this with (33) and (3) we deduce that the function $(r) = @A,d(ren)satisfies the differential equation (34)

n-1 r

II/" + -J/' -

k(k

+r nz - 2) $ = -AZII/.

But the constant multiples of the function J k + ( 1 , 2 ) , , - ,(Ar)/(h)("21n-1 are the smooth solutions to this differential equation. This, combined with Lemma 3.7, gives the formula in Lemma 3.6 up to a constant factor, which we shall not determine since it is not necessary for the application at hand. At any rate we have proved jim(x)= 0 for 1x1 > A, so by (19) j ' ( x ) = 0 for x = re,,, r > A. This proves Theorem 2.10. Remark. While the inclusion (9A)-

cxA

in Theorem 2.9 is obvious, it should be noted that the surjectivity ( g A ) -= X Afollows from Theorem 2.10. In fact, if F E X A ,the function &A, w ) = F(Aw) certainly satisfies (19) in 92; also, expanding F in

power series around 0, we obtain a power series $( a). = I, F(h)=

1AIPl(w), It0

where P I is a polynomial in (wl, . . . ,w,) of degree 1. If I < k, it is clear ~ S"-' if the from ( 5 ) that PI is orthogonal to the function o -, (u, w ) on vector u E C" is isotropic. Thus condition (ii) of Theorem 2.10 is fulfilled, so there exists an f ' E 9,(R") such that c#J(A,w ) = f(Aw). But then F = .f, so Theorem 2.9 is also proved. Naturally, it can be proved much more directly.

541

4. The Hyperbolic Plane H 2

29

$4. The Hyperbolic Plane HZ

1. Non-Euclidean Fourier Analysis. Problems and Results

Let D be the open disk I z ( c 1 in R2 with the usual manifold structure but given the Riemannian structure (cf. [DS]. Chapter I. 49)

if u and u are any tangent vectors at z E D. Here ( , ) denotes the usual inner product on RZ. This Riemannian manifold is usually called the PoincarP model of the hyperbolic plane H 2 . We shall now recall briefly some of its geometric properties (cf., [DS], pp. 93, 548). Since (u,u)2 (u, u>(u, 0)

- (u, V l 2 - (u, 4(0, 4

the angle between vectors in this Riemannian structure coincides with the Euclidean angle. The length of a curve y ( t ) ( a It IB) in D is, according to the general definition for a Riemannian manifold, defined by

UY)=

J

B

r‘(t)>”2 dt

(r’(03

a

and the distance between any two points z, w E D defined by d(z, w ) = inf L(y), Y

the infimum being taken over all curves joining z and w. If y ( t ) = ( ~ ( t )y ,( t ) ) and s(z) is the arc length of the segment y ( t ) (a It I7 ) , we get from (1)

(g)2

= (1

- x(z)2 - y ( z ) z ) - 2 [ ( $ ) z

+&)’I,

which is the meaning of the customary terminology ds2 =

+

dx2 dy2 (1 - x2 - y y

In particular, if y(a) = o (the origin), y(P) = x (point on the x-axis) and we denote by yo the line segment from o to x, we get from

30

[Intro.

GEOMETRIC FOURIER ANALYSIS

the inequality U Y O ) IUY).

This shows that straight lines through the origin are geodesics. Also,

Consider now the group

SU(1, 1) =

,:(I :)

1

: laI2 - ]biz = 1 ,

which acts on D by means of the maps (4)

az

g:z+bz

+b + if’

(zE D).

The action is transitive; the subgroup fixing o is SO(2). so we have the identification (5)

D

=

SU(1, l)/S0(2).

The Riemannian structure (1) is preserved by the maps (4). In fact, let z(t) be a curve with z(0) = z, z’(0) = u. Then

z’(o)

(bz

+ ify

at g(z),

and the desired relation

-

( 9 * u, 9 U>&) = ( u , u>z

follows by a simple computation. The mappings (4) are conformal and map circles (and lines) into circles and lines. It follows that the geodesics in D are the circular arcs perpendicular to the boundary lzl = 1. If zl,z2 are any points in D,the isometry Z-b-

z - 21 1 - Z1Z

maps z1 to o and zz to the point (zz - zl)/(l - Zlzz). Following this with a rotation around o, this point is mapped into J z z- z 11/11 - Zlzzl, so we deduce from (3)

§41

4. The Hyperbolic Plane H 2

31

Writing the Riemannian structure (2) in the general form (cf., Chapter 11, $2, No. 4)

and putting as usual ij = )det(gij)), gij = (gij)-' (inverse matrix), the Riemannian measure

f -,

If&

dx1 . . . dxn,

and the Laplace-Beltrami operator

become, respectively,

(7)

d z = (1 - x 2 - y2)-' d x dy,

L

= (1 - x2 -

y2)2

(iz -

+. i?$)

By general theory (Chapter 11, $2) they are invariant under all isometries; for the isometries (4) above (which together with z -+ Z generate all isornetries of D) this can of course be verified directly. It is also easy to prove directly the special case of Chapter 11, Corollary 4.11, that each SU(1, 1)-invariant differential operator on D is a polynomial in L. With these preparations we now consider Problems A, B, and C. We shall define some eigenfunctions of L, motivated by the Euclidean case. If p E C and w E S"- l , the function (9)

+

e"(x.o)

( x E R")

has the following properties: (i) It is an eigenfunction of the Laplacian LRnon R". (ii) It is a plane waue with normal w ; that is, it is constant on each hyperplane perpendicular to w. A hyperplane in R" is orthogonal to a family of parallel lines. The geometric analog for D is a horocycle, i.e., a circle in D tangential to the boundary B = aD;in fact such a circle 5 is orthogonal to all geodesics in D tending to the point of contact b. If z is a point on 5 we put

(10) ( z , b )

= distance

from o to 5

(with sign; to be taken negative if o lies inside 5).

32

[Intro.

GEOMETRIC FOURIER ANALYSIS

This “inner product” ( z , b) is in fact the non-Euclidean analog of (x, w), which geometrically means the (signed) distance from 0 E R” to the hyperplane through x with normal w. We shall write ( ( z , b) for the above horocycle through z and b. By analogy with (9) we consider the function ep. : z

(1 1)

Lemma 4.1.

+ e”(z*b) ,

ZED.

For the Laplacian L we have Lep. b = d p - 2 ) e ~b,.

Proof.

For t~ R let

)

(

cosh t sinh t a , = sinh t cosh t ESU(1, 1). Then by (3) we find (12)

d(o, a, o) = d(o, tanh t) = t.

If b, is the point of B on the positive x-axis, we therefore have (13)

e,,,,(tanh t) = e”.

The horocycles tangential to B at b, are the orbits of the group N = (ns =

(+ 1

is

is

)

-is :s E R}. 1 - is

In fact, the orbit N . o is such a horocycle and so is the orbit N a , . o = a,N o (a, normalizes N). Consider now functions on D which, like are constant on each of these horocycles. Because of its N-invariance, L maps the class of such functions into itself. The restriction of L to such functions is a differential operator Lo in the variable t. Since the isometry a, satisfies a, tanh T = tanh(t + T), the invariance of L under a, means that Lo is invariant under the translation z + T + t. Thus Lo has constant coefficients, so ep‘ is an eigenfunction for it; hence by (13), e,,bo is an eigenfunction of L. The eigenvalue can be calculated by expressing L in the coordinates a,ns.o + (t, s); in fact, one finds d2 d Lo = 2 - 2--. dt dt A simpler computation of the eigenvalue is suggested later (see also

Exercise B2). Reformulating the lemma, we have Lz(eW+ 1) (2. b ) = - ( n 2 +

l)eW++”(z..b) ,

AEC.

§41

4. The Hyperbolic Plane H 2

33

We now define a Fourier transform on D motivated by the Euclidean case (1) in 52. Again d z will denote the surface element (7). Definition. I f f is a complex-valued function on D, its Fourier transform is defined by

f(z)e(-i'+

f ( I , 6) =

(14)

')('vb)

dz

D

for all I E C , bE B, for which this integral exists. In order to state the next theorem we call a C"-function $(I, b) on C x B, which is holomorphic in I , a holomorphic junction of uniform exponential type R if for each N E Z + sup e - R l l m l l (1 + I W N l W611 < alEC,beB

Here Im I denotes the imaginary part of A. We can now state rather explicit answers to Problems A, B, and C . Theorem 4.2.

(i) ! f . f ~ - W then) ,

where db is the circular measure on B normalized by j d 6 = 1. (ii) The mapping f + f is a bijection of 9 ( D ) onto the space of'holomorphic junctions $(I, 6) of unijorm exponential type satisfying the functional equation

I,

e(il+

(iii)

l)(Z.b)$(I,b) db =

I

e(-il+l)(z.b)

$( - I , b ) db.

The mapping f + f extends to an isometry of' L2(D)onto

L 2 ( R + x B. (271)- ' A tanh(inI) d I db). Remarks. The three parts of this theorem are non-Euclidean analogs of the inversion formula, the Paley-Wiener theorem, and the Plancherel formula, respectively, for the Fourier transform on R". There are however some interesting differences:

(a) While the functional equation in (ii) gives genuine restrictions on the Fourier tansform $(I, 6) = f ( A , b), the analogous condition for the Euclidean Fourier transform f would be e"(x*my(do)do =

e-i'(x*my( - ~ w )do.

34

[Intro.

GEOMETRIC FOURIER ANALYSIS

FIG.I

This condition is however quite trivial so imposes no restrictions on .f. The appropriate Euclidean analog is condition (ii) in Theorem 2.10. (b) Comparing formulas (14) and (15) with their Euclidean analogs (l), (2) in 52 we see the kernel which does not have any Euclidean analog. If w is the point on the horocycle <(z, b) closest to 0, we have from (3), e 2 ( ' S b ) ,

But IwI can be found by using the cosine relation on the triangles b) and (0,z, c), where c is the center of the horocycle. We obtain (see Fig. 1) (0,z,

cos(z0b) =

1

+ 1212 - Iz - b y --

lzl2

+ (i(1 +

2121

IwI))2

IZl(1

-

(+(I

+ Irr>I>

so

which is just the classical Poisson kernel 1 - lzl2 1 - r2 P(z, b) = ___ I z - b12 - 1 - 2 r c 0 ~ ( 4- 6) + r2

if z = reie, b = @", The Poisson representation u(z) =

~

jB

1 - JzJ2 F(b) db I z - b12

-- I w l ) ) 2

941

4. The Hyperbolic Plane H 2

35

of u hurnioriic jiinction u(z) with continuous boundary values F can thus be written u(z) = j;2(’*b)F(b) db and can therefore, in accordance with ( 1 9 , be viewed as a formula in nonEuclidean Fourier analysis.

We shall see later that this leads to a simple proof of Schwarz’s limit theorem

lim u(z) = F(b) u-b

at each point b of continuity for F (Theorem 4.25). It also leads to group-theoretic proof of the classical Fatou theorem that a bounded harmonic function on D has boundary values along almost all radii (cf. Theorem 4.26). It should be observed that because of (8) the Euclidean and the nonEuclidean harmonic functions coincide. We also note that (16) can be used to give a slightly simpler proof of Lemma 4.1. In order to describe the solution to Problem B we need the concept of an analyric jirncrional (hypeyfirnction) on a compact analytic manifold. Let d ( B ) denote the space of analytic functions on the boundary B, considered as an analytic manifold. For each such manifold B, d ( B ) carries a certain natural topology; in the case of the circle it can be described rather simply as follows. Let U be an open annulus containing B, X ( U ) the space of holomorphic functions on U topologized by uniform convergence on compact subsets. Since each analytic function on B extends to a function in X( V ) for a suitably chosen U,we can identify d ( B ) with the union u , , X ( U ) and give it the inductive limit topology. The elements of the dual space d ( B ) are called analytic functionals (or hyperfirnctions). Since the elements of d ( B ) generalize measures, it is convenient to write

For A E C, let I n @ denote ) the eigenspace

with the topology induced by that of &D).

36

[Intro.

GEOMETRIC FOURIER ANALYSIS

Theorem 4.3. The eigenjunctions of the Laplace-Beltrami operator L on D are precisely the junctions

. if iA # - 1, -3, -5,. . . , then the where I E C and T E ~ ’ ( B ) Moreover, mapping T + j is a bijection of d ( B ) onto b,(D). We shall see later (85) that the hyperfunction T can be recovered from f by a suitable limit process at least for Re(il) > 0. We shall also prove (Theorem 4.24) that T is a distribution on B if and only if f grows at most exponentially with the distance d(o, z). For 1 E C let T, denote the representation of SU(1, 1) on the eigenspace b,(D). For Problem C we then have the following result. Theorem 4.4. The eigenspace representation T, is irreducible if and only if

il

+ 1 $22.

In the remainder of this section we give proofs of these theorems. In addition to the subgroup K = SO(2) c SU(1, 1) = G we have considered the subgroups

It is clear from (12) that the mapping t -+ a,. o is an isometry of R onto the geodesic A Let us now verify that the mapping 4 : x n, . o is an isometry of R onto the horocycle N . 0. For this note that -+

so.

X

4(x)= x + i’

d4

i ( x + i)2’

- =-

dx

such that

proving the statement. Let bo be the point of intersection of B with the positive x-axis. Consider the horocycles (, = Na, . 0, = Na,.,, . o and the geodesics n, A . o and n,+&,A . 0, indicated in Figure 2. Note (by matrix computation) that a;’n,a, = nx,,-z, and that a, is an isometry of D mapping

041

4. The Hyperbolic Plane H 2

37

FIG.2

N . o onto Nu, sides

a

0. It

follows that the shaded “rectangle” in Figure 2 has

At,

e-21Ax,

At,

e-zcr+Ar’Ax

such that the surface element d z is given by Jbf(z) d z = / R x , I ( . x a l . o)e-2‘dx dt

and the arc element d o , on the horocycle

is given by

do,= e - 2r d x .

The formula for d z can also be proved by writing [ = n x u l .o =

sh t - ixe-’ ch t - ixe-”

which gives dz

i d[r\di“ = e - 2r dx dt. 2 (1 -

=-

ri,’

The integral formula above can also be written

38

[Intro.

GEOMETRIC FOURIER ANALYSIS

Rotating by 8 we can therefore write the Fourier transform (14) in the form

f(n, ,ie)

=

J:(-

i l + I)1

(j C(tanh I eie. ele)

and this will be useful later.

2. The Spherical Functions and Spherical Transforms

A sphericul Junction on D is by definition a radial eigenfunction of L. If z E D,r = d(o, z), we know from (12) that z = Iz I eie = tanh r eie.

In the coordinates (r, 8 ) (geodesic polar coordinates) L has the form

a

L = - + 2 coth 2r - + 4 sinhP2(2r)2. dr2 ar ao a2

Thus a spherical function

a2

4 satisfies

84 = -(A2 + 1)4. + 2 coth 2r dr2 ar By expanding in a power series in sinh 2r or by Theorem 2.7, Chapter 11, we see that all smooth solutions of (18) are proportional-hence the constant multiples of the function (19) In particular, spherical functions are #O at z = o ; normalizing spherical functions 4 so that &o) = 1, we see that (19) gives all the spherical = since both functions satisfy (18). functions. Also, If we substitute z = tanh r ei@,b = eie into (16), we obtain from (19), (20)

41(u,.o ) = - J (cosh 2r - sinh 2r cos 8)-(1’2)(i’+ do. 1

211

-R

Now let G = SLr(1, l), K = SO(2) have Haar measures dg and dk normalized by dk = 1,

J’EC,(D)

4. The Hyperbolic Plane H

941

39

(cf., Chapter I, 91). The spherical function q5A satisfies the functional equation (21)

J$d

z E D, 9 E G.

. z ) dk = 4A(9 oW,(z),

In fact, as a function of z the left-hand side is a radial eigenfunction of L. Hence it is a constant multiple of $JA(z).The constant is found by putting z = o. Definition. Iff is a radial function on D, its spherical transform is defined by

whenever this integral exists. Let @(D) denote the space of radial functions in 9 ( D ) . We shall now prove an inversion formula and a Plancherel formula for the spherical transform. The spectral theory of the ordinary differential operator (18) (cf., Weyl [1910], Coddington and Levinson [1955], pp. 25 1-256) does indeed suggest an inversion formula

f(z) = Jf(Q#md P ( 4 , where the “spectral measure” is determined by the asymptotic behavior of $JA at so. Specifically this is given by Harish-Chandra’s c-function A -,c(A) displayed in the next theorem. Theorem 4.5.

!f Re(iA) > 0, then the limif ,.(A) = lim

1)r

&iA+

4A(ar * 0)

r++m

exists and c(A) = 7 1 - 1 ’ 2

r(+iA) T(+(iA

+ 1))’

Proof. Since $ A ( a - r .o) = #A(ar.o) = $ J - A ( a ro), . we can replace 1 and r in the integral (20) by -A and - r . Using the substitution u = tan#,

*do

=

(1

+ u Z ) - l du,

we obtain (23) $JA(ar. o) =

cosh 2r

+ sinh 2r 1 +u2

du

40

[Intro.

GEOMETRIC FOURIER ANALYSIS

Assuming Re(i1) > 0 we shall show that the integral tends to a limit as r --f + 00. Let A = 5 + iq, so that Re(il) = - q > 0. Select E (0 < E < f) so small that 1 + 2 ~ > q 0. Then the integrand above has the majorization ( 1 + e - 4 r u 2 ) - ( 1 / 2 ) ~ - 1/2(1 + u 2 ) ( 1 / 2 ) v - 1 / 2

I (1 + e - 4 r t 1 2 ) - ( 1 / 2 ) v + ~ v (1 + U2)(1/2)v2 (1 + u 2 ) - ( 1 / 2 ) v + E v ( l

+ u2)(1/2)v-1/2

112

=

(1 +

u2)-(1/2)+cv

3

and the last expression is integrable. This gives the formula lim

e(-iA+

1)r

+A(ar.0) = -

Putting t = (1

I,

(1 + u2)-(112)(iA+

1)

du.

l r -

r++m

+ u2)-',

this becomes

and the result follows. Remark.

The formula defines c as a meromorphic function on C.

Using now the integral formula at the end of No. 1 we obtain from (19) the following expression for the spherical transform (22):

f(A) Theorem 4.6.

=

1 R

e-i.'(e-f / Rf ( n x a t - o ) d x

The spherical transform f

+f

is inverted by the $ormula

Moreover,

(26) Proof. In Chapter IV we shall give a proof of this theorem valid for

a general symmetric space. This is based on detailed study of the spherical function, including Theorem 4.5. For the present case, G = SU(1, l), we give a different proof, which, in addition to brevity, has some interesting features which we shall comment on later. The starting point is formula (24). The function f is radial, so that with ( as above f ( 0= m n h s)

4. The Hyperbolic Plane H 2

§41

41

if (tanh s)’

=

(sh r)’ + x2eC2‘ c% = (ch t)’ + x’e”’

’

This relation holds if

+ x’e”‘.

(ch s)’ = (ch t)’ Defining F on [l,

by

00)

F((ch s)’) = j”(tanh s), we have F‘((ch s)’) = f‘(tanh s)(2 sh s ch3 s)-

’,

so that. since f’(o) = 0. limu+ F’(u) exists. Furthermore,

f(A)

=

1

e - i ’ l e - r ( j F((ch c ) +~x’e”’) dx dt

R

e-i.‘(

=

)

R

I

F((ch t)’

+ y2) d y ) dt.

R

R

The integral equation

4(u) =

S.F(. +

Y2> dy

(u 2 1)

(equivalent to Abel’s integral equation) can be solved as follows:

.(‘:I

+ z 2 ) dz = =

S.S,

F’(u + y 2 + z 2 ) dy dz

211 JoaF‘(u

+ r2)r dr = 71

so that

- nF(u) =

h

+‘(u

+ 2 2 ) dz.

Thus f(o)

=

F(1) =

- -

-

1 ; IR4’((ch T)’)

ch T dr.

42

[Intro.

GEOMETRIC FOURIER ANALYSIS

On the other hand, we have by the Euclidean Fourier inversion formula

1 4((ch r)Z) = - / f ( I ) e ' " dA 2n R =

1 J f ( I ) cos It d l , 2n R

since f ( A ) is even. Differentiating with respect to t , we obtain - 4'((ch

~ ) ~ch) t2 sh t

1 f ( l ) lsin At d l . 271 JR

=-

Using the formula sin At dt = x t a n h E ) , which is obtained by integrating the function z tour R u ( R + xi), we get from the above

--+

e'"/sh z over the con-

But if l e R, we have by simple identities for the gamma function,

(28)

Ic(A)(-' = c ( I ) - ' c ( - I ) - ' =

2

so we have proved

(29)

,f(o) = c

Jf(I)lc(l)l-2dI,

c =

(2x2)-'.

R

Given g E SU(1, 1) let us apply (29) to the average F ( z ) = J i f ( y k . z ) dk,

Using the functional equation (21) we find

z E D.

4. The Hyperbolic Plane

§41

H2

43

so that

F(4 = +A(S

*

43(4.

Since F(o) = f ( g . o), we get the desired formula,

For the second part of Theorem 4.6 consider for f 1 , f 2 ~ 9 ' ( D ) the convolution (fi

* f a g .o) =

J fl(h .o)fz(hc

*

0)

dh.

It has spherical transform

= .f1(4.f2(4

because of (21). Applying (29) to the function

JD

we obtain (26), so Theorem 4.6 is proved. Ranges. In analogy to the subspace @(D) c 9 ( D ) let L:(D) denote the space of radial functions in L2(D). Recall that a holomorphic function F on C is said to be an enrire ,function of exponential type R if for each integer N 2 0

(31)

sup e - R l l m I l (1 ISC

+ 14)NIF(~>I < 30.

Let H ( C) denote the space of all such functions, X , ( C ) the subspace of even functions in &?(C). As special cases of Theorem 4.2 to be proved in the next subsection we have the following statements about the ranges of the spherical transform: (32) (33)

9 ' ( D ) " = X,(C);

44

[Intro.

GEOMETRIC FOURIER ANALYSIS

In (32) f has support in CI(B,(o)) if and only i f f has exponential type R ; in (33) the mapping f - 3 is an isometry between the spaces indicated.

3. The Non-Euclidean Fourier Transform. Proof of the Main Result We shall now prove Theorem 4.2, which includes the inversion formula, the Paley-Wiener theorem, and the Plancherel formula for the Fourier transform (14) on the hyperbolic plane D. As before, let dg denote the Haar measure on the group G = SU(1, I), normalized by

Extending the convolution on D we put

(fl * . f 2 ) ( 4

* defined earlier, for two functions =

fl,

f2

1

f 1 ( S . o ) f i ( S - . z ) dg

c

whenever this integral exists. Equation (42) below shows how the Fourier transform behaves with respect to convolution. Lemma 4.7.

Ler f~ C,(D).Then

f*

O l ( Z ) = Je(il+')(z.b)

30,b) db.

Proof. We first observe the geometric identity

(34)

( g * z , g . b )= ( z , b )

+ (g.0,g.b)

for each g E SU(1, 1). For this we consider the horocycles ( ( g .o, g . b) and <(g. z , g .b). As remarked earlier in connection with Figure 2, these horocycles cut segments of equal length of the parallel geodesics (0,g b) and ( g .o. g . b). Now the identity (34) is clear from Figure 3. Taking z = g - . o the identity implies ( 9 - 0,b ) = - ( g - o, g . b). Since ( g - ' . z , b ) = ( ~ , g . b ) +( g - ' * o , b ) = ( z , g * b )- ( g * o , g * b ) ,

we obtain for the spherical function e(il+l)'-
db.

§41

4. The Hyperbolic Plane H Z

45

FIG.3

On the other hand, when the conformal mapping g acts on the boundary B, the Jacobian is given by (344 where P is the Poisson kernel. In fact, if

a simple computation shows

@-do

1 -

lei'

14'

-

a("

Using the non-Euclidean form of it given by (16), we get by the above

Hence, changing variables in the formula for we obtain the symmetric formula

(35)

+A(g- 1j . o) =

J:l-iA+

4Aand

1)<8.0.b)e(il+l)
The lemma follows by integrating against . f ( g . 0).

putting z db.

=

h.0,

46

GEOMETRIC FOURIER ANALYSIS

[Intro.

Consider now for a fixed g E G the radial function

From (29) we have

(36) But =

f i ( i ) = (f*+a)(S

*

0).

Since fl(o) = .f(g. o), we get from Lemma 4.7 and (36)

Using now formula (28) for I c(L)I-~,we obtain the inversion formula in ( 1 5). For the determination of the range of the Fourier transform f ( z ) + , f ( L , b) [Theorem 4.2, (ii) and (iii)] we first introduce a definition. Definition. The point A E C is called simple if the mapping F +f of L2(B) into Cm(D)given by

(37)

is one-to-one. Proposition 4.8.

The point L is nonsimple if arid only if

A

= i(l

+ 2k),

kEZ+.

Remark. The “conceptual” version of this criterion is that the nonsimple points A are the poles of the denominator of the c-function.

We first prove a lemma (cf. Takahashi [1963]).

4. The Hyperbolic Plane H 2

§41

Let F(6') be inregruble on 0 I 6' I x and let

Lemma 4.9.

Jon =

H(t) =

I f -.s $ Z + , then H Proof. dP -

dtP

47

(cosh t - sinh t cos 6')-sF(0) do.

0 if and only i f F

= 0.

We have, by induction on p, (cosht - sinh t cos O)-' P

2 c$(s)(cosh

t -

sinh r cos 6')-"q(sinh t - cosh t cos 6')4

q=O

where the c:(s)

are constants and cXs) = (- l)Ps(s

+ 1 ) . . . ( s + p - 1).

If --s $ Zt, then C ~ ( S ) # 0 for all p. Thus if H = 0 implies successively

I0'F(6') cosp 6' dO

=

0,

p

=

=0

the equation H ( P ) ( 0 )

0, 1,2, . . . ,

so F = 0 (by the Stone-Weierstrass theorem), proving the lemma. For Prop. 4.8 let := (tanh r)ei4, b = eio. Then formula (37) reads

Now if A. = i ( l + 2 k ) , k € Z + , so that -i(iA. + 1) = k, the function F(eie) - e(k+l w gives j ' ZE 0, so A. is not simple. On the other hand, suppose --+(l

+ U)$Z+.

1f.f = 0 in (38), the proof of Lemma 4.9 implies JO2'cosp 6' F(e"&+'))d6'

(39)

= 0,

p E Z'.

Expanding F in a Fourier series

(39) implies successively a

so F

=

eip&

0. as desired.

+ a - p e - i P ) = 0,

p = o , 1 , 2 )...,

48

[Intro.

GEOMETRIC FOURIER ANALYSIS

Proposition 4.10.

I f - 1 is simple, then the ,function space on B,

{f@, -1:.f E W ) }

is dense in L ~ ( B ) . Proof.

If this were not so let F f 0 in L 2 ( B ) satisfy

s,

f(1,b)F(b) db

=

0

for all f~ 9 ( D ) .

This implies

J;(-il+l)<'.b)F(b) = 0

(z E D),

contradicting the simplicity of -1. Passing now to the proof of Theorem 4.2, part (iii), we derive from Lemma 4.7 for 1 real

J f * 4Mfmdz

(40)

D

=

JBlf(n,@ I 2 db.

Here we multiply by I c ( A ) ~ - ~use , Lemma 4.7 and the inversion formula, and obtain

If(1, b)121c(1)l-2dA db

I j ( z ) l 2 dz = c JRXB

=

2c

J

If(1, b)121c(1)l-2dLdh,

R+ XB

the latter identity coming from the evenness in 1, which is clear from (40) and (28). Now suppose F E L2(R+ x B, I c(1)I - 2 d1 db) were orthogonal to the range, i.e., (41)

?(A, b)F(1, b)A tanh(inA) d A db = 0, R+ XB

If I) E 9 ( D ) , we have (42)

( f * +)*(A, b ) = f(a b ) W ) .

In fact, n

n

f E %D).

4. The Hyperbolic Plane H 2

$41 Writing now

(g.z,b)

=


49

+ (g.o,b),

relation (42) follows. The functions 4 vanish at cc on R [by (24)]. They form an algebra closed under complex conjugation because

( $ ) - ( I > = ($(I))-

>

R,

= +A. Finally, the algebra separates points on R / Z , because by if $ ( I 1 ) = $(A2) for all $ E ~ ' ( D )we , have c$A, = 4L2, so that applying L we get I : 1 = I: 1. Thus the algebra 9 ' ( D ) - is dense in the space of even continuous functions on R, vanishing at co. Replacing f ( I , b) in (41) by f ( I , b)$(A), we can therefore conclude that there exists a null set N , c R+ (depending on f ) such that if I # N , , then the function 6 + F(1, b) lies in L2(B) and

+

+

/ / ( I , b ) f ( I , b) db = 0.

(43)

For each n E Z + let & E 9 ( D ) be the smoothed-out characteristic function of the ball B,(O) c D, i.e., 4"(z) = 1 for d(o, z ) c n - ( l / n ) , 4,,(z) = 0 for d(o, z ) > n + (l/n), and 0 I 4(z) I 1 for all z. Let 'Dl denote the countable set of all functions on D of the form &(z)p(x. y), where n E Z' and p is a polynomial with rational coefficients. Let N = UfSm N , , so (43) holds for all I + N and all f ~ ' D l .Now if , f 9~( D ) , we can choose n E Z + such that j' = &f, and then a suitable subsequence ,fk = 4"fk (k = 1,2,. . .) from 1111 converges uniformly to .f. It follows that for each I E R. $(I, b) converges to f ( I , b) uniformly on B. Thus (43) holds for all f~9 ( D ) and all A E R + - N . But then Prop. 4.10 implies

F(1, b)P(b)db

=0

for all P E L2(B)

J.4

for I E R+ - N . Hence jcF(A, b) dI db = 0 for each "rectangle" C in ( R + - N ) x B. whence F E 0 almost everywhere on R + x B. This proves Theorem 4.2. part (iii). Finally we prove Theorem 4.2, part (ii), characterizing the image (9(D))-.For this we must consider generalizations of the spherical function. Proposition 4.11. Let m E Z . The eigenfunctions f of L satisfying the homogeneity condition

(44)

j'(e'%) = eimef(z)

50

[Intro.

GEOMETRIC FOURIER ANALYSIS

are the constant multiples

0s the

functions

where 1 E C and x,,,(ei4)= eim4 Proof. In view of (17) and (44) the function F(r) = f(tanh

(45)

d2F dr2

-

1')

satisfies

dF

+ 2 coth(2r) - 4m2 ~ i n h - ~ ( 2 r )+F (A2 + l)F = 0 dr

for a suitable 1. Expanding F in a power series of sinh(2r), we can prove that the analytic solutions to (45) are proportional. In fact, writing OD

F(r) =

1ak sinhk(2r), 0

we find the recursion formula ((k

+ 2)2 - m2)ak+2= -[&(A2 + 1 ) + k(k + l ) l a k ,

from which the statement is readily derived. On the other hand, the functions @,,,(tanh r) and @-,,(tanh r ) are solutions, and by Prop. 4.8 they are not both 0. The analyticity of F being automatic, this proves the proposition. Another proof comes from Theorem 2.7 in Chapter 11. The function can be expressed in terms of the hypergeometric function. Using (16) and Erdelyi er al. [1953] (Vol. I, p. 81), we obtain, writing v = &(iA + I ) ,

Using now the transformation formula (47)

F(a, b ; C ; Z) = ( 1

-

z)E-"-~F(c- U, c - b ; C ; z),

we obtain the following invariance property of

@A,m.

Proposition 4.12. Let pm denote the polynomial pm(x) = (&x Then

+ l ) ) ( t ( x + 1 ) + I ) * * . ( $ ( x+ 1) + Iml - 1 ) .

4. The Hyperbolic Plane H 2

441

51

Since the coefficients 2 coth(2r) and -4m2 ~ i n h - ~ ( 2 rin) (45) have expansions 2 coth(2r) = -2

+4

m

e-4nr

0

and m

-4m2 ~ i n h - ~ ( 2 = r)

+ 16m2 - 16m21eC4"** 0

it is reasonable to try to find both solutions to (45) by means of such expansions. As r + co the differential equation (45) approaches the equation

+

F"

+ 2F' + (A2 + l)F = 0,

which has solutions e'*ia-l)r. This suggests finding a solution to (45) of the form

Substituting this into (45) and equating coefficients to e(ialed to the recursion formula (49) n(n - ilz)r,(lz) =

C

T,,-2k(lz)(2n- 4k - ilz

-2n)r

, we are

+ 1 + 4m2k).

1s k s [ ( l / 2 ) n l

Putting To = 1, this relation defines r,(lz)recursively as a rational function on C. Note that r,,(A)= 0 for n odd. In order to show that (48) converges we use (49) to estimate the growth of r,(lz)in n. Lemma 4.13. Let ilz42Z and let h > 0. Then there exists a constant

Ka,h such that

Ir,(n)[IKa,henh

(50)

fir

nEZ+

Proof. (by induction on n). First we select constants c1 > 0, c2 > 0 such that for all even n

In(n - U)l 2 cln2,

12n

+ 1 + 2m2n - ilzl I c2(n + 1).

Then by (49),

with c = 2c2c;'. Let N o be an integer such that

c(1 - e - 2 h ) - 1 < N o

52

[Intro.

GEOMETRIC FOURIER ANALYSIS

and select K = KA,h such that

I r,(A)I IKenh

for n IN o .

Let N EZ', N > N o . Assuming (50) holds for n c N , we have

< KeNhN-'c(l - e C Z h ) - ' IKeNh, and this proves the lemma by induction. Since h > 0 is arbitrary in Lemma 4.13, the estimate shows that the series for OA(r)converges, can be differential term by term, and provides a solution to (45). But then @ - A is another solution, and if i A # 2 Z , these is a solutions are linearly independent. It follows that for these A, linear combination of @A and @ - A ; say

+

CI(A)@A c- ' ( A ) a ) - A , Here C1 and C - l are holomorphic for i A # Z . In fact, we can evaluate the terms in (51) at points r l , r2 at which takes different values and then solve with respect to Cl(A) and C-,(A). To determine them we generalize Theorem 4.5 as follows.

(51)

@A,m

=

Theorem 4.14. If' Re(iA) > 0 and F E C(B), then (with b, as in Fig.

4 lim

e(-iA+

r++m

p +I ) ( a , . o . b)F(b)db = c(A) F(b0).

1)r JB

Proof. We have from (16), if b = eie, z = a, o = tanh r, a

J;(iA+

l)(Z.b)f'(b)

db

1 " (cosh 2r - sinh 2r cos O)-(1i2)ciA+ l ) F(eie)do. 2n -"

=-

J

Using the substitution u

=

tan(@),

the integral becomes for y(u)

$ d o = (1 =

F((J~~),

+ u2)-

du,

4. The Hyperbolic Plane H 2

541

53

Because of the estimates in the proof of Theorem 4.5 we take limr+m of this last integral by taking the limit under the integral sign. The result is m

I , ( I + u 2 ) - ( 1 / 2 ) ( i ~ +1 )g(0) du = .c(MO), so that the theorem is proved. Theorem 4.15 . The generalized spherical firnction iA# 2 2 given by the following expansion:

is f o r

m

OA,,(tanh r ) = C,(A)C T,(A)e(iA--'-2n)r 0 00

+ C-,(A)C r n ( - A ) e ( - i A - 1 - 2 n ) r , 0

where

ro(A)= 1, and r,(A)is given by the recursionjormula (49). Proof.

From (51) we see that m

e(-iA+

O,,,(tanh r ) = C,(A) C r,(A)e-2nr

1)r

0 m

+ C-I(A)e-2iArC r,(-A)e-2nr. 0

Letting r -+ + co and using Theorem 4.14. we obtain C,(A) = c(A) for Re(iA) > 0; by analytic continuation this holds for i A # 22. Handling e(iA+l)(r)@-A,,,(tanh r ) similarly and using Prop. 4.12, we obtain

so that the formula for C-,(A) follows by analytic continuation. . functional In order to prove Theorem 4.2, part (ii), let . f ' ~ g ( D )The equation

54

[Intro.

GEOMETRIC FOURIER ANALYSIS

is immediate from Lemma 4.7. Moreover, as proved in No. 1,

where do,is the arc element on the horocycle indicated. This shows by the Paley-Wiener theorem on R that f has uniform exponential type. It now remains to prove that each $ with the properties in Theorem 4.2, part (ii), has the form f for some f ~ ! 2 ( D ) .For this purpose we define the function f~ C ( D ) by

It is clear from (16) that the derivatives of e""l)(r*b) with respect to z are bounded by a polynomial in I [uniformly for (z, b) varying in a compact set]. Because of the rapid decrease of $(I, b) in I , we have therefore , f € & ( D ) . Assuming $ of uniform exponential type R , we shall prove . f ( z ) = 0 for d(o, z ) > R . For z and I fixed we expand the functions 8 -P .f(eioz),fl -,$(A, deb) into convergent Fourier series and evaluate at 0 = 0. This gives . f ( z )=

1fm(z),

me2

$(A ei+) =mCe Z$m(I)eim6,

where

In the first formula we substitute expression (52) for 1: Then since $ ( I , b ) is rapidly decreasing in I , we can interchange the integrations JdH and ( d A db. This leads to the integral

Jo2nJo2=

ei+)e(ii+

l ) ( e f e z , e i Q ) - im0

e

But (eiez,e'") = ( z , ei(4--B)),so putting 0 written

JJ$(I,eib)e(ii+ I )

(53)

4 - ( this integral can be

( z . e ~ ; ) e i m F L le~- i r n +

= (2x)'mA. r n ( z W m ( I ) .

Thus we have obtained

=

ded4.

(14

4. The Hyperbolic Plane H Z

941

55

and since ,/;(&'z) = eimu/&), it suffices to prove ,fm(ur.o)= 0 for r > R. For this we use the expansion of Theorem 4.15. The following lemma is crucial. Lemma 4.16.

For

= f I we have

JI.c'(A)rn(&I)c&i"~m(A) 1 c ( ~ 1)- 2 n~ = o Proof. We consider first the case for A E R, the integral equals

I:

=

jbr r > R.

+ l . Since Ic(A)12

=

c(A)c(-A)

Jjm(-,qrfl( - A ) ~ ( A ) - 1 e - i ~ rl l ~ .

(54)

If the function I)~(-A)r,(-A)c(A)- were an entire function of exponential type R [like t j m ( - A ) ] , this integral would vanish for r > R as a consequence of the classical Paley-Wiener theorem. However, both r,(-A) and c(A)-' have poles. Fortunately there is a half-plane where both are holomorphic. In fact, by the recursion formula for the function r,(A) its poles are among the points i A E Z + ; in particular, the function A + r,(-A) is holomorphic for A in the lower half-plane. By Theorem 4.5 the poles of A -,c(A)-' are the poles of r($(iA + I)), so, it is also holomorphic in the lower half-plane. We therefore use Cauchy's theorem to shift the contour downward: For vl < 0 (55)

J),,,(-t)r,(-t)c(t)=

* e - ic* (it

/2m(-t - iq)rn(-t

- iq)c(<

+ iq)-le-i@eqrd t .

For this to be permissible we must check the behavior of the integrand at m. First, r,(A) is rational. Second, c(t + iq)-l can be estimated by means of the known asymptotics for the gamma function (Magnus and Oberhettinger [1948], p. 5 ) ; for example,

where K 1 and K 2 are constants. This implies by Theorem 4.5, (56)

Ic(t

+ iq)l-l

IC ,

+ C215 + iqll",

<E

R, q < 0,

where C , and C, are constants. But by (31) for t+bm(-A) the rapid decrease of $,,(-A) cancels out the growth in r, and c at 'x?, so

56

[Intro.

GEOMETRIC FOURIER ANALYSIS

the contour shift above is permissible. Denoting the right-hand side of ( 5 5 ) by Q(r), we have

I Q(r)l < CeRIq1erq, where C is a constant (independent of q). Letting 4 + - 30, we deduce Q(r) = 0 for r > R. This proves the lemma for the case E = + 1. For E = - 1 the integral becomes (57)

and again we would like to shift the integration downward. However, the denominator pm( -in) vanishes for I = -i, - 3 i , . . . , which at first seems to cause complications. Fortunately, $,(A) turns out to have zeros at these points. We shall, in fact, prove

To see this we integrate the relation

p +1 ) ( z . b > = 1

(59)

m(z)x

-m(b)

m

against

$(A, b ) . This gives $(A, b ) e ( i a + l ) ( z * b ) db

=

1@

1.m(z)$m(J)

m

=

1 @,,,,(tanh r)$,(l>eime m

if z

=

tanh r eie. By our assumption about $, the left-hand side is even in

A, so that @A,

m(tanh r ) $ m ( I ) = Q, - A. m(tanh

r>$m(

- 1).

Now (58) follows from Prop. 4.12. But then the integral (57) reduces to (541, which was already handled, so Lemma 4.16 is proved. Now we substitute the expansion of @,,,(tanh r ) into the integral @,,,(tanh r)$,,,(l)lc(l)l-z d l . Integrating term by term and using Lemma 4.16, we obtain the desired result, fm(tanh r ) = 0 for r > R . The term-by-term integration is justified by the following lemma.

4 . The Hyperbolic Plane H 2

$41 Lemma 4.17.

57

The r-coeflcients satisfj

Ir,(A)lI c(l + nd),

A E R, n g Z + ,

for suitable constants c and d.

Proof. The recursion formula (49) shows that Put a,@) = r2,,(A), so that

rn(A)= 0 if

n is odd.

n- 1

2n(2n - iA)a,(A) =

1ar(A)(4r- iA + 1 + 4rn2(n - r)).

r=O

We write this in the form n- 1

r=O

and then there exists an integer N such that

n- 1

nb, = N

1b,,

bo = 1,

r=O

we have from (60) and (61) by induction

la,(A)( I b,

(63)

for n E Z + , A E R.

Now (62) implies n-2

(64)

nh,

=

N

1b, + Nb,-,

=

(n - 1

+ N)b,-l,

0

so

b, =

(n + N - l)! n ! ( N - l)! .

The lemma is now obvious from (63). We have now proved that the function ,f defined by (52) satisfies (66)

f(z) = 0

for d(o, z ) > R.

It remains to prove f = $. By the inversion formula [Theorem 4.2, part (i)] we have for d, = f - $

58

[Intro.

GEOMETRIC FOURIER ANALYSIS

Since the inner integral is even in A, we can here replace R by - R + . Integrating the resulting relation against an arbitrary F E 9 ( D ) , we obtain

J-

&-A,

b)F(1,b)lc(A)l-2d l db = 0.

R+ X B

But we have seen that the functions F form a dense subset of L2(R+ x B); hence 4 = 0 on ( - R ' ) x B. But

s.

$(A, b)e(iA+1 ) ( z . b ) db =

s,

$(-A, b p - i A + I ) < z . b ) db,

so, since each 1E R is simple, r j G 0. Theorem 4.2 is now completely proved. 4. Eigenfunctions and Eigenspace Representations. Proofs of Theorems 4.3 and 4.4

Given AEC we consider now the representation Ti of G on the eigenspace

=

SU(1, 1)

+ l),f}.

b , ( D ) = { f Cm(D): ~ L j = -(A2

If g E G , T,(g) is the linear transformation of b,(D) given by (TXg).f) ( z ) = f ( g -

*

2).

We shall now prove Theorem 4.4:

+

TAis irreducible if and only if iA 1 $ 2 2 . (67) Remark. T', is irreducible if and only if A, is not a pole of the denominator of the function c(A)c(-A). Lemma 4.1 8.

The point

A E C is simple

i f and only if the ,functions

form a dense subspace of Lz(B).

This is an immediate reformulation of the definition of simplicity. Lemma 4.19. The mapping F

is continuous.

f of L 2 ( B ) inro b,(D) given by

4. The Hyperbolic Plane H 2

§41

59

The topology of 6 ( D ) is described in $1. The lemma follows immediately by using Schwarz's inequality. For I E C simple let & denote the space of functions J' defined by (68) as F runs through L'(Z3). Then 4 is a Hilbert space if j' is given the L2 norm of F . Suppose I E C is simple. Then

Lemma 4.20.

& is dense in ga(D).

The proof is analogous to that of Lemma 2.8, so we omit it. Suppose now iI + I$2Z. This means that both I and - I are simple. Let 0 # V c G,(D) be a closed invariant subspace. Then V contains an element ,f such that ,f(o) # 0; averaging over the rotations around 0, we conclude 4 , V. ~ Now by (35) .

Since - I is simple, we conclude from Lemmas 4.18 and 4.19 that the closed subspace of &,(D) generated by the various linear combinations (69) contains X,. But then Lemma 4.20 shows V = b,(D). Hence T,, is irreducible. On the other hand, suppose T, is irreducible. Since T,, = T-a, since the condition il + 1 4 2 2 is invariant under I + - I , and since either I or - I is simple, we may assume that I is simple. Let 0 # E c & be a closed invariant subspace of the Hilbert space 4. By the irreducibility of T,, E is dense in 4 ( D ) . Let .f * x,,,(z) = 2n I

Jo2R

f ( e - io z)X,,,(eie)do.

Then E being closed in & and invariant, (70) Since E is dense in

(71)

E * x,,, c E.

4 ( D ) and the map f + f * x,,, is continuous E * x,,, is dense in

* x,,,.

But by Prop. 4.11 this last space equals CaA.,. Thus (70) and (71) imply a,,,,E E for each m, whence E = &. Thus G acts irreducibly on 8: hence the functions (69)are dense in &. Using Lemma 4.18, we conclude that - I is simple. This finishes the proof of Theorem 4.4. Finally, we prove Theorem 4.3 giving the integral representation of the eigenfunctions of L by means of analytic functionals. Let .f be any eigenfunction of L and select 1 E C simple such that &f=

-(A2

+ I).f.

60

[Intro.

GEOMETRIC FOURIER ANALYSIS

For fixed z E D we develop the function 0 +f(eiez) in an absolutely convergent Fourier series (72)

f(eiez) =

2 cn(z)eino, n

where

Then by Prop. 4.1 1 c,(z) = [email protected](z),

UnEC.

Using the transformation formula

for the hypergeometric function (Erdelyi et al. [1953], Vol. I, p. 64), we derive from (46) with v = i(iA + 1)

where r = IzI. We know from (72) that

2 Ian*l,n(r)I

(74)

< 00

n

and we want to deduce from this that (75) Let us choose k e Z + such that k > IvI and put x = r2(r2 - l ) - ' . Then for n > 0 F(v, 1 - v; n

+ 1 ;x) = pk(n, x) +

P k + l(n,

x),

where pk is the kth Taylor polynomial pdn, x) = I

+ v(l(n +- 1)v) ~

+ ...

V(V+ l ) * * * (+ v k - 1)(1 - v ) . * * (l v (n I ) . . . ( n+ k ) k !

+

and Pkf, is the remainder. Using the general formula

+ k - 1 ) Xk

441

4 . The Hyperbolic Plane H 2

61

for the remainder term in a Taylor series and the Euler integral formula for the hypergeometric function

(Re c > Re b > 0), we obtain for n

+ 1 > Re(1 - v) > 0 x (1 - s ) k ( l

- srx)-"-k-l

dsdt.

The definition of P k + , and formula (73) show that P k + , ( n , x) is holomorphic in v except for V E - Z + and these points are excluded. since 1 is simple. Thus the last formula holds by analytic continuation for n + Re v > 0, since k > I v I . Since x < 0, 0 5 s, f 5 1, and k > 1 v(, we have 1(1 - s r x ) - v - k - '

I I 1,

so

X

r(2 - Re v + k)T(n + Re v) r(2 + n k )

+

Using the asymptotic property

of the gamma function, we have lim n-m

T(n

+ Rev) = 1,

I U n + v)l

where Ck is a constant. Now we put

62

[Intro.

GEOMETRIC FOURIER ANALYSIS

Then by (74)

whence

We have v and k fixed. Let 0 < ro < 1 be arbitrary and put xo = rg/(ri - 1). Then select N,, such that

This can be done since lim pk(n, xo) = 1. n-. m

But [ ~ ( nx)l , I1, so by (77) m

2Ibnr:n-(k+')I < a. 0

Since ro was arbitrary, this proves m

C0 lanlrn< 00,

0 I r < 1.

Now (75) follows since n appears in (73) only as In I. Now given a hyperfunction T E ~ ' ( Bwe ) define its Fourier series by T

-C

aneine

if a,, = Jo2ne-inedT(0).

n

Then we have the following result. Lemma 4.21. A hyperfunction is uniquely determined by series. The series C aneine n

is the Fourier series of a hyperfunction T Ed ' ( B ) if' and only (78)

1 IanIrInl <

00

.for all

n

Proof. Suppose we have a Laurent series (79)

F(z) =

1 b,z" n

o I r < 1.

if'

its

Fourier

WI

4. The Hyperbolic Plane H 2

converging in an annulus containing B : I z I both series

=

63

1. This is equivalent to

c bnzn, c b-,zn a3

W

0

0

having radii of convergence > 1 or equivalently, (80)

lim supIb,llln = p < 1,

lim supIb-,,I"" = y < 1;

n

n

all such numbers p, y, 0 I p, y < 1, can occur. By the definition of the inductive limit, the series (79) converges in the topology of d ( B ) . Thus if T Ed ' ( B ) has Fourier coefficients (an), we have

c bna-,.

T ( F )=

n

In particular, T is determined by its Fourier series. Because of (80) and the arbitrariness of j and y, we deduce

c

~ a n ~
< 1.

n

On the other hand, if this condition is satisfied, we can define a linear form T on d ( B ) by T ( F )=

c xnb-,. n

To see that T is continuous we represent d ( B ) as the union m

d(B)=

IJ

z n t

n= 1

where & is the set of holomorphic functions in the annulus 1 - (l/n) < 1zI < 1 + (l/n). By the definition of the topology of d ( B ) we must prove that for each n, the restriction TI& is continuous (for the topology of uniform convergence on compact sets). Because of (78) the functions W

fl(z) = 1akzk, 0

W

,f2(z) =

1a - k ~ k 1

are holomorphic in the unit disk Iz( < 1. For the curves 1 C 1 : I z l = l - - , 2n

1 C 2 : I z l = l + - 2n

64

[Intro.

GEOMETRIC FOURIER ANALYSIS

we have

from which the continuity of TlX is obvious. This proves the lemma. Going back to the proof of Theorem 4.3, we know now that there exists a hyperfunction T with Fourier series T

-1

aneine.

n

But given z E D the function 1 - zz

v = &(iA + I),

is holomorphic in an annulus II containing B and can be expanded there in a Laurent series. The restriction of this series to B is the Fourier series (59). Since the Laurent series converges uniformly on compact subsets and since the injection X ( U )-+ d ( B ) is continuous it follows that (59) converges in the topology of d ( B ) . Hence we can apply T to it term by term and deduce from (72)

I

etid+l)
1 Qd.,,(z)u,,, = f(z), m

which is the desired integral representation of f. On the other hand, we must verify that if T Ed ’ ( B ) , the function

is an eigenfunction of L. The continuity of f is clear, since z, + zo implies e(iA+l)(zn*b) + etid+l ) ( ‘ O s b ) in the topology of d ( B ) . Next we use the following result whose proof is entirely analogous to that of Prop. 2.5. Let dk be the normalized Haar measure in K = SO(2). Proposition 4.22. the junctional equation

A function

-

f satisfying Lf = - ( A 2

f ( g k az) dk = .f(g o M A ( z ) ,

(82) K

+ 1)f

g E G, z E D.

satisfies

91

4. The Hyperbolic Plane H Z

65

Conversely, a continuous ,function f sutisjjing this jirnctional equation is automatically ofclu.s.s C" and sutisjies Lf = -(AZ + 1 ) f . The integrand in (81) satisfies the functional equation (for each b); hence to show that f does too, it suffices to show the commutation

JK(L

e ( i l + l)<&,:.b)

d T ( b ) dk

)

=

I(S,

e(iA+ I ) < d . z , b )

1

' / k [/T(b).

Thinking of the integral over K as a limit, this follows just as the continuity of 1' above. To conclude the proof of Theorem 4.3 it remains to remark that if A is simple, then the mapping

T E d'(B) + ,f'E &,'( B) given by (81) is one-to-one. For this note that f and T are related by

(83)

a,,@,.,,(z),

.f(z> = n

T

- 1 vine,

and

do.

(84)

By the simplicity of 1 each @, is $0, so that .f determines the sequence (a,) which by Lemma 4.21 determines T. This finishes the proof.

For Re(i1) > 0 the hyperfunction T can be recovered from ,f somewhat more directly; in fact we have as a special case of Theorem 4.14, lim

(85)

e(l -i*"a,,@A,n(eiH tanh

t ) = c(A)u,ein0.

1-+m

Under a mild condition on A we shall now prove that the eigenfunction j ' ( z ) grows at most exponentially with the distance d(o, z ) if and only if the hyperfunction T is a distribution. We shall need the following analog of Lemma 4.21. Lemma 4.23.

The series

i s the Fourier series o f a distribution T on B if and only

(87)

JM,I

.fOr some cmstants C untl 1.

IC(1

+ In[)',

nEZ,

lf

8

66

[Intro.

GEOMETRIC FOURIER ANALYSIS

Proof. If T is a distribution, it is continuous on 9 ( B ) in the topology of the norms llFllk = sup,,,,,,kldrF/tiBII, k = 0, 1,2,. . . . Hence there exists an I2 0 and a constant C such that I T(F)I ICIIFII,. This proves (87). On the other hand, if a,eine is a series satisfying (87), then for k E Z + large enough the series

1

1a,(l + it2)-kein" n

is absolutely convergent to a continuous function G. The mapping t

T:F

r2n

J

G(O)(I - L)kF(0)dO 2n 0 is then a distribution with Fourier series a,ein". Given ~ E C let , & x ( D ) denote the subspace of functions . J E $ ~ , ( D ) satisfying an inequality +

I.f(z)l I C e U d ' U ~ z ' ( z E D )

(88)

for some constants C and a. Theorem 4.24.

Assitine iA $ Z. Then the nirrppiiiq T

j ' ( z )=

l)(z.b)

-+.1; where

[IT(/,),

H

is u bijection

of the space 9 ' ( B ) (of distribirtions 011 B ) onto

&T(D).

Proof. Consider the Fourier expansion (59). If L , is the Laplacian

d 2 / d 0 2 on B, we have ( - ,rB)g(eM

+ 1 )<:.

b) ) =

c

@,..(z)i12p;C-n(b)3

n

so that (89)

@,,,(z)i12P = J(-L,)g(dii+

)%Ah)dh.

1)(2.b>

B

If z (90)

=

tanh

t pi&,

h = pie, we have by (16)

e2"('*') = [ch(2t) - sh(2t)cos(b, - O)]-".

From the definition of ( z , b ) it is easily seen that l(z, b)l I d(o, z )

for all

z , h.

Thus if we apply Lf; to (90), it follows quickly that 2v(z, b ) < Ce[ldlo.z ) , I W e )I -

4. The Hyperbolic Plane H'

941

where C and a are constants depending on p and

(91)

I@A,"(z)n'PI

67 11.

Thus by (89)

I CC" d ' o - 2 ) .

Suppose now T E ~ ' ( Bhas ) Fourier coefficients u,. Since (59) converges in the topology of B(B), we can apply T to it term by term and deduce that

But by Lemma 4.23 u, is bounded by a power of II and since (91) holds for each p E Z + , it follows that , f e B:(D). Conversely, suppose 3, satisfies the conditions of the theorem and suppose j ' G':(D). ~ Let T Ed ( B ) represent f , in the sense that ,f(z> =

1

e(iA+l ) ( z * b )

c/~(b).

B

Since 3, is simple, we know by a previous remark that T is unique. In order to prove that the u, satisfy the condition of Lemma 4.23 we shall establish a modified version of (85). For this we need the following relation : F(u, b ; c; X) =

r(e)r(e r(c

-

h) F(a,b;a b - c - h)

-

+

+ 1; 1

-

x)

+

(cf., Erdelyi et a/. [1953], Vol. I, p. 108) valid for - c $ Z ' , u b - c$Z. Using this on (46) we obtain after simple manipulations [with L! = +(iA I)]

+

@A.

.,(tanh

I)

valid if 2v $ Z (which is equivalent to our assumption about A). Moreover, if , F 1 denotes, as usual, the confluent hypergeometric function, we have for each 1x1 < 1

(93)

lim F(u m-+m

+ m,b ; c ; x / m ) = ,F,(b, c, x)

68

[Intro.

GEOMETRIC FOURIER ANALYSIS

(Erdelyi et al. [1953], Vol. I, p. 248). We construct now the sequence tm + co by (94)

ch2 t , = Iml/a,

where a > 0 is a number to be determined later. Using (93) and the relations

lim (tanh t,)"' = e-(1'2)", n++m

we deduce from the formula for @,, lim (ch t,)2-2Y@A,,(tanht,) I4+m

1

+ r(2vUV)- I ) 1Fl(1 - v,2 - 2v,a) . Since (cf. loc. cit. p. 278)

lim ,Fl(b, c, x ) e - x x b - c = r ( c ) l u b ) x++w

and since 2v # 1, the two terms inside the square brackets behave differently for a large a. Hence we can select a such that the last limit is #O. Then for a suitable constant Co # 0 we have (95)

I @A, rn(tanh trn I 2 Co I m I(1/2)(Re(iA)- 1 )

On the other hand, putting z

=

tanh t , eie in (92), we obtain

which by the assumption , f ~ b ? ( D has ) a bound Clmf' for some constants C and b. But (95) and (96) then imply (97)

I a,

12

(c/co)1

( b + ( 1 / 2 ) (1 - M i l ) )

so by Lemma 4.23, T is a distribution and the theorem is proved. Remark. While the case ilz = 1 was excluded, the theorem is valid for this case. The proof is even easier. In fact, we have here

@,.,(tanh t ) = tanhlmlt ,

441

4. The Hyperbolic Plane H 2

69

so that lim @,,,(tanh t,)

=

e-(112)a.

Iml+m

Hence (95) is obvious and (97) follows as before. Thus the ordinary harmonic functions

which are Poisson integrals of distributions, are characterized by the growth condition lu(z)l 4 Cead'"*LJ. It is of interest to compare this with the classical characterization (Herglotz, Evans, Charatheodory, cf., Nevanlinna [19363) of Poisson integrals of measures as the harmonic functions u satisfying sup [ 0 2nl

u(rei+)ld 4 < m.

rcl

5. Limit Theorems To conclude this section we prove two classical theorems from potential theory. Theorem 4.25. Let F he continuous on B and define the harmonic ,firnction 11 on D by the Poisson integrul u(z) =

Then lim u(z)

1-

S, 1~

=

1212

~

-

F(bo)

b12

F(b) db.

fir

b, E B.

z-bo

This classical theorem of H. A. Schwarz is usually proved by reducing it to the case F(b,) = 0; then B is divided up into an &-arcBo around bo and its complement. Then in similar fashion 11 decomposes into 11 = u,, ( u - u,). The first term is small near bo because of the continuity of F at b,; the second term is small since the kernel is small for z near b,, Ih - bol 2 E > 0. Because of formula (16) for the Poisson kernel, the theorem is a special case of Theorem 4.14 (with iA = 1). For this case we can give an even more elementary proof. Because of (34a) u can be written

+

u(g . o) =

It suffices to consider the case z

--t

JF(y . b) db. b,, bo = 1.

70

[Intro.

GEOMETRIC FOURIER ANALYSIS

Let u, be the one-parameter group considered before and k , rotation by the angle Of,where 8, + 0 for t -, + m. Then 6217

I

J

u(k,u, . a) = u(eier tanh t ) =

~ ( k , u. f, i e )dU 2.n 0 eie + tanh t no. tanh t eie 1

+

Letting t

+

)

+ CQ,we get by the dominated convergence theorem 1 lim u(k,a,.o) = 2.n

,‘+a

J,

2n

F( 1 ) (10 = F(bo),

as desired. The proof works because the one-parameter group u, pulls the entire boundary B (except for the point - 1 ) to the point b , . We shall now prove the classical theorem of Fatou that a bounded harmonic function on D has radial boundary values almost everywhere on B. While the proof is hardly simpler than the classical one, its grouptheoretic features serve as a basis for the generalization to symmetric spaces. The theorem generalizes Theorem 4.25 in that continuity of F is dropped. Theorem 4.26. (Fatou) Let u be a bounded harrnonic junction D. Then.for almost all 9

lim u(reie)

011

exists.

r- 1

Proof. It is well known that u is the Poisson integral of a function F E Lm(B). Consider now the subgroup E of matrices

(

E = {t, = 1 -+i x ix

1 ix ix ) : x E R }

and the mapping

t,+t;1

+

1 2ix 1 - 2ix

=-

of E into B. It is a bijection of Z onto B - { - 1). Writing eie = ( I

+ 2ix)(l - 2ix)-’,

we obtain 4 dx ne = ___ 1 + 4x2’

941

4. The Hyperbolic Plane

HZ

71

so that

Now fix 5, E 2 and use this relation on the function f ( b ) = F(<,a,. b). Then, since a, 5,. 1 = a, (,a; . 1, we have

‘

Writing

4(y)= F ( 5 , . I),

we deduce that

We now use the Lebesgue differentiation theorem, which states that (99)

‘i”

lim h-02h

-h

14(y + x)

- &y)I dx = 0

for almost all y. Given E > 0 we select c > 0 such that, writing the righthand-side of (98) as the sum ( 100)

2 71 r- c

+;J;

XI

>c 1

the latter term is < E for all y and all t . In the first term we put x 1 = xe-2‘ and then write h = ceC2‘. Then the first term in (100) becomes

Using (99), we can thus conclude that for almost all y lim u ( ~ , u , .0)

exists.

,-a?

This means that u has a limit along the geodesic from 5,. o to the boundary point 5,. 1. From this it is easy to deduce that u has the same limit along the radius (o,c,. 1). For this we use the elementary inequality I(Vu)(z)I 5 C / R

for IzI < I

- R,

72

GEOMETRIC FOURIER ANALYSIS

[Intro.

Fiti. 4

Vu being the gradient of II and C being a constant. To prove this inequality we consider the Poisson formula

for the disk ( z J 5 R , which implies by differentiation

so, using Schwarz's inequality,

The desired inequality now follows by using (101) on the function M' 4 u(w + z ) in the disk I w I I R . As R decreases to 0, the two geodesics above come together at a faster rate, so combining the inequality above with the mean value inequality lu(z") - u(z')I IIVuIIz" - z'I we see that u has the same limit along both geodesics. This concludes the proof.

EXERCISES AND FURTHER RESULTS A. The Spaces Rn and Sn

1. Let H be a group acting on a space X. If ,f is a function on X and h E H, let . f h ( x ) = , f ( h - . x), x E X. A continuous function ,f is said all lie in a finite-dimensional to be H-finite if the translated functions space.

'

Exercises and Further Results

73

(i) For the sphere S"-' c R" show that a function f is O(n)-finite if and only if j ' is the restriction of a polynomial on R". Equivalently, f is a finite linear combination of spherical harmonics. (ii)* Consider the orthogonal group H = O(p, q + 1) acting on the quadric 2 2 cpq: .Y: + . . . + x p - x p + I

-

. . . - x p2 + y + 1

=

-1,

p20, q 2 0 ,

in R p c q + + Assume '. ( p , q ) # (1, 0). Show that a function ,f on C,, is Hfinite if and only if it is the restriction of a polynomial on R p + 4 + 1 (Helgason [1963], $3). (iii) Show that the statement (ii) fails for O(1, 1) acting on x: - xf = 1. (iv)* Let K be a compact group of linear transformations of a real vector space V, M the subgroup of K leaving a certain vector u # 0 in V and i the imbedding k M + k . u of K I M into V. Then the K-finite functions on K I M are precisely p i, where p is a polynomial function on I/ (ibid., $3). (v) For a noncompact analog see Chapter IV, Exercise AS. 2. For A E C let &,(R") denote the eigenspace

&,(R") = { f~&( R") : L R m f = - A z f } and let TAdenote the representation of the group M(n) of all isometrics of R" on d , ( R " )given by

(T,(s)I)(x)=

m-. I

XI,

9 E M(n),

x E R".

Using Theorem 2.7. Chapter I1 show that if A # 0 the O(n)-finite functions in Q ;.( R") are precisely f(x) =

Jp - P * " J ) F ( ~ iiw, ) I

where F is a K-finite function on S"-'. Deduce, generalizing the proof of Theorem 2.6, that T , is irreducible ifund only if

A

# 0

(cf., Helgason [ 19743, $8). 3. For the representation To (from Exercise 2) the space Xk = H i of harmonic polynomials of degree < k is an invariant subspace; in particular, To is not irreducible. Show, however, that the corresponding representation of M ( n ) on the factor space c@k/c@k-I is irreducible.

74

[Intro.

GEOMETRIC FOURIER ANALYSIS

*4. While the representation To of M(2) on the space S o ( R 2 ) of harmonic functions on R2 is not irreducible, there is an irreducible action on S o ( R 2 )of a larger group (or rather its Lie algebra): Let G = SL(2,C) denote the group of unimodular matrices acting on the one-point compactification of C by means of the maps g:z+-

uz

cz

+b + d'

Z€C

i) being an element of SL(2. C). If L = LRZrconsider the operator Lg: 4 + ( L ( 4 o 9)) o 4- ; show that L has the quusi-invariance (:

( ~ 4 4 ) ( ~= )

lcZ - 44(~4)(4,

so 4 harmonic * $ 0 y harmonic. Show that the corresponding action of the Lie algebra 9 of G on S o ( R 2 ) is scalur irrerltrcihle; that is, the only continuous operators commuting with the action are the scalar multiples of identity (cf., Helgason [1977a], also for a modified extension to I I dimensions).

5 . Let H k be as in $3. Show that (i) dim H , = ("ti-2) + (":!i3). (ii) LRn(IxIZJ/i) = cl.x12J-'h, / t € H k , whcrc c is thc constant 2j(2j + 2k + I I - 2). Deduce that if h E H , , Xjh

= hktl

+ (n + 2k -

wherc hk+ I E Hk+ 1. (iii) Let / I E H k . Using the mean-value functions, show that

theorem for harmonic

where Ck."

= (-

l),(n - 2)n(H 4- 2 ) . . . ( P I

+ 2k

-

4).

(Maxwell [1892], Hobson [1931], Coifman and Weiss [1971]).

75

Exercises and Further Results B. The Hyperbolic Plane

1. Derive the Iwasawa decomposition G G = SU(1, 1) by means of Figure 2. 2. (i) Show that the transformation

=

K A N for the group

is an isometry of D onto the upper half-plane H : u > 0 with the metric

du2 + d v 2 4u2 . (ii) Show that if

az

+b

- J b l z= I ,

g(z) = -

6z + ii’

then cgc - 1

[(a + a) - i(b (w)= [(b + 6 ) + i(a - a)]w

+ ( b + 6) - i(a - ii) + (a + ii) + i(b - 6)

and this can be written in the form c g - ‘(w)

=

uw ~

yw

+p + 6’

where

(i)!

e S L ( 2 , R).

In this upper half-plane model the isometry group is generated by

SL(2, R ) and the map w -+ ljw. (iii) Show that the circle S,(i) in the Riemannian manifold H equals the circle SsinhZr(cosh 2r) in the flat Riemannian manifold R2. (iv) Show that the Laplace-Beltrami operator on H is given by

and that, by analogy with Lemma 4.1, L ( p ) V )== p(p - 2)u(1/2h,

3. Prove that the mapping f - F,. where F,(t)

= r-1

s,I.(nXal.o ) dx

76

GEOMETRIC FOURIER ANALYSIS

[Intro.

is a bijection of @(D) onto the space @ ( R ) of even C"-functions on R of compact support. Also,

F,*,

=

F,

* F,.

(Takahashi [ 19631). 4. The spherical function 4A has the following properties: With 1 = 5 + i? rl E R), (i) di(z) is real for all z o
+

lim f ( < iq)

=

o

uniformly in lql I 1.

1e1-m

7 . If .f is a function on D and 5 a horocycle, put

.i'(<>= / . f ( x ) d 4 x ) , e

riw being the arc element on

5. Writing

,f(eie,t ) = f(<(tanh r e", e")). we have seen that

Deduce from Theorem 4.2 and Prop. 4. I2 the following: The mapping 1'- ,f is a bijection of 9 ( D ) onto the space of'firnctions E 9 ( S ' x R), which when expunded,

+

$(do,t ) =

c +,(t)e'"O,

nei!

where 4nE 9 ( R ) is even (cf., Helgason [1983a]). 8. Describe the orbits of A on D. 9. In the notation of Figure 5, where I' and t denote non-Euclidean distances in D, show x(r, t ) =

ch r t h f i sh r th t i

+ sh I' + ch r

Exercises and Further Results

77

FIG.5

and prove accordingly that

ID

f ( x )dx =

J R f ( x ( r , t ) ) cash 2t dt nr

JR

Remark. The diffeomorphism ( r , t ) Theorem 14.6, Chapter I.

-

x(r, 1 ) is a special case of [DS],

10. With the transform f - F , as in Exercise 3 show that if f ( ~= ) ch d(o, z ) - ~ ’ -

’

then F f ( t ) = nc(A)(ch t ) - i A (Koornwinder). C. Fourier Analysis on the Sphere

1. (Cf., Sherman [1975]) Let S” denote the unit sphere in R””. Fix (0,. . . ,O, 1) and B = { s E S ” :(a, s) = 0 ) (a is the North Pole and B the Equator). By analogy with (11) in @4 consider for k E Z’, b E B the functions a

=

(a f

,fb,k(S) =

sgn(S, ,)“-‘(a

(i) Show that -k(k II - 1).

+

SES”,

ib, .y)k.

ob.k(S) =

ek.b

and

.fk,b

+ ib, S ) - k - n + ’,

SES” -

B.

are eigenfunctions of Lsll with eigenvalue

78

GEOMETRIC FOURIER ANALYSIS

[Intro.

(ii) Let C: be the constant multiple of the ultraspherical polynomial Pi''2)("-1) in (8a) in $3 such that Ci(1) = 1. Then

[cf. (35) in $443, prove that

c;((s,

s')) =

eb,k(s).fa, k(s')

db.

B

(iv) Given f ~ g ( S " )vanishing in a neighborhood of B, define its Fourier transform ?(b* k, =

I

f(s)f6,k(s>

S"

ds*

Then [in analogy with Theorem 4.2, part (i)]

f(s)= fB

x z + eb.

k(s)f(b.k ) db!(b, k),

where dp(b, k ) = dim Hk(R"+') db in the notation of (6) in 53.

NOTES The representation (Theorem 4.3) of the eigenfunctions of the Laplace-Beltrami operator on H' was proved by the author ([1970a]. Chapter IV. $1. and [1974]). This fatter paper also contains the Euclidean analog. Theorem 2.1. which was extended to higher dimensions by Morimoto [1981]. Less specific version for R" had been given by Hashizume et a/. [ 19721. Different integral representations for constant coefficient operators on R" are given by Ehrenpreis [1969], Chapter VII, and Palamodov [1970] (Chapter VI). Lemma 2.2 is an immediate generalization of a classical Taylor series characterization of entire functions of exponential type (Boas [19543). The proof incorporates some simplifications kindly communicated by M. Cowling. Propositions 2.5 and 4.22 are special cases of a mean value theorem for homogeneous spaces (Helgason [1962a], Chapter X, Corollary 7.4; the case of zero eigenvalue had been proved by Godement [1952al). The eigenspace irreducibility (Theorem 2.6 and Exercise A2) is from the author's paper [1974]. Theorem 2.9 goes back to Paley and Wiener [1934]; see also Hormander [1963]. The modified version in Theorem 2.10 is from Helgason [1976], $11. (The theorem is misstated in this paper: "polynomial" in Theorem 1 1.1 should read "harmonic polynomial.")

Notes

79

The expansion of a function into its “Laplace series” of spherical harmonics is classical (see Hobson [1931], and references there). For convergence results in the spirit of Theorem 3.4 see, for example, Calderon and Zygmund [1957], Clerc [1972]. Inspired by PeterWeyl [1927], Cartan in [1929) interpreted the spherical harmonics group-theoretically and generalized the Laplace series to compact symmetric spaces. Further generalization was given by Weyl [I9341 (see Chapter V, @3-4, of this book). Another approach, more in line with our Fourier transform on H’. was initiated by Sherman [I975 19771; see Exercise CI. The classical decompositions ( 5 ) and (6) for harmonic functions will be generalized much further in Chapter 111. Theorem 4.2 is a special case of results from the author’s papers [1965b, 1970a], (Chapter 111, 55). and [1973a], $8, where a Fourier transform for symmetric spaces is defined and developed. The definition is sketched here at the end of Chapter IV, $7 (see Remark 3); the Paley-Wiener theorem for this transform is used in the answer to Problem B in Chapter 11, $4. The integral representation in Theorem 4.3 and the irreducibility criterion in Theorem 4.4 (both from Helgason [1970a], Chapter IV) have natural generalizations, some of which are described in Chapter 11, $4, No. 1. The proof of the inversion formula, (25) in $4, for the spherical transform is from Godement [1957a]; the more general Plancherel theorem for the group G = SL(2, R) was already given by Harish-Chandra [1952]. whereas the basic representation theory for SL(2, R) was developed by Bargmann [1947]. The general theory of spherical functions is the subject of Chapter IV, and we refer to the notes to that chapter for further references to the literature. The expansion in Theorem 4.15 is a special case of an expansion due to HarishChandra valid for all semisimple Lie groups (cf., Warner [1972], Vol. 11, Theorem 9.1.5.1). The functional equation in Prop. 4.12 is a special case of Lemma 6.2 in Helgason [1973a]; the proof of the Paley-Wiener-type theorem [Theorem 4.2, part (ii) and its generalization] also comes from this paper. An alternative proof was given by Torasso [1977]. Theorem 4.24 on the exponentially bounded eigenfunctions is due to Lewis [1978], as well as its extension to symmetric spaces of rank one. Lewis proved also for symmetric ) 8@) for the Poisson spaces of any rank a generalization of the inclusion P A ( 9 ’ ( B ) c transform P A :T + J. The converse inclusion was proved by Oshima-Sekiguchi [I9801 (for 1 outside some hyperplanes). The Schwarz limit theorem was extended by KarpeleviE [1965]; see also Lowdenslager [I9581 and Moore [I9641 for the case of bounded symmetric domains. Furstenberg’s basic paper [I9631 (as well as KarpeleviE [1965]) gives a Poisson integral representation of bounded harmonic functions on a noncompact symmetric space X . In a Mathematical Reoiews article on Furstenberg’s paper the author posed the problem of generalizing Fatou’s theorem (Theorem 4.26) to the symmetric space X in the following form: Let u be a bounded solution of Laplace’s equation Lu = 0 on X and let p E X be any point. Then for almost all geodesics y starting at p , lim u(y(t))

exists.

1-m

This was proved by Helgason-Korhyi [1968]. Many variations of this result have been proved, relaxing the boundedness condition and allowing more general convergence to infinity, see Koranyi [1970, 19793, for a survey. Some of the material of this chapter was published earlier in Helgason [1981] and some in lecture notes in DeMichele and Ricci [1982].

CHAPTER I

INTEGRAL GEOMETRY AND RADON TRANSFORMS This chapter deals with various features of group-invariant integration. In $1 it is described how an n-form on an n-dimensional manifold M gives rise to a measure on M. If M is a Riemannian manifold, orientable or not, this method gives a canonical measure on M . If M is a Lie group, or more generally a homogeneous space (satisfying a mild assumption), the process leads to an essentially unique invariant measure. In 55 we prove various integral theorems related to the Cartan. Iwasawa. and Bruhat decompositions of a semisimple Lie group. It was proved by Funk [I9161 and Radon [1917]. respectively. that a symmetric function on S’ can be determined by its great circle integrals and that a function on R2 can be determined by its line integrals. In 52 we give a self-contained elementary exposition of the Radon transform on R“. The emphasis is on inversion formulas and correspondence theorems for naturally defined function spaces. We also indicate applications to partial differential equations and to tomography. The theory of the Radon transform on R” suggest the general problem of determining a function on a manifold by means of its integrals over certain submanifolds. The diversity of results known for such problems suggests that a general theory would not go very far. However, the quoted examples S2 and R2 suggest the following viewpoint for generalizations. The set of points on S2and the set of great circles on S2 are both homogeneous spaces of the orthogonal group O(3). Similarly the set of points in R2 and the set of lines in R2 are both homogeneous spaces of the group M(2) of rigid motions of R2. This motivates our general Radon transform definition, which forms the framework of 53. Given two homogeneous spaces G / K and G / H of the same group G,the Radon transform ,/+fassociates to a function . / on G / K a function f on G / H in a canonical fashion. The theory of this transform has in certain cases significant applications to differential equations, group representations, tomography, and differential geometry. Another general problem indicated in $3 is the orbital integral problem for a homogeneous space G/H (H noncompact), special cases of which have played a long-term role in representation theory. In $6 we give a solution to this problem in the simplest cases O(I, n + l)/O(l, n), O(2, n)/O(l, n), and (L x L)/AL, where L = SL(2, R) and AL denotes the diagonal in L x L.

80

911

81

1. Integration on Manifolds

51. Integration on Manifolds 1. Integration of Forms. Riemannian Measure

Let S be a locally compact Hausdorff space. The set of real-valued continuous functions on S will be denoted by C(S), and C,(S) shall denote the set of functions in C(S) of compact support. A measure on S is by definition a linear mapping p: C,(S) + R with the property that for each compact subset K c S there exists a constant MK such that

I f ( x )I

1p ( f )I 5 XES

for all f~ C,(S) whose support is contained in K . We recall that a linear mapping p : C,(S) 6 R which satisfies p ( f ) 2 0 for f 2 0, f~ C,(S), is a measure on S. Such a measure is called a positive measure. For a manifold M we put d(M) = Cm(M), 9 ( M ) = d ( M ) n C,(M). Suppose M is,an orientable rn-dimensional manifold and let (U,,q5u)ueAbe a collection of local charts on M by which M is oriented (cf., [DS], Chapter VIII, $2). Let w be an rn-form on M . We shall define the integral syfo for each f e C , ( M ) . First we assume that f has compact support contained in a coordinate neighborhood Uuand let =

(x1(4),* . * xm(q)),

4E

9

uu

*

On U , , w has an “expression” (cf., [DS], Chapter I, 92, No. 4) WU, -

(1)

F,(xl, . . . ,x,) d x ,

.. A

A

dx,,

and we set

S,fa =

1

( f 0 4 , ’ ) ( ~ 1 .,,.x., ) F , ( x I , . . . , x r n ) d x l * . * d x , *

4dUd

On using the transformation formula for multiple integrals we see that if f has compact support inside the intersection U, n U, of two coordinate neighborhoods, then the right-hand side in the formula above is

1

( f o

4 i 1 ~ 1 ., .,. ym)F,(yl, .. ym) dy1.S *dym, . ?

&3VJ#)

if F , d y , A . . . A d y , is the expression for o on U,. Thus jy fo is welldefined. Next, let f be an arbitrary function in C,(M). Then f vanishes outside a paracompact open submanifold of M , and by Theorem 1.3, Chapter I. in [DS] f can be expressed as a finite sum f = A;..where

xi

82

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

each ji has compact support inside some neighborhood U , from our covering. We put

Here it has to be verified (Chevalley [1946], p. 163) that the right-hand side is independent of the chosen decomposition f = Cifi of f. Let f = C j g j be another such decomposition and select 4eC.M) such that 4 = 1 on the union of the supports of allfi and g j . Then r j = C 4, (finite sum), where each 4, has support inside a coordinate neighborhood from our covering. We have

and since each summand has support inside a fixed coordinate neighborhood,

For the same reason the formulas

a

a

imply that

from which we derive the desired relation

The integral J fa is now well-defined and the mapping

f-

J fa,

.fEC,(M),

M

is a measure on M . We have obviously Lemma 1.I. If

JM .fw = 0 for

all ,f E c,(M), then w = 0.

Definition. The rn-form w is said to be positive if for each a E A , the function Fa in (1) is >O on da(Va).

011

1. Integration on Manifolds

83

If w is a positive m-form on M , then it follows readily from Theorem 1.3, Chapter I, in [DS] that j M f w2 0 for each nonnegative function , f C,(M). ~ Thus, a positive m-form gives rise to a positive measure. Suppose M and N are two oriented manifolds and let @ be a diffeomorphism of M onto N . We assume that @ is orientation preserving, that is, if the collection of local charts (U,, & ) u E A defines the orientation on M , then the collection (@(U,), 4 , o of local charts on N defines the orientation on N . Let m denote the dimension of M and N . Let w be an m-form on N and @*w its rransform or pullback by @ ([DS], p. 25). Then the formula (2)

JMj’@*W

=

Jif 0

W1)w

holds for all , f € C , ( M ) . In fact, it suffices to verify (2) in the case when j’ has compact support inside a coordinate neighborhood U,. If we evaluate the left-hand side of (2) by means of the coordinate system 6, and the right-hand side of (2) by means of the coordinate system 4, @ - I , both sides of (2) reduce to the same integral. If M is a pseudo-Riemannian manifold, orientable or not, a measure can be defined on M as follows. Consider a local chart (U,, 6,) on M and, as before, let d,(q) = ( x l , . . . , x m ) , q E U,. Let g denote the pseudo-Riemannian structure and generalizing the definition in [DS], Chapter VIII, 52, put 0

For each function f in C,(U,) put ~ ( f= )

J

(f 4; ‘)(x1,* . 0

*

9

xm)&

dx1 dx2

* *

dxm.

$.(U.)

The expression on the right is invariant under coordinate changes. Using partition of unity as before, p(f) can be defined for all f~ C,(M). The result is a positive measure on M , which we shall refer to as the Riemunnian measure on M . We sometimes denote it by fidx,

... dx,.

Example. Let H 2 denote the hyperbolic plane, that is, the unit disk x2 y2 < 1 with the Riemannian structure

+

dx2 + dv2

84

[Ch. I

INTEGRAL GEOMETRY A N D RADON TRANSFORMS

(cf., [DS], Exercise G, Chapter I). The Riemannian measure is given by (1 - x 2 - y 2 ) - 2 dxdy.

In the geodesic polar coordinates (r, 8) (where x y = tanh r sin O), it is given by

=

tanh r cos 8,

sinh(2r) dr do. For an oriented Riemannian manifold we shall now describe the Riemannian measure a little differently. Lemma 1.2. Let M be a Riemannian manifold, oriented by means of a collection (U,, & ) u E A of local charts. Let &(q) = ( x l , . . . ,x,) for q E U,, and let J be dejned as above. There exists a unique m-form w on M such that

f-j

M

fa

is the Riemannian measure. On U,, w is given by

(3)

o=

,/; d x , A

-

a

A

dx,

Proof. We can define an m-form w on M by

w,(Xl,

(4)

. . . , X,)

=

l/m!

if qE U, and ( X i ) is any orthonormal basis of M , for which the matrix (aij) given by (d/dxj), = a i j X i has positive determinant. Clearly aq does not depend on the choice of ( X i ) . We have g i j = & akiakj,so that by the positivity, det(aij) = Moreover,

ci

&.

1

= - det(aij) = &dx,

m!

A

This w defined by (4) satisfies (3). Also (2) implies for f~ C,(Uu)

S,.b=

1

(14; 0

')(XI,

* * * 7

dx1* * dxm,

&dud

so that w gives the Riemannian measure. The uniqueness being obvious, the lemma is proved. Proposition 1.3. Let M and N be Riemannian manifolds and @ a difleomorphism of M onto N . For p E M let ldet dap1denote the absolute

411

85

1. Integration on Manifolds

value of the determinant of the linear isomorphism d o p :M , + N,(,, when expressed in terms of any orthonormal bases. Then for F E 9 ( N )

JJh)

(5)

dq = JMF(@@))ldetd@,,IdP,

if dp and dq denote the Riemannian measures on M and N , respectively. Proof. By partition of unity it suffices to prove ( 5 ) when F has support in a coordinate neighborhood. We may therefore assume M and N to be oriented and that @ is orientation preserving. In accordance with Lemma 1.2 let w and 6 be the m-forms on M and N , respectively, giving the Riemannian measures dp and dq. Then by (2), iff = F 0, 0

(1.@-

F ( q ) dq = JN

1)e

=

JN

J (F

cD)m*e.

M

But @*6' = det(d@) w because of (4), and now ( 5 ) follows. Remark. Proposition 1.3 shows, in particular, that the Riemannian measure is invariant under isometries.

2. Invariant Measures on Coset Spaces Let M be a manifold and @ a diffeomorphism of M onto itself. We recall that a differential form w on M is called invariant under @ if @*w = w.

Let G be a Lie group with Lie algebra 9. A differential form w on G is called left-inuariant if L,*w = w for all x E G, L, [or L(x)] denoting the left translation g --* xg on G. Also, R , [or R(x)] denotes the right translation g -,gx on G and right-invariant differential forms on G can be defined. If X E g, let 2 denote the corresponding left-invariant vector field = Sj deteron G. Let X I , . . . , X n be a basis of g. The equations ~'(2,) mine uniquely n l-forms wi on G. These are clearly left-invariant and the exterior product w = w 1 A A w" is a left-invariant n-form on G. Each 1- form on G can be written ha',where 1;. E 6 ( G ) ; it follows that each n-form can be written fw, where f E ~ ( G ) .Thus, except for a constant factor, o is the only left-invariant n-form on G. Let

cy=

4 :x

+

(x~(x),*

3

xn(x))

86

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

be a system of canonical coordinates with respect to the basis X I , . . . , X, of g, valid on a connected open neighborhood U of e (cf. [DS], Chapter 11, $1). On U , w has an expression

q,= F ( x , , . . . ,x,) d x ,

A

A

dx,

and F > 0. Now, if g E G , the pair ( L , U , 4 0 L,- I) is a local chart on a connected neighborhood of g . We put (4 L,- I ) ( x )= ( y l ( x ) ,. . . ,y,(x)) ( x E L, U ) . Since y,(gx) = x i ( x ) ( x E U n L, U ) , the mapping 0

L , : u + L,U has coordinate expression ([DS] Chapter I, $3, No. 1) given by (Yl,.

. ., y,)

= (x,,.

.

*

9

x,).

On L,U, w has an expression wL,u =

G(y1,. . ., y,) d y ,

A

*.

*

A

dy,,

so that the invariance condition w, = L:w,, ( x E U n L, U ) can be writ ten G ( y , ( x ) ,. . . * JJ,(X))(dYl A * * . A dy,), = G ( x , ( x ) ,. . . , x , ( x ) ) ( d x , A . . . A dx,)*. Hence F(x,(x), . . . , x , ( x ) ) = G ( x , ( x ) ,. . . ,x,(x)) and

4 - l is for x E U n L, U , which shows that the Jacobian of (4 L,> 0. Consequently, the collection (L, U , 4 0 I,,- I ) g o G of local charts turns G into an oriented manifold and each left translation is orientation preserving. The orientation of G depends on the choice of basis of g. If X i , . . . , X i is another basis, then the resulting orientation of G is the same as that before if and only if the linear transformation 0

X i + Xi

0

(1 5 i 5 n )

has positive determinant. The form w is a positive left-invariant n-form on G and except for a constant positive factor, w is uniquely determined by these properties. We shall denote it by d , g . The linear mapping of C,(G) into R given by f + f d , g is a measure on G , which we denote by pl. This measure is positive; moreover, it is left-inoariant in the sense that pl(.fo L,) = p , ( f ) for x E G, f~ C,(G). Similarly, G can be turned into an oriented manifold such that each R , (g E G ) is orientation preserving. There exists a right-invariant posi-

611

87

1. Integration on Manifolds

tive n-form d,g on G and this is unique except for a constant positive factor. We define the right-invariant positive measure pr on G by pr(f> =

ssd,Q,

~EcAG).

The group G has been oriented in two ways. The left-invariant orientation is invariant under all right translations R , (x E G ) if and only if it is invariant under all I(x) = L, 0 R,- (x E G). Since the differential dZ(x), satisfies dI(x), = dL,,,- 1 0 Ad(x) 0 dL,- 1 , the necessary and sufficient condition is det Ad(x) condition is always fulfilled if G is connected.

=- 0 for all x E G. This

Lemma 1.4. With the notation above we have d,g = c det Ad(g) d l g ,

where c is a constant. Proof. Let 0 = det Ad(g) d,g and let x E G . Then

(R,- ,)*0 = det Ad(gx- ')(R,- I)* d,g = det Ad(gx- ')l(x)* d,g. At the point g = e we have (I(x)*(d,g))e

=

det Ad(x)(d,g)e.

Consequent 1y,

(R,-,*0),

= det

Ad(e)(d,g),

=

0,.

Thus, 0 is right-invariant and therefore proportional to d,g. Remark. If G is connected it can be oriented in such a way that all left and right translations are orientation preserving. If d,g and dlg are defined by means of this orientation, Lemma 1.4 holds with c > 0.

Corollary 1.5. Let x, y E G and put d,(ygx) = (L,R,)* d,g, d,(xyg) = (L,R,)* d,g. Moreover, if J denotes the mapping g + g-', put d,(g-') = J*(dlg). Then d,(gx) = det Ad(x-') d,(g),

d,(xg) = det Ad(x) d,g,

d,(g-') = ( - l)dimG det Ad(g) d,g.

In fact, the lemma implies that c det Ad(g) d,g = d,g = d,(gx) = c det Ad(gx) d,(gx),

d,(xg) = c det Ad(xg) d,(xg) = c det Ad(xg) d,g.

88

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

Finally, since JR, = L,- J, we have (R,)* d,(g-') = (R,)*J* dig = (JR,)* d1g = ( L , - I J ) * dig = J* dig,

so that d,(g- ') is right-invariant, hence proportional to d r g . But obviously (di(g-'))e = (- 1IdimG(dlg)e,

so that the corollary is verified. Definition. A Lie group G is called unimodular if the left invariant measure pl is also right-invariant. In view of Corollary 1.5 we have by (2) PI(f

(6)

O

R,) = I det Ad(x) I Pl(f).

It follows that G is unimodular if and only if ldet Ad(x)I = 1 for all x E G. If this condition is satisfied, the measures pi and pr coincide except for a constant factor. Proposition 1.6. T h e following Lie groups are unimodular: (i) Lie groups G f o r which Ad(G) is compact; (ii) semisimple Lie groups; (iii) connected nilpotent Lie groups. Proof. In the case (i), the group { (det Ad(x)I : x E G} is a compact subgroup of the multiplicative group of positive real numbers. This subgroup necessarily consists of one element, so that G is unimodular. In the case (ii), each Ad(x) leaves invariant a nondegenerate bilinear form (namely, the Killing form). It follows that (det Ad(x))2 = 1. Finally, let N be a connected nilpotent Lie group with Lie algebra n. If X en, then ad X is nilpotent, so that Tr(ad X) = 0. Since

det eA = r T r A for an arbitrary linear transformation A, we obtain det Ad(exp X ) = eTr(Ad = 1. This proves (iii). Notation. In the sequel we shall mostly use the left invariant measure p,. The measure dg is usually called H a m measure on G. For simplicity we shall write p instead of pI and dg instead of d i g .

Let G be a Lie group with Lie algebra g; let H be a closed subgroup with Lie algebra c g. Each x E G gives rise to an analytic diffeomorphism ~ ( x ) : g H-P xgH of G / H onto itself. Let II denote the natural mapping of G onto G/H and put o = n(e). If h E H , (dt(h)), is an endomorphism of the tangent space (GIH),. For simplicity, we shall write dt(h) instead of (dz(h)), and dn instead of (dn),.

811

1. Integration on Manifolds

89

Lemma 1.7.

det(dt(h)) =

det Ad, (h) det AdH (h)

(h E H).

Proof. It was shown in [DS], Chapter 11, $4, that the differential dn is a linear mapping of g onto (GIH), and has kernel lj. Let m be any subspace of g such that g = t, + m (direct sum). Then dn induces an isomorphism of m onto (GIH),. Let X E m. Then

Ad,(h) X = dRhSince

7c 0

R , = II, (h E H ) and

(7)

II

0

L,

=

0

dLh(X).

z(g) n, (g E G), we obtain 0

h

dn 0 AdG(h)X = dz(h) 0 dn(X),

E

H, X

The vector AdG@)X decomposes according t o g = t, Ad,(h) X = X(h)h The endomorphism A,,: X

+ X(h),

E

m.

+ m,

+ X(h),.

of m satisfies

dn 0 Ah(X) = dz(h) dn(X),

X E m,

0

so that det A,, = det(dz(h)). On the other hand, exp AdG(h)tT = h exp t T h-

=

exp AdH(h) t T

for t E R, T E 6. Hence Ad,(h) T = AdH(h) T so that det Ad,(h) = det A),det Ad,(h), and the lemma is proved. Proposition 1.8.

Let rn = dim G / H . The ,following conditions are

equivalent: (i) GIH has a nonzero G-invariant rn-form w ; (ii) det Ad,(h) = det AdH(h) for h E H .

If these conditions are satisfied, then GIH has a G-invariant orientation and the G-invariant m-forrn w is unique up to a constant factor. Proof. Let w be a G-invariant rn-form on GIH, w # 0. Then the relation t(h)*w = w at the point o implies det(dz(h)) = 1, so (ii) holds. On the other hand, let X l , . . . , X , be a basis of (GIH), and let wl,. . . ,w"' be the linear functions on (GIH), determined by w i ( X j )= 6 , . Consider the element w1 A - . . A om in the Grassmann algebra of the tangent space (GIH),. Condition (ii) implies that det(dz(h)) = 1 and the element w' A A w"' is invariant under the linear transformation dz(h). It follows that there exists a unique G-invariant rn-form w on GIH such that w, = w1 A ... A a". If w* is another G-invariant rn-form on

90

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

G / H , then w* = ju, where f~ b ( G / H ) . Owing to the G-invariance, f = constant. Assuming (i), let 4: p ( x l ( p ) ,. . . ,x,(p)) be a system of coordinates on an open connected neighborhood U of o E G / H on which w has an expression wu = F ( x , , . . . ,x,) d x l

. . . A dx,,

A

with F > 0. The pair (r(g)U, 4 o z(g-')) is a local chart on a connected neighborhood of g o E G / H . We put (4 0 r(g- '))(p) = (yl(p), . . . ,y,(p)) for p E z(g)U. Then the mapping z ( g ) : U + z(g)U has expression ([DS], Chapter I, $3.1) (yl,. . . ,y,) = ( x i , .. . ,x,). On s(g)U, w has an expression w,(g)u = G(Y1,

*

'

* 9

Y,) dY 1

A

A

* * *

dy,

and since wq = z(g)*w,(,,, we have for q E U n z(g)U wq

= G(Yl(d, * * = G(xl(q),

Hence.F(x (q),. . . ,x,(q))

* 9

Ym(q))(dYl A

. . . ,x,(q))(dxl A = G(x . . . ,x,(q))

'

-

*

A

A

dy,), dx,),.

and

which shows that the Jacobian of the mapping (4 z(g-')) 0 4-l is >O. Consequently, the collection (z(g)U, 4 0 z ( g - l ) ) g E !of local charts turns G/H into an oriented manifold and each z(g) is orientation preserving. The G-invariant form w now gives rise to an integral l f w which is invariant in the sense that 0

JG,Hfw =

J

(f

O

W ) W

9

E

G.

GIH

However, just as the Riemannian measure did not require orientability, an invariant measure can be constructed on G / H under a condition which is slightly more general than (ii). The projective space P 2 ( R ) will, for example, satisfy this condition but it does not satisfy (ii). We recall that a measure p on G / H is said to be invariant (or more precisely Ginvariant) if p(f ~ ( g ) = ) p(f) for all g E G. 0

Theorem 1.9. Let G be a Lie group and H a closed subgroup. The relation

911

I . Integration on Manifolds

91

is a necessary and suflcient condition for the existence of a G-invariant measure >O on GIH. This measure dg, is unique (up to a constant factor) and (9)

s,

f(9) d9 =

s,,( 1,f

(gh)dh)dg,,

.fE Cc(G),

if the left-invariant measures d g and dh are suitably normalized. Formula (9) is illustrated in Fig. 6, where n: G + G / H is the natural mapping. We begin by proving a simple lemma. Lemma 1.10. Let G be a Lie group and H a closed subgroup. Let dh be a lef-invariant measure >O on H and put f(gH) =

JH f @ h )dh,

. f C,(G). ~

Then the mapping f + f is a linear mapping of C,(G) onto C,(C/H). Proof.

Let F E C , ( G / H ) ; we have to prove that there exists a function

f E C,(G) such that F = f: Let C be a compact subset of G/H outside which F vanishes and let C' be a compact subset of G whose image is C under the natural mapping n: G -+ G / H . Let C , be a compact subset of H of positive measure and put c = C' . C,. Then n ( c ) = C. Select f l E C,(G) such that f i 2 0 on G and fi > 0 on c. Then f1 > 0 on C (since C , has positive measure) and the function

if n(g)E C if 4 9 ) 4 c

lo belongs to C,(G) and f

=

F.

G

H

e

-1 G/ff

FIG.6

92

INTEGRAL GEOMETRY A N D RADON TRANSFORMS

[Ch. I

Now in order to prove Theorem 1.9 suppose first that the relation

Idet AdG(h)I = I det Ad&) 1,

h E H,

holds. Let ~ E C , ( G ) .Since we are dealing with measures rather than differential forms, we have by Cor. 1.5

But the relation (8) and the last part of Corollary 1.5 shows that

I det AdG(h) I dh =

JH4(ghso that

JG9(9)dg S H f @ h )dh Taking

=

JG

1"

&gh) dh,

f(s)ds JG4(i7h) dh.

4 such that &gh) dh = 1 on the support of f, we conclude that

In view of the lemma we can therefore define a linear mapping p : C,(G/H) + R by

P(F)

=

J f (9)dg

if F = f

G

Since p(F) 2 0 if F L 0, p is a positive measure on G / H ; moreover, ~ ( ( f > " " ' )=

s

G

fL'"'(g) dg

=

JGf (9)ds

=

df).

so that p is invariant. In order to prove the converse we shall make use of the theorem that the group G has (up to a constant factor) a unique positive left invariant measure. For this "uniqueness of Haar measure" see e.g., Weil's book [19401. If p is a positive invariant measure on G / H , the mapping f- p ( f ) is a positive left invariant measure on G. Owing to the uniqueness mentioned,

811

1, Integration on Manifolds

93

In view of the lemma this proves the uniqueness of p as well as (9). In order to derive (8), replace f ( 9 ) by f (gh,) in (9). Owing to Corollary 1.5 the left-hand side is multiplied by ldet Ad,(h,)I and the right-hand side is multiplied by Jdet AdH(h,)J.This finishes the proof of Theorem 1.9. Remark. If H is compact, condition (8) is satisfied; hence in this case GIH has a G-invariant measure.

In the remainder of this section we shall often have to calculate how invariant measures transform under mappings. To a certain extent these calculations can be reduced to the following general lemma. Lemma 1.11. Let G and S be Lie groups and H c G and T c S closed subgroups. Suppose that the coset spaces GIH and SIT have the same dimension m and that they possess positive invariant diflerential mforms, denoted dg, and dS,. Let o = { H } , 0’ = { T } . Let b, be a diflerentiable mapping of’GIH into SIT such that b, - { H } = { T } . Then $*(dST) = DdgH,

where D is a function on GIH computed as follows: Let X , , . . . , X , and Y,,. . . , Y, be fixed (but arbitrary) bases of the tangent spaces (GIH), and (S/T),., respectively, such that dgH(X1.. . . , x,) = dS,( Let g

E

r1..

.

. . y,).

G and select s E $(gH). Consider the linear mapping

A(g) = dz(s- ‘1

d$gH

of (GIH), into (SIT),. and put A ( g ) X j =

dt(g)o

xi aij(g)q. Then

D(gH) = det(ai&))* Proof. Let d,. . . ,urnbe the linear functions on (GIH),. determined by o i ( X j ) = dij; let e l , . . . ,dm be the linear functions on (SIT),. deter= d i j . Then the dual mapping mined by @(?)

‘(A(g))= r(g)* $* ?(s0

0

I)*,

which maps the dual of (S/T),. into the dual of (GIH), satisfies

‘(A(g))O’= C aij(g)wj. j

Now, due to the assumption about dg, and dS,,

(dgH),H = C t ( g -

‘)*(01 A

(ds,),, = C 7 ( S - ’)*(@

A

* *

A Om),

... A

em),

94

INTEGRAL GEOMETRY A N D RADON TRANSFORMS

[Ch. I

where c is a constant. Consequently, (+*(dST))nH= C+* 0 T(s-')*(8' A * A em) = c.r(g-')*'A(G)(B' A . . . A Om) = c det(aij(g))t(g- ' ) * ( w ' A . A urn) = det(aij(g))(dgH)gH*

Next we calculate the behavior of the Haar measure under the decomposition of a group into a product. Proposition 1.12. Let U be a Lie group with Lie algebru i t . Suppose u is a direct sum u = m + $, where m and t, are subalgebras o j u (not necessarily ideals). Let M and H denote the analytic subgroups o j U with Lie algebras m and t,, respectively. Suppose the mapping a : (m, h) + mh is a one-to-one mapping of M x H onto U . Then the positive lef-invariant measures dh, dm, du, can be normalized in such a way that

for all f E C,( U ) . Proof. From Lemma 5.2, Chapter VI [DS] we know that a is a diffeomorphism and

dac,,,,,(dLmY, dL,,Z) = dL,,,(Ad,(h-')Y

+ Z)

, Z E t,. Using Lemma 1.1 1 we see that for m E M ,h E H, Y ~ m and a*(du) = D(m, h) dm dh, where dmdh is the invariant measure on the product group M x H and D(m, h) is the determinant of the linear mapping

A(m, h): (Y, Z ) + Ad,(h-')Y

+Z

of m x t, into g. [Since we identify m x t, with g we can indeed regard A(m, h) as an endomorphism of g and thus can speak of its determinant.] Since Adu(h-')Y

+ Z = Ad,(h-')(Y + AdH(h)Z),

it is clear that

det A(m, h) = and the lemma follows from (2).

det AdH(h) det Ad,@)

611

95

1. Integration on Manifolds

We conclude this section with a useful “chain rule” version of Theorem 1.9. Proposition 1.13. Let G be a Lie group, H and N closed subgroups such that H c N c G. Assume that GIH and GIN have positive G-invariant measures &H and dgN. Then NIH has an N-invariant positive measure dn, which (suitably normalized) satisjies

(10)

I,,,

f ( g H )dgH =

I (IN,,

f ( g n H ) d n H ) dgN

GIN

for all f E C,(G/H). Moreover, if for some f E C ( G / H ) one side is absolutely convergent, then so is the other and both sides are equal. Proof.

By Theorem 1.9, the assumptions mean that ldet Ad,(h)I = ldet Ad,(h)l,

h E H,

ldet Ad,(n)( = (det AdN(n)(,

n E N.

Hence ldet AdN(h)I = ldet Ad,(h)I, so that dn, exists. Let njq: G

-,G/H,

EN:

be the natural maps. Let f E C,(G/H), g NIH = G/H

-

E

G + GIN

G, no E N . Since

nH(G - N ) ,

the subset NIH c GIH is closed, so that the function nH + f ( g n H ) on N / H has compact support. The integral

IN,,

f (gnH) dnH

is thus well-defined, and since it is invariant under g garded as a function F on GIN,

+ gn,,

it can be re-

which is continuous. To see that F has compact support let C c GIH be a compact set outside of which ,fvanishes. Select a compact set c‘ c G such that n,(C‘) = C. The integral (1 1) vanishes if 9 - l . C n N I H = 0 ; thus it vanishes if CH n gN = 0. Hence F vanishes outside the compact set nN(C‘).Thus we can form

l(f)=

GIN

(1

N/H

f ( g n H ) dnH)

dgN*

96

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

The functional f + Z(f) is a positive measure on G / H which is clearly G-invariant. Also, Z(f) > 0 i f f 2 0 and > O near the origin in G / H . Thus Z(,f) is a positive multiple of dg,, so (10) follows, with a suitable normalization of dn,. By standard measure theory, (10) extends to the case when ,f is the characteristic function of a Bore1 set; approximating by an increasing sequence of linear combinations of such functions, we see that (10) holds for all positive f e C ( G / H ) (possibly with both sides + a). Decomposing f E C ( G / H ) into two such functions, the proposition follows.

3. Haar Measure in Canonical Coordinates Let G be a Lie group with Lie algebra 9. Select neighborhoods N o of 0 in g and N , of e in G such that the exponential mapping exp: g + G gives a diffeomorphism of N o onto N , . Fix a Euclidean measure d X on g and let dg denote the left-invariant form on G such that (dg), = d X . Theorem 1.14. W i t h dg and dX as above we have for the pullback by exp

(exp)*(dg) = det

(',,'J; d X" ." )

Zf f E C ( G ) has compact support contained in the canonical coordinate

neighborhood N , , then

Proof. Since dg is left-invariant, formula (12) is an immediate consequence of Theorem 1.7, Chapter I1 in [DS]. Then (13) follows from (2) in $1 used on the function f 0 exp. $2. The Radon Transform on R"

1. Introduction It was proved by J. Radon in 1917 that a differentiable function on R3 can be determined explicitly by means of its integrals over the planes in R3. Let J(w, p ) denote the integral of f over the hyperplane (x, co) = p, co denoting a unit vector and ( , ) the inner product. Then

921

2. The Radon Transform on R"

97

where L is the Laplacian on R3 and dw the area element on the sphere S2 (cf. Theorem 2.13). We observe that the formula above contains two integrations dual to each other: first one integrates over the set of points in a hyperplane, then one integrates over the set of hyperplanes passing through a given point. This suggests considering the transforms f 43 4 + defined below. The formula has another interesting feature. For a fixed w the integrand x + J(w, (w, x)) is a plane wave, that is, a function constant on each plane perpendicular to w. Ignoring the Laplacian, the formula gives a continuous decomposition of f into plane waves. Since a plane wave amounts to a function of just one variable (along the normal to the planes), this decomposition can sometimes reduce a problem for R3 to a similar problem for R. This principle has been particularly useful in the theory of partial differential equations. The analog of the formula above for line integrals is of importance in radiography, where the objective is the description of a density function by means of certain line integrals. In this section we discuss relationships between a function on R" and its integrals over k-dimensional planes in R". The case k = n - 1 will be the one of primary interest. At some stages we use some elementary facts concerning distributions and Riesz potentials. Since this material will be familiar to many readers it is placed in an appendix to this section.

6

2. The Radon Transform of the Spaces 9 ( R n )and Y ( R n ) . The Support Theorem

Let f be a function on R",integrable on each hyperplane in R". Let P" denote the space of all hyperplanes in R", P" being furnished with the obvious topology. The Radon transform off is defined as the function 3 on P" given by

where dm is the Euclidean measure on the hyperplane 5. Along with the transformation f + 3 we consider also the dual tran?form 4 + d, which to a continuous function 4 on P" associates the function on R" given by

6

98

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

where d p is the measure on the compact set {< E P":x E 5 ) which is invariant under the group of rotations around x and for which the measure of the whole set is 1. We shall relate certain function spaces on R" and on P" by means of the transforms f + 3, 4 + later we obtain explicit inversion formulas. Each hyperplane E P" can be written = { x E R": ( x , w ) = p } , where ( , ) is the usual inner product, w = (ol, . . . ,w,) a unit vector, and P E R . Note that the pairs ( w , p ) and (-u,- p ) give the same 5 ; the mapping ( q p ) + 5 is a double covering of S"-' x R onto P". Thus P" has a canonical manifold structure with respect to which this covering map is differentiable and regular. We thus identify continuous (differentiable) functions 4 on P" with continuous (differentiable) functions 9 on S"- x R satisfying 4(w, p ) = 4 ( - w , - p ) . Writing f ( w , p ) instead of f(t;) and f, for the translated function x + f ( t + x), we have

4;

<

X
s

<

f(x

(x. 0 )= p

so (1)

s

S ( Y ) dm(y),

(Y. w) = P + ( I , w)

b, P) = 3 b P + ( 4

Taking limits, we see that if (2)

+ t ) dm(x) =

0)).

ai = d / a x i ,

a3 (0, P). (aif)*(w, PI = wi aP

xi

Let L denote the Laplacian 8: on R" and let 0 denote the operator 4(w, p ) + (a2/ap2)4(o,p), which is a well-defined operator on 8(P"). By analogy with the Laplacian L (cf. Chapter 11, Corollary 4.11) it can be shown (cf. Exercise C2 in Chapter 11) that if M(n) is the group of isometries of R", then 0 generates the algebra of M(n)-invariant differential operators on P". Lemma 2.1. The transforms f

3, qj -,d; intertwine L and 0,i.e.,

w-

= 0(3), (04)" = Ld;. Proof. The first relation follows from (2) by iteration. For the second

we just note that for a constant c

(3)

d ( x > = c Js"-,o(o, ( x , w ) ) dw,

where d o is the usual measure on S"-'. The Radon transform is closely connected with the Fourier transform / ( u > = sRnf(x)e-i(x*")dx,

u

E

R".

2. The Radon Transform on R"

021

99

In fact, if s E R, w a unit vector, then

so f ( s w ) = /-rnrnf(w, r)e-'" dr.

(4)

This means that the n-dimensional Fourier transform is the l-dimensional Fourier transform of the Radon transform. From (4),or directly, it follows that the Radon transform of the convolution

f(x) =

JRm - Y)f,(Y) dY

is the convolution (5)

f h P ) = JRh P - 4$hA

4 ) d4.

Although slightly greater generality is possible, we shall work with the space Y ( R " ) of complex-valued rapidly decreasing functions on R". We recall that f E Y ( R " ) if and only if for each polynomial P and each integer m 2 0,

(7)

SUP W E Sn - I , r e R

1(1

d'

+ lrlk)j-+(D4)(a

r ) < m.

100

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

Lemma 2.3. For each f E Y ( R " ) the Radon transform f ( w , p ) satis$es the following condition: For k E 2' the integral

can be written as a kth-degree homogeneous polynomial in wl, . . . , w,. Proof, This is immediate from the relation

(8)

1PCQ

PIPk dP =

R

s.

Pk dP

Lp

dx.

f ( x ) dm(x) = J-J(x)(x.

In accordance with this lemma we define the space

{

For each k E 2'.

4 E y(pn):

YH(P") =

L+(

w. p)pk dp is a

kth-degree homogeneous polynomial in wl.. . . , w,

With the notation 9 ( P " ) = C:(P") we write

9H(Pn) = YH(P") n 9 ( P " ) . According to Schwartz [1966] (p. 249), the Fourier transform f + f maps the space Y ( R " ) onto itself. We shall now, by analogy with the Schwartz theorem, determine the image of 9 ( R n )under the Radon transform. Theorem 2.4. The Radon transform f mapping o f Y ( R " ) onto ,4pH(P").

-+

f is a linear one-to-one

Proof. Since

it is clear from (4) that for each fixed w the function r -+ f ( w , r ) lies in Y ( R ) . For each wo E S"-' a subset of {wl, . . . , on}will serve as local coordinates on a neighborhood of wo in S"-'. To see that ~ E Y ( P " ) , it therefore suffices to verify (7) for C$ = f on an open subset N c S"-', where w, is bounded away from 0 and w l r. . . , w,- serve as coordinates, in terms of which D is expressed. Since

(9) u1 we have

= sw,,

a

...,

u,-1 = s w n- l,

a!

-( f ( s w ) )= s - -

awi

aui

SOi(1

- w:

U" =

-

* *

s(1

2 )1/2 - 0:- * . . - 0,1

- - 0,"- l i 2 -.a!

dun

921

2. The Radon Transform on R"

sup

(10)

I(1

oeN.s~R

+s

d'

~ ~ (Df)(w, ) s)

101

< co.

and obtain d'

(1 + r2k)-j-+ P A . f ( w ,r ) ) ) =

((is)' D,(f(sw)))eisrds.

211

Now (10) shows that f E Y ( P " ) ,so that by Lemma 2.2, f E Y H ( P " ) . Because of (4) and the fact that the Fourier transform is one-to-one, it only remains to prove the surjectivity in Theorem 2.4. Let + E Y ~ ( P " ) . In order to prove = f for some f E Y ( R " ) ,we put

+

m

@(s, w ) =

J-m+(w, r)e-irsdr.

Then @(s, w ) = @( -s, -a) and @(O, w ) is a homogeneous polynomial of degree 0 in wl,, . . ,wn,and hence is constant. Thus there exists a function F on R" such that F(sw) =

s,

+(w, r)e-irsdr.

While F is clearly smooth away from the origin, we shall now prove it to be smooth at the origin too; this is where the homogeneity condition in the definition of Y H ( P " )enters decisively. Consider the coordinate neighborhood N c S"-l above, and if h E Cm(R" - {0}), let h*(w,, . . . , on-,, s) be the function obtained from h by means of the substitution (9). Then

and

102

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

Hence

In order to use this for h = F we write m

03

F(sw) = f_,@(o, r ) dr

+ s-m@(o, r)(e-irs - 1) dr.

By assumption the first integral is independent of o.Thus using (7) we have for a constant K > 0

and a similar estimate is obvious for dF(sw)/as. The formulas above therefore imply that all the derivatives aF/aui are bounded in a punctured ball 0 c IuI c E, so that (since n > 1) we can conclude that F is uniformly continuous on 0 c IuI c E, hence continuous at u = 0. More generally, we prove by induction that

where the A have the form (12) k l , ...,k r ( O , s, = u j . k l , ...,kt(')sj-q* For q = 1 this is in fact proved above. Assuming (11) for 4, we calculate aq+'h

auil . . . a t + +,'

using the above formulas for a/aui. If A j , k l , . , , , k , ( o , s) is differentiated with respect to uiq+ we get a formula like (12) with q replaced by 4 + 1. If, on the other hand, the (i + j)th derivative of h* in (1 1) is differentiated with respect to uiq+],we get a combination of terms s-

a ' + ] +lh*

a'+]+lh*

amkl . . . amkr+ asj'

awkl. . . aoki asj+ 1

and in both cases we get coefficients satisfying (12) with 4 replaced by + 1. This proves (11) in general. Now

q

§21

2. The Radon Transform on R"

103

where t4

e,(t) = q!

t4+ 1

++. (4 + l)!

Our assumption on 4 implies that the first integral in (13) is a polynomial in ul,. . . , u, of degree sq - 1 and is therefore annihilated by the differential operator (1 1). If 0 I j I q, we have (14)

a]

(e,( - irs)) = I ( - ir)4(- irsy'-4e,-

j(

- irs) 1 I K j r 4 ,

where K j is a constant because the function t -,(it)-Pep(it)is obviously bounded on R (p 2 0). Since 4 E Y(P"), it follows from (1 l), (13), and (14) that each qth-order derivative of F with respect to u l , ..., u, is bounded in a punctured ball 0 < IuI < E. Thus we have proved F E Cm(Rn).That F is rapidly decreasing is now clear from (7) and (11). Finally, i f f is the function in Y ( R " ) whose Fourier transform is F , then f ( s w ) = F(so) =

hence by (4),

laa

d(w, r)e-i's d r ;

f = 4 and the theorem is proved.

To make further progress we introduce some useful notation. Let S,(x) denote the sphere {y: J y - X I = r } in R" and A(r) its area. Let B,(x) denote the open ball {y : ly - X I < r } . For a continuous function f on S,(x) let ( M ' f ) ( x )denote the mean value I

P

where d w is the Euclideap measure. Let K denote the orthogonal group O(n), dk its Haar measure, normalized by JK dk = 1. If y E R", r = J yI, then

In fact, for x, y fixed, both sides represent rotation-invariant measures on Sr(x), having the same value for the function f = 1. The rotations being transitive on S,(x), (15) follows from the uniqueness in Theorem 1.9. Formula (3) can similarly be written

104

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

if to is some fixed hyperplane through the origin. We see then that iff E Y(R"), flk the area of the unit sphere in Rk,

so that (17)

6.

We consider now the analog of Theorem 2.4 for the transform 4 -+ But ~ $ E Y ~ ( Pdoes " ) not imply $ E Y ( R " ) . [If this were so and by Theorem 2.4 we wrote 4 = 3, f € Y ( R " ) , then the inversion formula in Theorem 2.13 for n = 3 would imply J f ( x ) dx = 0.1 On a smaller space we shall obtain a more satisfactory result. Let Y * ( R " ) denote the space of all functions f E Y ( R " ) which are orthogonal to all polynomials, i.e., f(x)P(x)dx = 0

for all polynomials P.

JR.

Similarly, let Y * ( P " )c Y ( P )be characterized by

JR4(o,r)p(r) dr

=0

for all polynomials p .

Note that under the Fourier transform the space Y * ( R " ) corresponds to the subspace Y o ( R " )c Y(R")of functions all of whose derivatives vanish at 0.

Corollary 2.5. The transjorms f -+ f, 4 -+ Y*(R") onto Y*(P") and ofY*(P") onto Y*(R").

6

are bijections of

The first statement is clear from (8) if we take into account the elementary fact that the polynomials x 3 ( x , w ) ~span the space of homo8eneous polynomials of degree k (cf., Exercise Cl). To see that 4 + d is a bijection of Y * ( P " ) onto 9'*(R") we use (17), knowing that 4 = f for some f EY *( R") . The right-hand side of (17) is the convolution o f f with the tempered distribution 1xI-l whose Fourier transform is by Lemma 2.41 a constant multiple of IuI1-". (Here we leave out the trivial case n = 1.) By the general theory of tempered distributions [cf. (69) later], this convolution is a tempered distribution whose Fourier transform is a constant multiple of lul'-"f(u). But this lies in the space Yo@") since 3 does. Now (17) implies that = ( ~ ) " E Y * ( R "and ) that

6

021

4$ 0

2. The Radon Transform on R"

+- " -

105

4

if 4 0. Finally, the mapping 4 + is surjective because the function ((f) ) ( u ) = c l u l ' - y ( u ) (c a constant) runs through Y o ( R " ) as f runs through Y*(R"). We now turn to the space 9 ( R " ) = C:(R") and its image under the Radon transform.

Theorem 2.6. (The Support Theorem) Let f E C(R") satisfy the following conditions:

(i) For each integer k > 0, Ixl"f(x) is bounded. (ii) There exists a constant A > 0 such that

3(0 = 0

for 4 0 , 5 ) > A ,

d denoting distance. Then f(x)=0

for

1x1 > A .

Proof, Replacing f by the convolution 4 * f , where 4 is a radial C"function with support in a small ball B,(O), we see that it suffices to prove the theorem for f E b(R"). In fact, 4 * f is smooth, it satisfies (i), and by ( 5 ) it satisfies (ii) with A replaced by A + E. Assuming the theorem for the smooth case, we deduce that support (4 * f ) c Cl(BA+e(0))so letting E + 0, we obtain support ( f ) c Cl(B,(O)). To begin with we assume that f is a radial function. Then f ( x ) = F ( l x l ) , where F E b ( R ) and even. Then 3 has the form I(<)= f ( d ( 0 , t)), where F is given by

R(P) =

J

R"-1

NP2

+ lY 12)1'2)

MY)

(P 2 019

because of the definition of the Radon transform. In particular, F being even, f extends to an even function in b(R). Using polar coordinates in Rn- 1, we obtain (18)

f(p)=

Here we substitute s = ( p z becomes

/omF((p' + tz)1/z)tn-2dt.

+ t2)-'"

We write this equation for simplicity

and then put u = p - ' . Then (18)

106

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

This integral equation is very close to Abel's integral equation (Whittaker and Watson [1927]) and can be inverted as follows. Multiplying both sides by u(t2 - u ~ ) ( ' / ~ ) (and " - ~integrating ) over 0 Iu It, we obtain Sdh(u)(t' - U ~ ) ( ~ / ~ ) ( " du -~)U

=

f{JIg(s)[(u2 -

s2)(tZ

I

ds u du

-

0

= J;g(s){[=:[(t'

I

- u2)(u2 - s ~ ) ] ( ' / ~ ) (du " - ~ds. )

+

The substitution ( t 2 - s2)u = ( t 2 s2) - 2u2 gives an explicit evaluation of the inner integral, and we obtain

where C is a constant. Here we apply the operator d/d(t2) = (1/2t)(d/dt) ( n - 1 ) times, whereby the right-hand side gives a constant multiple of t - ' g ( t ) . Hence we obtain with another constant c

By assumption (ii) we have F(u-') = 0 if u - l 2 A, that is, if u IA - ' . But then (20) implies F ( t - ' ) = 0 if t 4 A - ' , that is, if t-' 2 A. This proves the theorem for the case when f is radial. We consider next the case of a general 1: Fix x E R" and consider the function gx(Y) =

J f(x + k . y ) dk K

as in (15). Then gx satisfies (i) and

+k x

+k

*

5) dk,

5 denoting the translate of the hyperplane k . 5 by

x. The triangle

inequality shows that d(0,x

+ k . 5 ) 2 d ( 0 , t ) - 1x1,

XER", k E K .

Hence we conclude from assumption (i) and (21) that (22)

Ox(5)

=0

if d(O,5) > A

+ 1x1.

021

2. The Radon Transform on R"

107

But g x is a radial function, so that (22) implies by the first part of the proof that

(23)

IKf(x+k.y)dk

=0

if l y l > A

+ 1x1.

Geometrically, this formula reads: The surface integral of f over Sl,,(x) is 0 if the ball BlYl(x)contains the ball BA(0). The theorem is therefore a consequence of the following lemma. Lemma 2.7. Let f E C(R") be such that for each integer k > 0,

SUP Ixl"ff(x)l < 03. X€R"

Suppose f has surface integral 0 over every sphere S which encloses the unit ball. Then .f(x) = 0,for 1x1 > 1. Proof. The idea is to perturb S in the relation

slightly and differentiate with respect to the parameter of perturbation, thereby obtaining additional relations. Replacing, as above, f with a suitable convolution $ * f , we see that it suffices to prove the lemma for f in d'(R"). Writing S = S,(X) and viewing the exterior of the ball B,(x) as a union of spheres with center x (see Fig. 7) we have by the assumptions

108

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

which is a constant. Differentiating with respect to x i , we obtain

We use now the divergence theorem

(26)

s

(div F)W dy =

BR(O)

1

( F , n>(s)dub)

SR(0)

for a vector field F on R", n denoting the outgoing unit normal and dw the surface element on S,(O). For the vector field F ( y ) = f ( x + y)(a/ayi) we obtain from (25) and (26), since n = R - ' ( s , , . . . , s"),

s s

+ s)si dw(s) = 0.

f(x

SR(0)

But by (24)

f(x

+ s)xi d o ( s ) = 0.

SR(0)

Adding the two last relations we get S,.f(S)Sidw(s) = 0.

This means that the hypotheses of the lemma hold for f ( x ) replaced by the function x i f ( x ) .By iteration d4s)=0

S,.f(S)P(S)

for any polynomial P, so f Theorem 2.6.

= 0 on

S. This proves the lemma as well as

Corollary 2.8. Let f E C(R") satisfy (i) and assume

m

=0

for all hyperplanes 5 disjoint from a certain compact convex set C. Then

(28)

f(x) =0

for x $ C.

In fact, if B is a closed ball containing C we have by Theorem 2.6, f ( x ) = 0 for x $ B . But C is the intersection of such balls, so (28) follows. Remark 2.9. While condition (i) of rapid decrease entered in the proof of Lemma 2.7 [we used I x l k f ( x ) ~L'(R") for each k > 03, one may

2. The Radon Transform on R"

421

109

wonder whether it could not be weakened in Theorem 2.6 and perhaps even dropped in Lemma 2.7. As an example, showing that the condition of rapid decrease cannot be dropped in either result, consider for n = 2 the function

f(x,Y) = (x

+ iY)-

made smooth in R2 by changing it in a small disk around 0. Using Cauchy's theorem for a large semicircle, we have JI . f ( x ) dm(x) = 0 for every line I outside the unit circle. Thus (ii) is satisfied in Theorem 2.6. Hence (i) cannot be dropped or weakened substantially. This same example works for Lemma 2.7. In fact, let S be a circle I z - z o ( = r enclosing the unit disk. Then d o ( s ) = -ir -, z

dz

- zo

so by expanding S, or by residue calculus,

Lz-~(z if

- zo)-'

(the residues at z = 0 and z

=

d z = 0,

zo cancel), so that we have in fact (s) d 4 s ) = 0.

We recall now that by its definition gH(pn)=

{

For each k E Z',

4 E g ( p ") :

s,

&o, p)pk d p

is a homogeneous

kth-degree polynomial in ol, . . . , on.

I.

Combining Theorems 2.4 and 2.6, we obtain the following characterization of the Radon transform of the space 9 ( R " ) = C:(R"). This can be regarded as the analog for the Radon transform of the Paley-Wiener theorem for the Fourier transform. Theorem 2.10. The Radon transjorm is a bijection of Q(R") onto 9H(P").

We conclude this subsection with a variation and a consequence of Theorem 2.6. Lemma 2.11. Let feC,(R"), A > 0, oo a fixed unit vector, and N c S"-' a neighborhood of wo in the unit sphere S"-' c R". Assume f(0,p) =0

for

OE

N , p > A.

110

INTEGRAL GEOMETRY A N D RADON TRANSFORMS

[Ch. I

Then (29)

f (x)

=

0

in the half-space (x, a),

> A.

Proof. Let B be a closed ball around the origin containing the support of $ Let E > 0 and let H, be the union of the half-spaces (x, W) > A + E as w runs through N. Then by our assumption

3(<)= o

(30)

if

4

c

H,.

Now choose a ball B, with center on the ray from 0 through -w0, with the point ( A + 2 . 9 ) ~on~ the boundary, and with radius so large that any hyperplane 4 intersecting B but not Be must lie in H , . Then by (30) f(<)= 0 whenever 5 E P", n B, = 0.

<

Hence by Theorem 2.6, f (x) = 0 for x .$ B,. ,In particular, f ( x ) = 0 for + 2 ~ since ; E > 0 is arbitrary, the lemma follows.

(x, coo) > A

Corollary 2.12. Let N be any open subset of the unit sphere S - ' . If f E C,(R") and f ( ~ , p )= 0 fbr P E R , W E N , then .f I 0.

Since f( -0, - p ) = f(o, p), this is obvious from Lemma 2.1 1.

3. The Inversion Formulas We shall now establish explicit inversion formulas for the Radon transform f +f and its dual 4 +

6.

Theorem 2.13. The function f can be recovered from its Radon transform by means of the following inversion formula:

cf = L'"- '"'((fj), . f E Y(R"), where c is the constant

Here L is the Laplacian on R". For n even the fractional power L("-')12requires a definition which will be given in course of the proof. To begin with, we recall the familiar fact that i f f E C2(Rn)is a radial function, i.e., f (x) = F(r), r = I x I, then

2. The Radon Transform on R"

921

111

This is immediate from the relations

Lemma 2.14.

=-

(i) LM' = M'L,for each r 0. (ii) For f c C 2 ( R " ) the mean value ( M ' f ) ( x ) satisfies the "Darboux equation"

that is, the function F(x, y) = (Mlylf)(x)satisfies

L,(F(x, Y ) ) = Ly(F(x9Y)). Proof. We prove this group-theoretically. using expression (1 5 ) for the mean value. For z E R", k E K , let T, denote the translation x -,x + z and Rk the rotation x + k . x . Since L is invariant under these transformations (cf. Chapter 11, 92, No. 1). we have, if r = J y J .

( L M ' f ) ( x )= / > ( f ( x =

+k

y)) dk =

s,

(Lf)(x

+k

y) dk

(M'Lf)(x) = jK[(Lf) Tx Rkl(y) dk

which proves the lemma. In order to prove Theorem 2.13 let f € Y ( R " ) . Fix a hyperplane &-, through 0 and an isometry g EM(^). As k runs through O(n).g k . to runs through the set of hyperplanes through 9 . 0 , and we have

so that

J t0

JK

J co

112

INTEGRAL GEOMETRY A N D RADON TRANSFORMS

[Ch. I

Hence (32) where

is the area of the unit sphere in R " - l , i.e., 2,$t1- 1 ) / 2 =

r<+(n - 1))'

Applying L to (32), using (31) and Lemma 2.14, we obtain (33) where F(r) = (M'f)(x). Integrating by parts and using F(0) =.f'(x), lim,.+mrkF("'(r) = 0, we get

More generally,

If n is odd, the formula in Theorem 2.13 follows by iteration. We now pass to the case of even n and use the definition of the fractional power L("-')l2 in terms of Riesz potentials Zy in (81) and (82). Using (17) in the present section, we have (34) Using Prop. 2.38 and the definition of L("-1)/2((85) No. 8), we obtain the desired formula, (35)

L ( " - y ( 3 ) " )= c j

in Theorem 2.13. We shall now prove a similar inversion formula for the dual transform $ -+ 8; on the subspace Y*(P")c Y ( P " ) . Theorem 2.15. W e have c$ = El("-'"'<(B;)^),

$ E Y*(P"),

2. The Radon Transform on R"

§21

113

where c is the constant

Here 0 denotes as before the operator d 2 / d p 2 , and its fractional powers are again defined in terms of Riesz's potentials on the one-dimensional p-space. If n is odd our inversion formula follows from the odd-dimensional case in Theorem 2.13 if we put f = 8, and take Lemma 2.1 and Corollary 2.5 into account. Suppose now n is even. We claim that (36)

(( -L)("-

= (-

1)/2f)A

a)("1)/2i f e Y*(R").

In fact, by Lemma 2.37 and Cor. 2.5 both sides belong to Y*(P"). Taking the 1-dimensional Fourier transform of ( ( - L ) ( " - ' ) / Y ) *we , get by

(4)7 (( -L)'"- "'2jy(sw) = 1st"- tf(sw).

This coincides with the Fourier transform of ( - 0 ) ' " - 1 so) that /2~ (36) is proved. Now Theorem 2.15 follows from (35) if we put in (36)

4

= 4,

f

=

@I",

g E Y*(Rn).

Because of its theoretical importance we now prove the inversion theorem (2.13) in a different form. The proof is less geometric and involves just the one-variable Fourier transform. Let 2 denote the Hilbert transform

(&'F)(t)

=

4x

F(P) dp, -mt-p

F E Y(R),

the integral being considered as the Cauchy principal value. For 4 E Y ( P " ) let A 4 be defined by

Note that in both cases ( h $ ) ( - w , - p ) function on P".

= (A$)(w,p),

Theorem 2.16. Let A be as dejned by (37). Then cf = (A!)",

f e Y(R")),

so that A 4 is a

114

INTEGRAL GEOMETRY A N D RADON TRANSFORMS

[Ch. I

where as before

Proof. By the inversion formula for the Fourier transform and by (4)

which we write as f (x) = ( 2 R ) - "

1.-

F ( o , x) do

1

=(24-

+(F(o, x)

+ F( -o,x)) do.

L n - I

Using f( - w, - p ) = 3(o,p), this gives the formula (38) f ( x ) = $2n)-"

1

sn-

Q)

L

do ~ ~ ~ ~ s l " - ' e i s ( x * O ) d s ~ - Q )p)e -dp. i'~~o,

If n is odd the absolute value on s can be dropped. The factor s"-' can be removed by replacing 3(o, p) by (- i)"- '(d"- l / d p n - ')f(o,p). The inversion formula for the Fourier transform on R then gives

as desired. Supposing now n is even, we let sgn s =

i'

-1

if s r 0 if s < O .

Then for a constant co, (39) f (x) = co

1

do I/gn

sn-1

s)eis(x*O) ds

sR

dn- 1

-f(q

dp"-

'

p)e-iSPdp.

The Cauchy principal value

is a tempered distribution whose Fourier transform is -ni sgn s (cf., Proposition 2.39). Hence (40)

(xF)-(s) = sgn s &).

2. The Radon Transform on R"

921

115

Thus sgn s can be removed in the integral above if we replace (d"- V P " -

'>3<,,P)

by A3
The inversion formula for the one-dimensional Fourier transform now gives the result. 4. The Plancherel Formula

We recall that the functions on P" have been identified with the functions 4 on S-' x R which are even: 4( -0, - p ) = 4(o,p). The functional

4

(41)

-+

J - - , JjW¶P) dw dP,

4 E C,(P"),

is therefore a well-defined measure on P", denoted d w d p . The group M(n) of rigid motions of R" acts transitively on P"; it also leaves the measure d o dp invariant. It suffices to verify this latter statement for the translations T in M(n) because M(n) is generated by them together with the rotations around 0, and these rotations clearly leave d o dp invariant. But

(4 W o ,P) = 4 h P + O

T)),

where q(w, T ) E Ris independent of p, so

JJ(4

O

T)(wY P) do dP = JJmA P =

+ 4(@, TI) dP do

JJ4(oyP) dP d o ¶

proving the invariance. In accordance with (81) and (82) the fractional power Ok is defined on Y ( P " )by

and then the one-dimensional Fourier transform satisfies

f(o, p ) = (2a)- Sf(so)eiSp ds

116

[Ch. I

INTEGRAL GEOMETRY A N D RADON TRANSFORMS

and n

')I4f extends to an isometry of Theorem 2.17. T h e mapping f + 0'"L2(R")onto the space L$lP-' x R ) of euen functions in L2(s"-' x R), the measure on S"- ' x R being

i(2n)' -" d w dp. Proof. From (44) we have from the Plancherel formula on R

(2n)JRI (- El)('- ')l43(wyp)I2 d p =

JR

1s 1'-

' If(sw) 1'

ds,

so that by integration over S"-' and use of the Plancherel formula for f ( x ) - f ( s w ) we obtain

JRLf(x) 1' d x = $(2n)' -"

J

I 0'"' ) / " f ( w p)l' , d o dp.

Sn-lxR

It remains to prove that the mapping is surjective. For this it would suffice to prove that if $EL'(S"-' x R ) is even and satisfies

LnI

/$(wy

P)( - 0)'"-1)/4f(w, P ) d o d p = 0

for all f ~ y ( R " )then , $ = 0. Taking Fourier transforms, we must prove that if II/ E L2(S"-' x R) is even and satisfies (45)

1s. - I R!

$(w, s)IsI("-')'~~(so)

ds dm = 0

for all J E 9'(R"), then $ = 0. Using the condition $(-a, -s) we see that

= $(my s),

0

J-, =

$(o, s ) ( s ~ ~ ' / ' ~ ( ""f(sw> ds d w

Is"-,Lrn

$(w, t)lt1('/2""- ')f(to)dt dw,

so (45) holds with R replaced by the positive axis R'. But then the function

satisfies

Y ( u ) ~ ( udu) = 0, so ' I'= 0 almost everywhere, whence

j~ Y ( R " ) ,

II/ = 0.

021

2. The Radon Transform on R"

117

Remark. Combining Theorem 2.16 with (46), we have an alternative Plancherel formula:

5. The Radon Transform of Distributions It will be proved in a general context in $3 (Proposition 3.6) that

(46)

JP!(<)4(C) d5 =

J .f(x)d(x) dx R"

for f E C,(R"), 4 E C(P") if d< is a suitable fixed M(n)-invariant measure on P". Thus d r = ydodp, where y is a constant, independent off and 4. With applications to distributions in mind we shall prove (46) in a somewhat stronger form. Since composite functions will be considered, it is more convenient to work with Bore1 measurable functions.

Lemma 2.18. Let f and 4 be Borel-measurable functions on R" and P", respectively. Formula (46) holds (with f and existing almost every-

4

where) in the following two situations:

(a) , f L'(R") ~ oanishing outside a compact set; (b) ,f E C,(R"), r$ locally integrable.

4 E C(P").

A h , d< = an-'dm dp.

Proof. We shall use the Fubini theorem repeatedly both on the product R" x S"-' and on the product R" = R x R"-'. Since fEL'(R"). we have for each w E S"- that f ( m , p ) exists for almost all p and that

'

JRm dx = J P < uP) dp. R

Also, since for each w, f ( w , p ) exists for almost all p , the Fubini theorem implies that f ( w , p ) exists for almost all (w, p ) E S"-' x R. Next we consider the Borel-measurable function (x, w ) + f ( x ) 4 ( o , (w, x)) on R" x We have

118

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

which in both cases is finite. Thus f(x)c$(w,(w,x)) is integrable on R" x S"-', and its integral can be calculated by removing the absolute values above. This gives the left-hand side of (46). Reversing the integrations, we conclude that &x) exists for almost all x and that the double integral reduces to the right-hand side of (46). The formula (46) dictates how to define the Radon transform and its dual for distributions. In order to make the definitions formally consistent with those for functions we would require 3(4) = S(J), g ( f ) = Z(3) if S and X are distributions on R" and P", respectively. But while f~ 9 ( R " ) implies 3~9 ( P n ) , a similar implication does not hold for 4 ; we do not even have r$ E Y(R")for Q, E 9 ( P " ) ; hence 3 cannot be defined by the formula above, even if S is assumed to be tempered. Using the notation 6 (resp., 9) for the space of C"-functions (resp., of compact support) and 9' (resp., 8')for the space of distributions (resp., of compact support), we make the following definition. Definition. For S E b'(R") we define the functional

3 by

3(4) = ~(4)for Q, E &(P"). For X E gr(P")we define the functional f: by k(f) = X(3) for f~ 9(R"). Lemma 2.19.

(i) For each X E 9 ' ( P " )we huve f. E W ( R " ) . (ii) For each S E 6'(R")we have

3 E S'(P").

Moreover, (iii) ( L S ) - =

13 (RE)" ,

=

LL

Proof. For A > 0 let gA(R") denote the set of functions f ~ 9 ( R " ) with support in the closure of BA(0). Similarly, let QA(P")denote the set of functions 4 E 9 ( P " ) with support in the closure of the "ball" D,(O) =

(5 E P" :d(0, t ) < A } .

The mapping f + 3 being continuous from 9,(R") to QA(Pn)(with the topologies from $1 of the Introduction), the restriction of 2 to each QA(R") is continuous, so (i) follows. That 3 is a distribution is clear from (3). Concerning its support select R > 0 such that S has support inside BR(0). Then if 4(m, p ) = 0 for p S R, we have &x) = 0 for 1x1 S R , whence 3(4) = S($) = 0.

2. The Radon Transform on R"

921

119

For (iii) we note by Lemma 2.1, ( L S ) - ( ~=) =

(~(8) = su.8) = s((o4n 3 ( W=m(4).

The other relation is proved in the same manner. We shall now prove an analog of the support theorem (Theorem 2.6) for distributions. For A > 0 let p,(O) be defined as above. Theorem 2.20. Let T Eb'(R") satisfy the condition

SUPP F = CKBA(0)). Then

SUPP( T ) = CI(BA(0)). Proof. For f~ 9(R"), 4 E 9 ( P " ) we can consider the mixed conuolution

(f

x

4)(8 =

J f(Y)d(S- Y ) dY, R"

where for ~ E P "5, - y denotes the translate of the hyperplane ( by -y. Then ( f x4

j

= f d

In fact, if tois any hyperplane through 0,

(f x

4nX)

= = =

By the definition of

S, dk jRn S(MX

+ k . to- y) dy

s, lRn + dk

f(x - Y M Y

k * 5 0 ) dY

( f * d><X>.

T,the support assumption on

is equivalent to

T ( 6 )= 0

for all 4 E !3(P") with support in P" - Cl(p,(O)). Let E > 0, let f E 9 ( R " ) be a symmetric function with support in CI(B,(O))and let 4 E 9(P") have support contained in P" - Cl(j,+,(O)). Since d(0, t - y ) 4 d(0, t) ( y ( ,it follows that f x 4 has support in P" - Cl(p,(O)); thus by the formulas above and the symmetry of 1;

+

(1* T)(& = T ( J * 8)= ~ ( ( xf 4 j ) = 0.

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

But then

(f * T1-W

=

(f * T)(&

= 0,

which means that (f * T)* has support in Cl(flA+c(0)). But now Theorem 2.6 implies that f * T has support in Cl(BA+e(0)).Letting E -,0, we obtain the desired conclusion, supp( T ) c Cl(B,(O)). Remark. For a strengthening of this result see Exercise B5.

We can now extend the inversion formulas for the Radon transform to distributions. First we observe that the Hilbert transform 2 can be extended to distributions T on R of compact support. It suffices to-put

F E 9(R).

2 ( T ) ( F )= T(%F),

In fact, &' being the convolution with a (tempered) distribution, the mapping F -, 2 F is a continuous mapping of 9 ( R ) into b(R) (cf. No. 8 below). In particular, %(T ) E 9 ' ( R ) . Theorem 2.21. The Radon transform S by the following formula:

CS = (AS)",

+

S ( S E &"(R")) i s inverted

SE~'(R"),

where the constant c equals

I n the case when n is odd we have also c~ = ~ ( 1 / 2 ) @ - 1(@I' ) 1. Remark. Since 9 has compact support and since A is defined by means of the Hilbert transform, the remarks above show that ASE~'(P"). Hence the right-hand side is well-defined. Proof. Using Theorem 2.16, we have

(AS)'(f 1 = (A$)($) =

=m

S((AP)')

j)

= cS(f).

The other inversion then follows, using Lemma 2.19. Let M be a manifold and dp a measure such that on each local coordinate patch with coordinates ( t l ,. . . , t,) the Lebesgue measure dr, . . .dt, and dp are absolutely continuous with respect to each other. If h is a function on M locally integrable with respect to dp, the distribution 4 + h 4 d p will be denoted q.

923

121

2. The Radon Transform on R"

Proposition 2.22.

(a) Let f ' L'(R") ~ vanish outside a compact set. Then the distribution TJ has Radon trangorm given b y

T; = Tj.

(47)

(b) Let

4 be a locally integrable function

on P". Then

(T,)"= Td.

(48)

6

Proof. The existence and local integrability o f f and was established during the proof of Lemma 2.18. The two formulas now follow directly from Lemma 2.18.

As a result of this proposition the smoothness assumption can be dropped in the inversion formulas. In particular, we can state the following result. Corollary 2.23. (n odd)

The inversion formula

holds,for al1.f E L'(R") of compact support, the derivative interpreted in the sense of distributions. Examples. If ,u is a measure (or a distribution) on a submanifold S of a manifold M, the distribution on M given by 4 -,p ( 4 l S ) will also be denoted by p.

(a) Let 6, be the delta distribution f

-, f (0)

on R". Then

so that

(49)

8,

=n ,

the normalized measure on S"-'

91 x R.

' msn-

1,

considered as a distribution on

(b) Let todenote the hyperplane x, = 0 in R", and 6;,, the delta dison P". Then tribution 4 -, $(to)

so that (50)

the Euclidean measure of 5,.

4 0 = mro,

122 .(c) Let by (47)

INTEGRAL GEOMETRY A N D RADON TRANSFORMS

ze be the characteristic function of the unit

[Ch. I

ball B c R". Then

(d) Let R be a convex region in R" whose boundary is a smooth surface. We shall obtain a formula for the volume of R in terms of the areas of its hyperplane sections. For simplicity we assume n odd. The characteristic function zn is a distribution of compact support and ( ~ n ) is - thus well-defined. Approximating zn in the L2-norm by a sequence (*,) c 9(R), we see from Theorem 2.17 that a~l'z)("')$,,(w,p) converges in the L2-norm on P". Since

J'$~(s)Q(od t

=

J'$n(x)d(x) dx,

it follows from Schwarz's inequality that I,?,,-,(xn)- in the sense of dis'I$,, converges as a distribution to tributions, and accordingly d(l'z)("-l)((~n)~). Since the Lz-limit is also a limit in the sense of distributions, this last function equals the Lz-limit of the sequence l )+n. From Theorem 2.17 and Corollary 2.23 we therefore have the following result. Theorem 2.24. Let Q c R" (n odd) be a convex region as above and V(R) its volume. Let A(w, p ) denote the (n - 1)-dimensional area of the intersection of R with the hyperplane (x, o)= p . Then

Furthermore,

in the sense of distributions. 6. Integration over d-Planes. X- Ray Transforms

Let d be a fixed integer in the range 0 < d c n. Since a hyperplane in R" can be viewed as a disjoint union of parallel d-planes, parametrized by R n - l - d , it is obvious from (4) that if f~ Y ( R " ) has 0 integral over each d-plane in R",then it is identically 0. Similarly, we can deduce the following consequence of Theorem 2.6.

2. The Radon Transform on R"

623

123

Corollary 2.25. Let f E C(R") satisfy the ,following conditions: (i)

For each integer m > 0, Ixlm'(x)

is bounded on R"

(ii) For each d-plane tdoutside a ball 1x1 < A. we have

dm being the Euclidean measure. Then for 1x1 > A.

f(x) = O

We now define the d-dimensional Radon transform ,f + f by

Because of the applications to radiology indicated in No. 7b), the onedimensional Radon transform is often called the X-ray transform. We can then reformulate Corollary 2.25 as follows. Corollary 2.26. Let J g E C(R") satisfy the rapid decrease condition : For each m > 0, Ixlmf(x) and Ixlmg(x) are bounded on R". Assume for the d-dimensional Radon transforms

f= d ( 0 whenever the d-plane

lies outside the unit ball. Then f(x) = g(x)

for

1x1 > 1.

We shall now generalize the inversion formula in Theorem 2.13. If 4 is a continuous function on the space of d-planes in R", we denote by 6 the point function i ( x >=

J-., 4(<) dP(t),

where p is the unique measure on the (compact) space of d-planes passing through x, invariant under all rotations around x, and with total measure 1. If a is a fixed d-plane through the origin, we have by analogy with (16), (53)

$(x) =

IK

4(x

+k

a) dk.

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INTEGRAL GEOMETRY A N D RADON TRANSFORMS

[Ch. I

Theorem 2.27. The d-dimensional Radon transform in R" is inverted by the formula (54)

cf = L'"2'd((f)"),

f EY(R"),

where

Proof. We have by analogy with (32)

<3>'<x>

=

s, ( s,

f(x

+k

a

Y) M Y ) ) dk

Hence

(f)'(x)

= fld

(M'f)(x)

rd- dr,

so, using polar coordinates around x, (55)

( 3 ~= )n,JRn I X- Y i d - w ) dy.

The theorem now follows from Prop. 2.38. As a corollary of Theorem 2.10 we now obtain a generalization, characterizing the image of the space 9 ( R " ) under the d-dimensional Radon transform. The set G(d, n) of d-planes in R" is a manifold, in fact, a homogeneous space of the group M(n) of all isometrics of R". Let Gd,"denote the manifold of all d-dimensional subspaces (d-planes through 0) of R". The parallel translation of a d-plane to one through 0 gives a mapping n of G(d, n) onto Gd.,. The inverse image n-'(a) of a member O E G ~is, ~naturally identified with the orthogonal complement d.Let us write ( = (6,x") if o = n(5) and x" = '6 n 5. Then (52) can be written f(0,

x") = Ju f(x'

For k E Z f we consider the polynomial (57)

+ x") dx'

021

2. The Radon Transform on R”

If u = U ” E

(r-‘,

JRn

125

this can be written Juf(x’

f(x)(x, u”Ikdx =

+ x”)(x”, u ” ) ~dx’ dx“,

so the polynomial n

P#,k(d’) =

J

f(o, x”)(x”, u ” ) dx” ~

d

is the restriction to o* of the polynomial Pk. By analogy with the space BH(P”)in No. 2 we define the space ) G(d, n ) of comgH(G(d,n)) as the set of C”-functions $(<) = ~ J X ” on pact support satisfying the following condition.

(H): For each k E Z+ there exists a homogeneous kth-degree polynomial Pk on R” such that for each o E Gd,” the polynomial Pu,k ( d ’ ) =

IL

40(Xn)(Xn,U”)k dx”,

U” E CT’,

coincides with the restriction Pk I CJ’. Corollary 2.28. The d-dimensional Radon transform is a bijection of 9 ( R ” ) onto gH(G(d, n)). Proof. For d = n - 1 this is Theorem 2.10. We shall now reduce the case of general d to the case d = n - 1. It remains just to prove the surjectivity in Corollary 2.28. Let 4 E 9,,(G(d, n)). Let w E R” be a unit vector. Choose a d-dimensiona1 subspace (r perpendicular to o and consider the (n - d - 1)-dimensional integral (58)

Yu(w9

PI =

J

4 ~ ( ~dn-d” ) 1(x”),

p

R.

(W.X”)=p.X”EuL

We claim that this is independent of the choice of o. In fact, JR yu(WPIPk

dP =

J Pk ( J(bu(x”) dn-d- 1(Xff))d p R

=

LL

~,(x’~)(x’~, dx“ = Pk(w).

If we had chosen another o, say ol, perpendicular to w, then by the above, Y,,(w,p) - Y,,(w,p) would have been orthogonal to all polynomials in p ; having compact support it would have been identically 0.

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

Thus we have a well-defined function Y(w, p ) = Y,(o, p ) to which Theorem 2.10 applies. (It should be noted that Y is smooth; in fact, for w near a fixed w o ,we can let u depend smoothly on w, so that Y,(w, p) is smooth.) From Theorem 2.10 we get a function f~ 9(R")such that (59)

WO, P ) =

I

(X.

f ( x ) dm(x). w) = p

It remains to prove that (60)

9,(X'')

=

J, f ( x ' + x") dx'.

But as X" runs through an arbitrary hyperplane in u* it follows from (58) and (59) that both sides of (60) have the same integral. By the injectivity of the (n - d - 1)-dimensional Radon transform on c*, Eq. (60) follows. This proves Corollary 2.28. 7. Applications A. Partial Difierential Equations The inversion formula in Theorem 2.13 is very well suited for applications to partial differential equations. To explain the underlying principle we write the inversion formula in the form

(61)

f(x)=

yc-

lY2

( J-"- fb, I

(x, w))dw)

9

where the constant y equals +(2ni)'-". Note that the function f,(x) = f(w,(x, w)) is a plane wave with normal w ;that is, it is constant on each hyperplane perpendicular to w. Consider now a differential operator

with constant coefficients a,,...kn,and suppose we want to solve the differential equation (62)

DU = f,

where f is a given function in Y(R").To simplify the use of (61) we assume n to be odd. We begin by considering the differential equation (63) Dv = f,, where .f, is the plane wave defined above and we look for a solution v which is also a plane wave with normal w. But a plane wave with

021

2. The Radon Transform on R"

127

normal w is just a function of one variable; also, if u is a plane wave with normal w, so is the function Du. The differential equation (63) (with v a plane wave) is therefore an ordinary differential equation with constant coefficients. Suppose v = u, is a solution and assume that this choice can be made smoothly in w. Then the function

is a solution to the differential equation (62). In fact, since D and L("-l)l2 commute, we have

L-

D~ = yL("-1)/2

S,.

D ~ do, = y ~ ( " - 1 ) / 2

-1

f , d w = f:

This method only assumes that the plane wave solution u,, to the ordinary differential equation Du = f, exists and can be chosen so as to depend smoothly on w. This cannot always be done because D might annihilate all plane waves with normal w. [For example, take D = d2/(dx, dx,) and w = (1. O).] However, if this restriction to plane waves is never 0 it follows from a theorem of Trbves [1963] that the solution u,, can be chosen to depend smoothly on w. Thus we can state Theorem 2.29. Assuming the restriction D, of D to the space of plane waves with normal w is # O for each w formula (64) gives a solution to the diferential equation Du = f ( f E Y ( R " ) ) .

The method of plane waves can also be used to solve the Cauchy problem for hyperbolic differential equations with constant coefficients. We illustrate the method by means of the wave equation in R", au2 au u(x, 0 ) = 0, Lu = 2 - (x, 0) = f ( x ) , at at y

f E Y ( R " ) being a given function.

Lemma 2.30. For each x E R"

Proof. The hyperplane perpendicular to x with distance p from 0 intersects s"-' in an ( n - 2)-sphere of radius (1 - p2)l/'. The integrand is constant ( x l p on this sphere and from Fig. 8 we have

128

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

FIG.8

It follows that

which implies the lemma. Using (66) and the identities (88) and (89) in No. 8 we can derive the identity

for n odd. Suppose u , is a plane wave with normal o satisfying the wave equation a2

Lu, = - u at2 m 3 ignoring the initial condition. Then the function Z(XYt )

=

s,"-,

u,(x,

0do

is also a solution and so is the convolution 4x9 t ) =

The time derivative satisfies

s..

S(Y)Z(X - YY t ) dY.

f(Y)Z,(X - Y , OMY,

2. The Radon Transform on R"

§21

129

so in view of (67) we try to determine u, such that

j--* I(x9 all do.

Z,(X, 0) =

For this we put

o2

+

u,,(x, t ) = 4t(<x,0) + sgn((x, 0) t), and have then obtained the following result. Theorem 2.31. Let n be odd. A solution to the Cauchy problem (65) is given by

u(x, f) =

We shall now put this into another form for n = 3. Since u, satisfies the wave equation, the formula can be written

As for Lemma 2.30 we compute the S2 integral by slices perpendicular to x - y. This gives

L2sgn{(x-y,o)+t}do=2n

s g n { l x - y l p + t}dp,

which, since t 2 0, equals 1

27c

J-, sgn{p + Ix

- yl-'t}

dp =

4nJx- y J - ' t

if t < J x - y J if t > Ix - y l .

Hence u(x, t ) =

- -a 4nat l a

-

- - -4n at 1

J;

J: s,,,,

= - (4nt')t-

4n

f(Y)lX -Yl-'dY

x--yl>l 00

dp

P - f(Y) d4Y)

'(Mtf)(x).

This proves the following result of Poisson. Corollary 2.32. given by

For n = 3 the solution to the Cauchy problem (65) is u(x, t) = t(M'f)(x).

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INTEGRAL GEOMETRY A N D RADON TRANSFORMS

[Ch. I

Remarks. (i) A similar formula holds for arbitrary n ; the solution to (65) is given by

(cf. Tedone [1898], Asgeirsson [1937], John [1955]. Chapter 11). A formula of this type is derived in Exercise F1,Chapter 11. (ii) Huygens's Principle. The formula in Corollary 2.32 shows that u(x, t ) is determined by the values o f f on the sphere S,(x) in R3. This phenomenon, called Huygens's principle, remains true for R" (n odd) in a slightly weaker form. In fact, if the differentiations 8"- 2/atn- are carried out, we end up with a linear combination with polynomial coefficients

which shows the following: For n odd, the solution u(x, t ) to the Cauchy problem (65) is determined by the values off in an arbitrarily thin shell around S,(x). For even n the integral will remain, so u(x, t ) will require knowing f in the ball B,(x). B. Radiography The classic interpretation of an X-ray picture is an attempt at reconstructing properties of a three-dimensional body by means of the X-ray projection on a plane. In modern X-ray technology the picture is given a more refined mathematical interpretation. Let B c R3 be a body (for example, a part of a human body) and let f ( x ) denote its density at a point x . Let t be a line in R3 and suppose a thin beam of X rays is directed at B along 4. Let lo and I denote respectively, the intensity of the 'beam before entering B and after leaving B. For physical reasons we then have the formula (Cormack [1963]), log(~o/O=

I,

f ( x )d m W ,

the integral f(<) o f f along 5. Since the left-hand side is determined by the X-ray picture, the X-ray reconstruction problem amounts to the determination of the functionf by means of its line integrals?((). The inversion formula in Theorem 2.21 gives an explicit solution of this problem. If Bo c B is a convex subset (for example, the heart), it may be of interest to determine the density f outside B , using only X rays which do

2. The Radon Transform on R"

021

131

not intersect B,. The support theorem (Theorem 2.6 and Corollaries 2.8 and 2.26) implies that f is determined outside B, on the basis of the integrals f(()for which ( does not intersect €3.,

8. Appendix. Distributions and Riesz Potentials In this subsection we collect a few notions and facts from distribution theory as well as some results about Riesz potentials which have been used in this section. In the Introduction, 51, we reviewed the definition of a distribution on R" and of the support of a distribution. If F is a locally integrable function on R". the distribution + j F(x)+(x) d x (4 E 9 ( R " ) ) is denoted TF. If T E 9 ' ( R ) . the derivative T' is defined as the distribution

+

4

+

- T(+'),

4 EW).

If F E C'(R), then by partial integration, the distributions TF,and (TF)' are seen to coincide. A linear form on the space Y ( R ) is called a tempered distribution if it is continuous in the topology given by the norms (6). The restriction of a tempered distribution to 9 ( R ) is a distribution, and since the subspace 9 ( R ) c Y ( R ) is dense, two tempered distributions coincide if they coincide on 9 ( R ) . A distribution of compact support is tempered. Let Y ' ( R ) denote the space of tempered distributions. If TEY ' ( R ) , its Fourier transform P i s the linear form on Y ( R ) defined by EY(R), T(+)= T(d9, where

+

&u) = R

+(x)e-'""dx.

+

Since + 4 is a homeomorphism of Y ( R ) onto itself, TE Y ( R ) . Moreover, since jFc$ = jF+ [+, F E Y ( R ) ] , the distributions (TF)-and T? coincide. Since (&'(x) = 211+(-x), we have i = 2116. We also have the rules (68)

(T)-=27tZ

(T')-= i x r (uT)-=

i(T)',

where 7 is the image of T under the image x + - x . Since distributions generalize measures, it is sometimes convenient to write

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INTEGRAL GEOMETRY A N D RADON TRANSFORMS

[Ch. I

for the value of a distribution T o n the function 4. If TE&'(R), then ?; as defined above, is the distribution given by the function u-+j e-ixud T ( x ) . If S and T a r e two distributions at least one of which is of compact support, their convolution is the distribution defined by

4

+

J J 4(x + Y ) d S ( x ) dT(Y), R R

4 E W).

This distribution is denoted S * T If f~ 9 ( R ) , T E ~ ' ( R )the , distribution T, * T has the form T,, where g(x) =

JR

f(x -Y)

so we write for simplicity g = f * T The mapping .f -+ f * T is continuous from 9 ( R ) into d(R). If S and T both have compact supports, so does S * 7: Also, and i= have the form = T,, = 7;, where s, t E&(R), and, in addition, ( S * T)-= We express this in the form

r,.

( S * T)-=

(69)

ST

This formula holds also in the cases S E Y(R),

T E Y'(R),

S E &'(R),

T E .Y'(R),

and in both cases S * T is tempered. These notions (distributions, tempered distributions, derivative, Fourier transform, and convolution) generalize in an obvious manner to several variables. We thus have function spaces and distribution spaces satisfying the following canonical inclusions:

9 ( R " ) c .Y(R") c d(R")),

n

n

n

&'(R")c Y ' ( R " ) c 9'(R"). We shall now study some useful examples in detail. If ci E C satisfies Re c1 > - 1, the functional x; :4 +

s,"

x a 4 ( x )dx,

4 EY(R),

is a well-defined distribution. The mapping c1 4 x ; can be extended to a holomorphic distribution-valued function in the region Recr > - n - 1,

ci

# - 1 , -2 ,..., - n ,

2. The Radon Transform on R"

023

133

by means of the formula (70)

x"++) =

Jo'

[

xn- 1

xu +(x) - 4(0) - x4'(0) - . . +

(n - l ) !

~

#"-')(O)]

dx

In this manner a + x: is a meromorphic distribution-valued function on C,with simple poles at a = - 1, - 2 , . . , , We note that the residue at a = -k is given by Res x: = lim (a + k)x; = ( - l ) k - ' 6 ( k - 1 ) . (k - l)! u= - k a- - k ~

Here 6(")is the hth derivative of the delta distribution 6. We note that x*+ is always a tempered distribution. Next we consider for Re a > -n the distribution ru on R" given by

ru:4 --*

J

4 E 9(R").

$(x)l x 1' dx,

R"

Lemma 2.33. The mapping a + ru extends uniquely to a meromorphic mapping from C to the space Y ( R " ) of tempered distributions. The poles are the points

a

=

-n -2h

(hEZ+),

and they are all simple. Proof. We have for Re a > -n

(72)

r'(4)

=

som

a,,

(M'4)(0)tu+"- dt.

Next we note that the mean value function t + (M'4)(O) extends to an even C"-function on R, and its odd-order derivatives at the origin vanish. Each even-order derivative is nonzero if 4 is suitably chosen. Since by (72)

'

ru(4 ) = 0,t"+' " - (M'4), (73) the first statement of the lemma follows. The possible (simple) poles of rQ are by the remarks about x: given by a + n - 1 = - 1, - 2 , . . . . However, if tl + n - l = -2, -4, . . . , formula (71) shows, since (M'4(0))ch)= 0 (h odd) that P(+) is holomorphic at the points

a = -n

- 1,

-n - 3, ... .

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

The remark about the even derivatives of Mr4 shows, on the other hand, that the points a = - n - 2h ( h E Z') are genuine poles. We note also from (71) and (73) that Res ra = lim (a + n)r" = QnS.

(74)

a= - n

a+ - n

We recall now that the Fourier transform T + Tof a tempered distribution T o n R" is defined by

n 4 ) = W), 4 E . m w . We shall now calculate the Fourier transforms of these tempered distributions r". Lemma 2.34.

(76)

W e have the identities

(r2h)'= (2n)"(-L)hS,

h E Z'.

Proof. We use the fact that if $ ( x ) = e - ( 1 / 2 ) 1 xthen 12, ,J(u) = (2n)(n/2)e-(1/2)I~12,

so that by the formula j j j = j fg we obtain for (77)

~&x)e-(1/2)rl"12 dx = ( 2 ~ ) "t-"l2 '~

4 E Y ( R n ) ,t > 0,

Sd(u)

e-(1/2)t-'b412due

We multiply this equation by t-(1/2)a-1 and integrate with respect to t. On the left we obtain, using the formula

the expression

On the right we similarly obtain

The interchange of the integrations is valid for a in the strip - n < Re a c 0, so that (75) is proved for these a. For the remaining ones it follows by analytic continuation. Finally, (76) is immediate from the definitions.

423

2. The Radon Transform on R"

135

Lemma 2.35. T h e action of the Laplacian is given by (78)

Lr"

(79) For n

= a(a

+n -2

)

~- a - n~+ 2 # 2~ Z + ,

Lr2-" = (2 - n)R,6, =

n # 2.

2 this "Poisson equation" is replaced by L(1og r ) = 27th.

(80)

Proof. For Re a sufficiently large, (78) is obvious by computation. For the remaining ones it follows by analytic continuation. For (79) we use the Fourier transform and the fact that for a tempered distribution S, (- LS)-= r2S.

Hence, by ( 7 9 ,

Finally, we prove (80). If q5e9(R2),we have, putting F(r) = (M'q5)(0),

J

(L(log r))(q5) = J log r (L+)(x) dx = R=

log r (27tr)(MrL4)(0)dr. 0

Using Lemma 2.14, this becomes

som

log r 2nr(F"(r) + r - 'F'(r))dr,

which by integration by parts reduces to [log r (27tr)F)(r)]$ - 2n [See Exercise D4 for a simpler proof.]

som

F'(r) dr

=

27t F(0).

We shall now define fractional powers of L, motivated by the relation (- L f m = I u

12ml

so that formally we should like to have a relation (( - L)!f)-(u) = Iu

12p3(u).

Since the Fourier transform of a convolution is the product of the Fourier transforms, formula (75) (for 2 p = - a - n) suggests defining (-L)pf = P P ( f ) , where I y is the Riesz potential

(81)

136

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

with

Writing (82) as H , ( y ) - ' ( J * r Y - " ) ( x )and assuming f E Y ( R " ) ,we see that the poles of rS-" are cancelled by the poles of r(iy),so that (Zyf)(x) extends to a holomorphic function in the set C , = {y E C:y - n 4 2 2 + } . We also have by (74) and the formula for 0,

I o f = lim IYf = f .

(84)

s-0

Furthermore, by (78) and by analytic continuation, (85)

I y L f = LIY = IY-'f,

fE Y(R"),

E

C,.

We now prove an important property of the Riesz potentials. Proposition 2.36. The ,following identity holds:

for f~ Y(R"), Re a, Re j3 > 0, Re(a

I"(I!f) = I a + @ f

+ p) < n.

Proof. We have

The substitution u = ( x - z ) / l x - yl reduces the inner integral to the form

(86)

J x-

yp+@-"JRJui.-nlw

- u l P - n du,

where w is the unit vector ( x - y ) / l x - y l . Using a rotation around the origin, we see that the integral in (86) equals the number (87)

c,,(o(,p) =

JRn

1 u la-' 1 el - u 1 p - n

dc,

where el = (1,0,. . . ,O). The assumptions made on a and p ensure that this integral converges. By the Fubini theorem the exchange of order of integrations above is permissible and

021

2. The Radon Transform on R"

137

It remains to calculate c,(a, p). For this we use the following lemma. As in Corollary 2.5 let Y * ( R " ) denote the set of functions in Y ( R " ) which are orthogonal to all polynomials. Lemma 2 . 3 7 . Euch laleaves the space Y'*(R") invariant. Proof. By continuity it suffices to prove this for those a for which n, - a $22'. But then, if JE Y*,

a - n satisfies the assumptions of Lemma 2.34, that is, a -

(89) since (4 * S)- = $3 for 4 E 9, S E 9'. But f has all derivatives 0 at 0 and so does f ( u ) I u - ' , so the lemma is proved. We can now finish the proof of Proposition 2.36. Taking . f b ~ Y ' *we can put ,f = I f f o in (89) and then (P(rqf,))- = (rqf,)-(U)I u I- = J b ( l l ) 1 u I-=-I3 = (ra+Pf,)-(U).

This shows that the scalar factor in (88) equals 1, so Proposition 2.36 is proved. In the process we have obtained the evaluation

We now prove a variation of Proposition 2.36 needed in the theory of the Radon transform. Proposition 2.38. Let 0 c k < n. Then

I-k(r"f)

=f

f i r all f~ 9 ( R " ) ,

Proof. By Proposition 2.36 we have

(90)

I*(Ikf)= Za'kf

for 0 c Re a c n - k.

We shall now prove, following a suggestion of R. Seeley that the function 4 = ~"fsatisfies

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[Ch. I

INTEGRAL GEOMETRY A N D RADON TRANSFORMS

For each N we have an estimate If(y)I I c N ( 1 constant. Thus we have

1

JR!(Y)lx

- ylk-" d Y (

(1

cNJ

+IY~)-~,

where

cNiS a

+ IYI)-Nlx - Y l k - " dY

IYl5(1/2)1Xl

(1 lYl t ( 1 / 2 ) 1x1

+IYI)-~~x

In the first integral we have Ix - ylk-" use the inequality (1

-

ylk-"dy.

I ~ x I " - ~and in the second we

+ Iyl)-N 1(1 + I y I ) - N - k + " ( l+ Itxl)k-".

Taking N large enough, both integrals on the right-hand side will satisfy (91), so (91) is proved. We claim now that la($)(x), which by (90) is holomorphic for 0 < Re a < n - k , extends to a holomorphic function in the half-plane Re LY < n - k. It suffices to prove this for x = 0. We decompose 4 = 41 + 42, where 41 is a smooth function identically 0 in a neighborhood 1x1 < E of 0 and 42E 9 ( R " ) . Since 41 satisfies (91), we have for Re LY < n - k,

=

c

~wlxla+k-n-l

dlxl <

00,

so la41is holomorphic in this half-plane. On the other hand, la& is holomorphic for all a E C , . Now we can put LY = - k in (90) and the proposition is proved. Proposition 2.39.

Let S denote the Cauchy principal value

Then S is a tempered distribution and I

S(s) = -ni sgn s =

Proof.

is the function -ni, xi,

s 20 s

< 0.

We have xS = 1, so by (68), 2nd =

i

= (xs)" =

i(S)',

But (sgn s)' = 26, so

S

= -ni sgn s

+c

(C = constant).

3. A Duality in Integral Geometry

431 But

s' and

139

sgn s are odd, so that the result follows.

For later use we formulate now a result for the analogs of the Riesz potential corresponding to the quadratic form Q ( X ) = x: - x.' ( X = ( x , . . . . , x , ) ) on R". Let D denote the retrograde cone D =

{X

E

R": Q ( X ) > 0. x 1 < 0 }

and for f~ 9 ( R " ) consider the integral I

P

(92) where

(93) The following result is proved in Riesz [1949], Chapter 111. Theorem 2.40. The inregral J ' f which converges for Re y 2 n extends to a holomorphic ,function y + J ' f on all of C. Moreover, J o f = .f(0).

Remark. The result remains valid if f , together with each of its partial derivatives, depends holomorphically on y (cf., loc. cit., Lemma IV, p. 62).

$3. A Duality in Integral Geometry. Generalized Radon Transforms and Orbital Integrals 1. A Duality for Homogeneous Spaces

The inversion formulas in Theorems 2.13, 2.15, 2.16, and 2.19 suggest the general problem of determining a function on a manifold by means of its integrals over certain submanifolds. In order to provide a natural framework for such problems, we consider the Radon transform .f +3 on R" and its dual $ + from a group-theoretic point of view, motivated by the fact that the isometry group M ( n ) acts transitively on both R" and on the hyperplane space P". Thus

4

(1)

R" = M(n)/O(n), P" = M(n)/Z2 x M(n - I),

where O(n) is the orthogonal group fixing the origin OER" and Z2 x M(n - 1) is the subgroup of M(n) leaving a certain hyperplane to through 0 stable (Z2 consists of the identity and the reflection in this hyperplane).

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

We observe now that a point g , O ( n ) in the first coset space above lies on a plane g2(Z2 x M(n - 1)) in the second if and only if these cosets, considered as subsets of M(n), have a point in common. This leads to the following general setup. Let G be a locally compact group, X and E two left coset spaces of G,

X = G/H,,

(2)

E = G/H,,

where H x and H , are closed subgroups of G. It will be convenient to make the following assumptions:

(i) variant (ii) (iii)

The groups G, H , , H , , H , n H , are all unimodular (the left inHaar measures are right-invariant). If h , H E c H H H X ,then h x E H E . If h , H , c H , H , , then h,E H , . The set H , H , c G is closed.

We note that (iii) is satisfied if one of the subgroups H , , H E is compact. All the assumptions are satisfied for the pair of coset spaces in (1). Let us, for example, check the first part of (ii). If h , H , c H , H , , we obtain by applying both sides to the origin, h , toc to,so that h , E H E . Two elements X E X ,< E Eare said to be incident if as cosets in G they intersect. We put j; =

{tE H : x and < incident},

g = { xEX

: x and

5 incident}.

Using the notation Ag = g A g - ' for g E G , A c G, we have the following lemma. Lemma 3.1.

Let y E G, x = g H , , 5 = g H H . Then

(a) 2 is an orbit q f ( H x ) " and we have the coset space itlrntiJicution

2 (b) [ is an orbit qf

=

(HX)"(Hx n H E Y ;

arid

e

=

(H,)g/(Hx nHEY.

Proof. By definition

2

=

{ y H s : y H , ng H , #

which can be written (3)

R = {ghx HE: hx E H , } .

01,

931

3. A Duality in Integral Geometry

141

This is the orbit of the point g H , in E under the group gH,g-'. The subgroup leaving the point g H , fixed is (gH,g-') n (gH,g-'). This proves (a); (b) follows in the same way. Let xo = {H,} and to= { H E }denote the origins in X and tively. Then by (3) 2=g.Zo,

z, respec-

t=g.[,,

where . denotes the action of G on X and on E. Lemma 3.2. T h e maps x

+2

and ( + [ are one-to-one.

Proof. Suppose xl, x2 E X and il = Z2. Let g l , g2 E G be such that x1 = g l H x , x2 = g 2 H x . Then by ( 3 ) g1 .2,, = g 2 - i o ,so that writing g = g ; ' g 2 , we have g R, = f,. In particular, g to€go, so that, since

Zo is the orbit of toE E under H,, we have g to= h,. tofor some h , H,. whence h x ' g = h,E H E . It follows that h,.X, = 2,. so that h,H,-t,

E

= H,.t,.

that is, h,H, c H , H,. By assumption (ii) h, E H,, which gives x1 = x 2 . This proves the lemma. In view of this lemma, X and E are homogeneous spaces of the same group G in which each point in z can be viewed as a subset of X and each point of X can be viewed as a subset of E.We say X and E are homogeneous spaces in duality. The terminology is suggested by the familiar duality between R" and P" in projective geometry. The maps x + Z and l + [ are also conveniently described by means of the following double fibration, G / ( H , n HE)

(4) X = G/H,

E = G/H,,

where the maps p and x are given by p(gH, n H,) = gH,, n ( g H , n H E ) = g H z . Then by ( 3 ) we have

[ = p(n-'(<)).

i = x(p-'(x)), Lemma 3.3.

Each 2 c E is closed and each

tcX

is closed.

Proof. If p,: G + G / H , is the natural mapping, we have (PSI-

'(s - (20)) = (9:gHE # H , H,} 1

=

G - H , HE.

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INTEGRAL GEOMETRY A N D RADON TRANSFORMS

[Ch. I

In particular, ps(G - H,H,) = E - Zo, so that by using (ii) and the fact that p , is an open mapping, we deduce that Zo is closed. By translation each 2 is closed and similarly each g is closed. Examples. (i) Points Outside Hyperplanes. We saw before that if in the coset space representation (1) O(n) is viewed as the isotropy group of 0 and Z,M(n - 1) is viewed as the isotropy group of a hyperplane through 0, then the abstract incidence notion is equivalent to the naive one: x E R" is incident to 5 E F if and only if x E 5. On the other hand, we can also view Z,M(n - 1) as the isotropy group of a hyperplane Sa at a distance 6 > 0 from 0. [This amounts to a different imbedding of the group Z , M ( n - 1) into M(n).] Then we have the following generalization.

Proposition 3.4. The point X E R" and the hyperplane SEP" are incident if and only if distance (x, 4 ) = 6. Proof. Let x = g H x , 5 = yH,, where H , = O(n), H , = Z , M ( n - 1). Then if g H , n y H , # 521, we have gh, = yh, for some h, E H , , h, E H , . Now the orbit H,.O consists of the two planes 5; and parallel to
Y

*

Sa =

S.

On the other hand, if distance (x, 5 ) = 6, we have g .O E y . (ti u 4;) = yH, a 0 , which means g H , n yH, # 0. (ii) Unit Spheres. Let g o be a sphere in R" of radius one passing through the origin. Denoting by C the set of all unit spheres in R", we have the dual homogeneous spaces (6)

R" = M(n)/O(n),

C = M(n)/O*(n),

where O*(n)is the set of rotations around the center of g o . Here a point x = gO(n) is incident to the sphere o = yO*(n) if and only if x E g. (iii) Complex Flag Manifolds. Consider the complex n-space C" and a fixed set of integers 0 < d , < . < d, < n. A Jag in C" of type ( d , , . . . , d,) is a sequence of subspaces L , c L , c .. . c L, c C",where dim Li= di (1 Ii Ir). The set F d , d , of all flags in C" of type ( d , , . . . , d,) is a complex manifold in a natural way. In fact, the general linear group GL(n, C) acts transitively on it with a complex isotropy group. Consider, in particular, the manifolds F , , of flags of type (1, 2) in C4, the complex projective space F , = P3(C) (of complex lines in C4), and

931

3. A Duality in Integral Geometry

I

143

the Grassmann manifold F , = C,,,(C) of complex two-dimensional subspaces of C4.The group U(4) acts transitively on these manifolds F l z , F , , and F,, and our double fibration (4) becomes

F , , = W 4 ) / ( W ) x U(1) x W 2 ) )

F , = W4)/(W) x U ( 3 ) )

F,

=

U(4)/(W) x

W))

-,i, ( + { mean the following: map x -,R associates to a complex

and our maps x

(A) The line the set of two-dimensional complex subspaces c C4 containing it. (B) The map 5 + [ associates to a two-dimensional complex subspace of C4 the set of complex lines contained in it.

These maps A, B have been utilized by Penrose; they are called Penrose correspondences in the paper Wells [1979], to which the reader should refer for an account of their applications.

2. The Radon Transform for the Double Fibration In accordance with the unimodularity assumption (i), we fix invariant measures dg. dh,, dh,, and dh on the groups

G. H , . H , . and H = H x r\ H , , respectively. By the existence theorem for invariant measures on homogeneous spaces (Theorem 1.9) there exists a unique H,-invariant measure dp = d(h,), on 2, = H , / H satisfying

s,, (

S(h,) dhx = JH.l(hXh) dh) 44hX H ) (7) JH, for all , f C~, ( H x ) . Similarly, there is defined an HE-invariant measure dm on = H , / H . We shall now see that by translations we obtain consistently defined invariant measures on each 2.

to

Lemma 3.5. There exists a nonzero measure on each R, coinciding with dp on 2,, such that whenever g .R, = 2,, the measures correspond undei. g . A similar statement holds ,for 4.

Proof. If R = g.R,, we transfer the measure dp on 2, over to R by means of the homeomorphism ( -, g . (. This gives an (H,)g-invariant measure on 2, but since such measures are only determined up to a constant factor, we must still prove independence of the choice of g . But if

144

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

9’. Z0 = g 20, we have (g xo)” = (g’ . xo)”, so that by Lemma 3.2, g . xo = g ’ . x o , whence g ~ g ’ l f , . Since d p on KO is H,-invariant, the lemma follows.

e

The measures on 2 and defined by Lemma 3.5 will be denoted dp and dm, respectively. We also denote by dgHx and d g H E , respectively, the G-invariant measures on X and 6 which are normalized by the relations

(

(8)

J / ( g ) dg = Jx J H r ( g h X ) d h X ) dgHx,

(9) We shall also put d x = dgH,, d t = dgHs for simplicity. We now define the Radon transform f + f a n d its dual 4 + by the formulas

4

~ 4 E Cc(Z). Since g is closed, the restriction .fie belongs to for , f C,(X), C,([). Thus the integrals (10) are well-defined. Formulas (10) can also be written in group-theoretic terms: (1 1)

f(gHE,) =

J

f(ghBHX) d(hdH,

d ( g H X ) = J H X l H 4 ( g h X H dd(hX)H.

HaIH

Proposition 3.6. Let JE C,(X), 4 E Cc(Z). T h e n f and

4 are contin-

uous and

4)

Proof. The continuity off (and of is immediate from (1 1). Next we consider the double fibration (4), where p ( g H ) = g H x , n ( g H ) = gH,. We fix a G-invariant measure dgH on G/H such that for all FEC,(G)

(12)

J/(d

(J-/(gh)

= JG/,

d h ) dg,.

By the “chain rule” for invariant integration (cf. Proposition 1.13 for Lie groups), we have (13)

Q ( g H ) dgH = C JG/H

/

dgHx / H x / H Q ( g h x H )d ( h x ) ~ v

G/Hx

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3. A Duality in Integral Geometry

145

where QEC,(G/H) is arbitrary and c is a constant independent of Q. Combining (12) and (13), we obtain

which by (7) equals F(ghx) dhx * By comparing this with (8) we obtain c = 1. We consider now the function

p = ( f o P)(4 4 on G/H. We integrate it over G / H in two ways corresponding to the two fibrations in (4).This amounts to using (13) and its analog for H,. The result is O

so that the right-hand side of (15) reduces to f ( x ) $ ( x ) d x . Treating (16) similarly, we obtain the proposition. Figure 9 illustrates the equality of (15) and (16). The result shows how to define the Radon transform and its dual for measures and, in case G is a Lie group, for distributions. Definition. Let s be a measure on X of compact support. Its Radon transform is the functional 3 on Cc(E) defined by (17) Similarly, 5 is defined by

3(4) = 44).

(18) 8f)= 431, fc C,(X), if (r is a compactly supported measure on Z.

146

LNTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

/

\ FIG.9

-

Lemma 3.7. (i) If s is a compactly supported measure on X , 3 is a measure on =. (ii) If s is a bounded measure on X and 2o h a s j n i t e measure, then 3 as dejned by (1 7) is a bounded measure. Proof. (i) The measure s can be written as a difference s = s + - sr\

of two positive measures, each of compact support. Then 3 = ? - s is a difference of two positive functionals on C@). Since a positive functional is necessarily a measure, 3 is a measure. (ii) We have

so for a constant K,

and the boundedness of 3 follows.

If G is a Lie group, then (17), (18) with,f€g(X), & E 9 ( Z ) serve to define the Radon transform s --* 3 and the dual CJ 4 6 for distributions s and CT of compact support. We consider the spaces 9 ( X ) and B ( X ) [= Cm(X)] with their customary topologies (Chapter 11, $2, No. 2). The duals 9 ( X ) and b ' ( X ) then consists of the distributions on X and the distributions on X of compact support, respectively.

3. A Duality in Integral Geometry

531

147

Proposition 3.8. The mappings

f E 9 ( X ) +fE (b E 9(E) + f$

qq, E

&(X)

are continuous. In particular, s E B ‘ ( X ) =3 E 9’(E), 0E

&(Z)

*; E 9 ’ ( X ) .

Proof. We have

Let g run through a local cross section through e in G over a neighborhood of to in Z. If ( t l , . . . , t,,) are coordinates of g and ( x , , . . ., x,) the coordinates of x E then (19) can be written in the form

to,

A

F(tl, . . . , t n ) =

J

F ( t , , . . . ,t , ;

XI,.

. . , x,) dxl

a

*

*

dx,.

Now it is clear that f ~ b ( E and ) that f-f is continuous, proving the proposition. The results for the Radon transform on R” in 82 now suggest the following general problems: (A) Relate function spaces on X and on Z by means of the integral transforms , f +,f $ + In particular, relate the supports of .f and of$ ( B ) Relate directly the functions f and (1)”on X , the functions (b and on Z. ( C ) In case G is a Lie group, let D ( X ) and D(Z) denote the algebras of G-invariant differential operators on X and S, respectively. Does there exist a map D + 6 of D ( X ) into D(Z) and a map E --* B of D(E) into D ( X ) such that

4.

(4)-

Theorems 2.4, 2.6, 2.10, 2.13, 2.15, Lemma 2.1, and Corollary 2.5 give answers to these questions for X = R”, E = P”.But while the problems can be posed for the general double fibration (4), one cannot expect complete solutions in this generality. For example, iff+j’is the Radon transform on S2 defined by integration over geodesics, the function (3)’will, in contrast to necessarily be a symmetric function. Here however, assumption (ii) is not satisfied.

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In the next section we consider Problems A and B in detail for the case when X is a two-point homogeneous space, or equivalently, an isotropic Riemannian manifold. Examples. (i) Let G denote the group SL(2, R ) of 2 x 2 matrices of determinant one and r the modular group SL(2,Z). Let N denote the unipotent group (h T), where n E R , and consider the homogeneous spaces

X

(20)

=

G/N,

B = G/T.

Under the usual action of G on R2 N is the isotropy subgroup of (l,O), so that X can be identified with R2 - (0), whereas B is of course threedimensional. In number theory one is interested in decomposing the space L 2 ( G / r ) into G-invariant irreducible subspaces. We now give a rough description of this in terms of the transformsf-tfand 4 + d;. As customary we put ra,= r n N ; our transforms (10) then take the form

f(sV = 1 f(sr), nr,

&gN) =

4 ( g W dnr,. JN/rm

Since N I T , is the circle group, &gN) is just the constant term in the Fourier expansion of the function nr, + $(gnr). The null space L j ( G / r ) in L 2 ( G / r ) of the operator 4 - t $ is called the space of cusp forms. According t o Proposition 3.6 they constitute the orthogonal complement of the image C,(X)^. We have now the G-invariant decomposition

Lz(G/r)= L,Z(G/I') 0 L j ( G / r ) , where (- denoting closure) (21)

and as mentioned above,

L i ( G / r ) = (CC(X)-)*. It is known (cf., Selberg [1962], Godement [1966]) that the representation of G on L:(G/r)/C is the continuous direct sum of the irreducible representations of G from the principal series, whereas the representation of G on L:(G/r) is the discrete direct sum of irreducible representations each occurring with finite multiplicity. (ii) The determination of a function in R" in terms of its integrals over unit spheres (John [1955]) can be regarded as a solution to the first half of Problem B for the double fibration (6).

$31

3. A Duality in Integral Geometry

149

3. Orbital Integrals As before, let X = G / H , be a homogeneous space with origin o =

{Hx}. Given X,EX let G,, denote the subgroup of G leaving xo fixed, i.e., the isotropy subgroup of G at xo. Definition. A generalized sphere is an orbit G;x, in X of some point x E X under the isotropy subgroup at some point xo E X . Examples. (i) If X = R", G = M(n), then the generalized spheres are just the spheres. (ii) Let X be a locally compact group L and G the product group L x L acting on L on the right and left, the element (Il, 1 , ) ~L x L inducing the action 1 + llll;' on L. Let A L denote the diagonal in L x L, so that L = ( L x L)/AL. If I,EL, then the isotropy subgroup of I, is given by (L x

Ul,

= (10, e)AL(I,

l,

el,

and the orbit of 1 under it by ( L x L),, * 1 = l o ( l ; ' l ) L , that is, the left translate by lo of the conjugacy class of the element 1;'l. Thus the generalized spheres in the group L are the leji (or right) translates of its conjugacy classes. Coming back to the general case X = G / H x = G/Go we assume that Go, and therefore each G,,, is unimodular. Writing the orbit G;x, as Gxo/(Gxo)x,we see that if (G,,), is unimodular, the orbit G;x , has an invariant measure determined up to a constant factor. We can then consider the following general problem (following A, B, C above). (D) The Orbital Integral Problem. Determine a function f on X in terms of its integrals over generalized spheres. Remarks. In this problem it is of course significant how the invariant measures on the various orbits are normalized. (a) If Go is compact, the problem above is rather trivial because each orbit G,, . x has finite invariant measure, so that f(xo) is given as the limit as x -+ xo of the average of ,f over G,, x. (b) Suppose that for each X,EX there is a G,,-invariant open set C,, c X containing xo in its closure such that for each x E C, the isotropy group (Gx0), is compact. The invariant measure on the orbits G,, . x (xo E X, x E Cxo)can then be consistently normalized as follows: Fix a Haar measure dg, on Go. If xo = g 0,we have G,, = gGog- and

'

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can carry dg, over to a measure dg,, on G,? by means of the conjugation z + gzg- l ( z E Go). Since dg, is biinvariant, dg,, is independent of the choice of g satisfying x , = g . o and is biinvariant. Since (G,,), is compact, it has a unique Haar measure dg,,,, with total measure 1, and now dg,, and dg,,,, determine canonically an invariant measure p on the orbit G,, - x = G,,/(G,,), (Theorem 1.9). We can therefore state Problem D in a more specific form.

(D’) Express f ( x o ) in terms of the integrals

J

Gxo.s

fb)dP(P),

X E

Go.

For the case when X is a Lorentzian manifold of constant curvature, the assumptions above are satisfied (with C,, consisting of the “timelike” rays from x,) and an explicit solution to Problem D’ can be given (cf. Theorem 6.17 in this chapter). (c) If in Example (ii) above L is a semisimple Lie group, Problem D is a basic step in proving the Plancherel formula for the Fourier transform on L (see notes to 56 at the end of this chapter).

54. The Radon Transform on Two-Point Homogeneous Spaces. The X-Ray Transform Let X be a complete Riemannian manifold, x a point in X and X, the tangent space to X at x. Let Exp, denote the mapping of X, into X given by Exp,(u) = y,,(l), where t + y,,(t) is the geodesic in X through x with tangent vector u at x . A connected submanifold S of a Riemannian manifold X is said to be totally geodesic if each geodesic in X which is tangential to S at a point lies entirely in S. The totally geodesic submanifolds of R” are the planes in R”. Therefore, in generalizing the Radon transform to Riemannian manifolds, it is natural to consider integration over totally geodesic submanifolds. In order to have enough totally geodesic submanifolds at our disposal, we consider in this section Riemannian manifolds X which are two-point homogeneous in the sense that for any two-point pairs p , q E X,p‘, q‘ E X satisfying d(p, q ) = d(p’, 4’) (where d = distance), there exists an isometry g of X such that g p = p‘, g - q = q’. A Riemannian manifold X is said to be isotropic if for each p E X and each pair of unit tangent vectors u, u to X at p there exists an isometry of X leaving p fixed and mapping

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4. Two-Point Homogeneous Spaces

151

u to u. It is well known that a Riemannian manifold is two-point homogeneous if and only if it is isotropic (see, e.g., [DS], pp. 535, 585). We now start with the class of Riemannian manifolds with the richest supply of totally geodesic submanifolds, namely, the spaces of constant curvature.

1. Spaces of Constant Curvature

Let X be a simply connected complete Riemannian manifold of dimension n 2 2 and constant sectional curvature. Lemma 4.1. Let x E X , V a subspace of the tangent space X , . Then ExpJ V ) is a totally geodesic submanifold of X . Proof. For this we choose a specific imbedding of X into R"", and assume for simplicity that the curvature is E (= f 1). Consider the quadratic form B,(x) = x: + ' * ' + x; + Ex;, 1

and the quadric Q, given by B,(x) = E. The orthogonal group O(B,) acts transitively on Q,. The form B, is positive definite on the tangent space R" x (0) to Q, at xo = (0,. . . ,O, 1); by the transitivity B, induces a positive definite quadratic form at each point of Q,, turning Q, into a Riemannian manifold, on which O(B,) acts as a transitive group of isometries. The isotropy subgroup at the point x o is isomorphic to O(n) and it acts transitively on the set of two-dimensional subspaces of the tangent space (Q,),.. It follows that all sectional curvatures at xo are the same, namely, E, so by the homogeneity, Q, has constant curvature E. In order to work with connected manifolds, we replace Q - , by its intersection Q ? , with the half-space x , + ~> 0. Then Q+ and Q T 1 are simply connected complete Riemannian manifolds of constant curvature. Since such manifolds are uniquely determined by the dimension and the curvature (see, e.g., [DS] pp. 227, 250, 564), it follows that we can identify X with Q + , or Q?,. The geodesic in X through x o with tangent vector (1,0,. . . ,0) will be left pointwise fixed by the isometry (Xl,X2,...,X,,X,+,)-'(Xl,

-x

2,...,

-x,,x,+1).

This geodesic is therefore the intersection of X with the two-plane x2 = = s, = 0 in R"". By the transitivity of O ( n ) all geodesics in X through xo arc intersections of X with two-planes through 0. By the transivity of O(Q,) it then follows that the geodesics in X are precisely the nonemptji intersections of' X with two-planes through the origin. 9

.

1

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Now if V c X x o is a subspace, Expxo(V) is by the above the intersection of X with the subspace of Rn+' spanned by V and xo. Thus Expxo(V)is a quadric in V + Rx', and its Riemannian structure induced by X is the same as induced by the restriction B,I(V + Rx'). Thus, by the above, the geodesics in Expxo(V)are obtained by intersecting it with twoplanes in V + R x o through 0. Consequently, the geodesics in ExpXo(V) are geodesics in X, so Expxo(V)is a totally geodesic submanifold of X. By the homogeneity of X this holds with xo replaced by an arbitrary point x E X. The lemma is proved. In accordance with the viewpoint of $3 we consider X as a homogeneous space of the identity component G of the group O(Q,). Let H, denote the isotropy subgroup of G at the point xo = (0,. . . ,0, 1). Then H, can be identified with the special orthogonal group SO(n). Let k be a fixed integer, 1 5 k In - 1; let toc X be a fixed totally geodesic submanifold of dimension k passing through xo and let H, be the subgroup of G leaving to invariant. We have then (1)

X

-

s = G/H,,

= G/H,,

E denoting the set of totally geodesic k-dimensional submanifolds of X. Since xo E to,it is clear that the abstract incidence notion boils down to the naive one, in other words: The cosets x = g H , t = yH, have a point in common if and only if x E t, A. The Hyperbolic Space We take first the case of negative curvature, that is, E = - 1. The transform f + 3 is now given by

(2)

3(t>= J f ( 4 dm(x), t

( being any k-dimensional totally geodesic submanifold of X (1 I k) with the induced Riemannian structure and dm the corresponding measure. From our description of the geodesics in X it is clear that any two points in X can be joined by a unique geodesic. Let d be the distance function on X,and for simplicity we write o for the origin xo in X. Consider now geodesic polar coordinates for X at 0 ; this is a mapping

Expo Y

+

(r, 81,. . .,en- A,

where Y runs through the tangent space X,, r = I Y ( (the norm given by the Riemannian structure), and (el,.. .,On- 1) are coordinates of the unit vector Y/l Y I. Then the Riemannian structure of X is given by (3)

ds2 = dr2

+ (sinh r)'

dc2,

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4. Two-Point Homogeneous Spaces

153

where do2 is the Riemannian structure n- 1

1

gij(e1,.

. . 0,-

1)

dei d e j .

i. j = 1

on the unit sphere in X,. This is easy to see by using the model (15) below for X. Then a point with distance r from the origin o is given by

r

x i = tanh - oi

2

n

where

o? = 1 1

[cf. Introduction, Eq. (12), $4, where the curvature is -41. Substituting this and

(

dx, = - cosh 2

r

oi dr - tanh - d o i

2

into (15) and using oid o i = 0, we obtain the expression (3). The surface area A(r) and volume V ( r ) = JL A(t) dt of a sphere in X of radius r are thus given by

(4)

A(r) = Q,,(sinh r)n-',

V ( r ) = R, J?hn-l

t dt,

so V ( r ) increases like e("-')'. This explains the growth condition in the next result, where d(o, 5 ) denotes the distance of o to the manifold 5. Theorem 4.2. (The Support Theorem) Suppose f~ C ( X ) satisfies

(i) For each integer m > 0, f(x)emd(o,x) is bounded. (ii) There exists a number R > 0 such that

f(5)

=0

for d(o,5) > R.

f(x)=0

for d(o, x) > R.

Then

Taking R = 0, we obtain the following consequence. Corollary 4.3. The Radon transform f+f is one-to-one on the space of continuous functiorts on X satisfying condition (i) of "exponential decrease." Proof of Theorem 4.2. Using smoothing of the form

J.(g)f(g- -

x) 4

[ 4 ~ 9 ( G ) dg , being the Haar measure on GI, we can (as in Theorem 2.6) assume that , f ~&X).

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FIG. 10

We first consider the case when f in (2) is a radial function. Let P denote the point in 5 at the minimum distance p = d(o, 5 ) from o, let Q E 5 be arbitrary, and let q = 40,QX

r = d(P, Q). Since 5 is totally geodesic, d(P, Q) is also the distance between P and Q in t. Consider now the totally geodesic plane n through the geodesics oP and oQ as given by Lemma 4.1. Since a totally geodesic submanifold contains the geodesic joining any two of its points, x contains the geodesic PQ. The angle oPQ being 90" (see, e.g., [DS], Chapter I, Lemma 13.6), we conclude by hyperbolic trigonometry (cf. Exercise C2 and Fig. 10) (5)

cosh q = cosh p cosh r.

Since f is radial, it follows from ( 5 ) that the restriction,fIt is constant on spheres in 5 with center P. Since these have area Rk(sinh r)k-l, formula (2) takes the form

Since f is a radial function, it is invariant under the subgroup H, c G which fixes o. But H, is not only transitive on each sphere S,(o) with center o, it is for each fixed k transitive on the set of k-dimensional totally geodesic submanifolds which are tangent to S,(o). Consequently, f(t) depends only on the distance d(o, 5). Thus we can write f ( Q ) = f'(cosh qh for certain 1-variable functions F and

(7)

f= &cosh fi, so that

P)

by ( 5 ) we obtain

F(cosh p cosh r)(sinh r)k- dr.

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4. Two-Point Homogeneous Spaces

155

Writing here t = cosh p , s = cosh r, this reduces to (8)

P(t) =

/lmF(tS)(Sz

- l)('iZ)k-' ds.

Here we substitute u = ( t s ) - ' and then put o = C-'. Then (8) becomes U-'E(U-')

= i&

{F(U-')U-k}(U2

- U2)("2)k-'

du.

This integral equation is of the form (19) in $2, so that we get the following analog of (20) in $2:

Now by assumption (ii), E(cosh p ) = 0 if p > R . Thus E ( v - ' ) = 0 if 0 < u < (cosh R ) - ' . From (9) we can then conclude that F(u-') = 0 if u < (cosh R ) - ' , which means f ( x ) = 0 for d(o, x) > R. This proves the theorem for f radial. Next we consider an arbitrary J E 8 ( X ) satisfying (i) and (ii). Fix x E X , and if dk is the normalized Haar measure on H,, consider the integral FAY) = J - x / ( g k * Y) dk,

YE

x,

where g E G is an element such that g . o = x. By Theorem 1.9, F , is the average o f f on the sphere with center x passing through g - y . The function F , satisfies the decay condition (i) and it is radial. Moreover,

We now need the following estimate: (1 1)

4 0 , gk

*

5) 2 40, 5) - 4 0 , g

For this let xo be a point on triangle inequality

*

0).

5 closest to k - ' g - ' . o . Then by the

d(0,gk.t)= d(k-'g-'*o,t) 2 d(o, xO) - d(o, k - ' 9 2 d(o, 5 ) - d(o, g * 0).

- 0)

Thus it follows by (ii) that

P,(5) = 0

if d(o, 5 ) > d(o, x)

+ R.

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Since F, is radial, this implies by the first part of the proof that

d(0, y) > d ( o , g . 0)

(13)

+ R.

But the set { g k . y: k E H,} is the sphere $ , ( o , y ) ( g . 0) with center g o and radius d(o, y); furthermore, the inequality in (13) implies the inclusion relation (14)

BR(o)

O)

Bd(o,y)(g

for the balls. But considering the part in B R ( o ) of the geodesic through o and 9 . 0 , we see that conversely relation (14) implies (13). Theorem 4.2 will therefore be proved if we establish the following lemma. Lemma 4.4. Let f~ C ( X ) satisfy the conditions:

(i) For each integer m > 0, f ( x ) e m d ( 0 9isX )bounded. (ii) There exists a number R > 0 such that the surface integral

whenever the sphere S encloses the ball

BR(0).

Then f(x)=0

for d(o, x ) > R .

Proof. This lemma is the exact analog of Lemma 2.7, whose proof, however, used the vector space structure of R". By using a special model of the hyperbolic space we shall nevertheless adapt the proof to the present situation. As before we may assume f is smooth, i.e., f~ &X). Consider the unit ball { X E R": x f < 1) with the Riemannian structure

17

(15)

ds2 = P ( x , , . . . ,x,J2(dx:

+

* *

+ hi),

where P(XI,

... , X " )

= 2(1

- x: - * . * - x i ) - ' .

We know from Exercise C1 that this Riemannian manifold has constant curvature -1, so we can use it for a model of X. This model is useful here because the spheres in X are the ordinary Euclidean spheres inside the ball. This fact is obvious for the spheres X with center 0. For

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157

the general statement it suffices to prove that if T is the geodesic symmetry with respect to a point (which we can take on the x,-axis), then T ( Z ) is a Euclidean sphere. The unit disk D in the x,x,-plane is totally geodesic in X, hence invariant under T. Now the isometries of the nonEuclidean disk D are generated by the complex conjugation x,

+ ix, + x1 - ix,

and fractional linear transformations, so they map Euclidean circles into Euclidean circles. In particular, T(Z n 0)= T ( Z ) n D is a Euclidean circle. But T commutes with the rotations around the x,-axis. Thus T ( Z ) is invariant under such rotations and intersects D in a circle; hence it is a Euclidean sphere. The totally geodesic hypersurfaces in X are the spherical caps perpendicular to the boundary. After these preliminaries we pass to the proof of Lemma 4.4. Let S = S,(y) be a sphere in X enclosing BR(o),and let B,(y) denote the corresponding ball. Expressing the exterior X - B,(y) as a union of spheres in X with center y, we deduce from assumption (ii) (16)

s

f ( x ) dx = Jx.f(x) dx,

WY)

which is a constant for small variations in r and y. The Riemannian measure dx is given by (17)

dx = p” dxo,

where dxo = dx, . - .dx, is the Euclidean volume element. Let ro and yo, respectively, denote the Euclidean radius and Euclidean center of S,(y). Then S,,(y,) = S,(y), B,,(yo) = B,(y) set-theoretically, and by (16) and (17) (18)

,f(xo)p(xo)” dx, = const. s&&d

for small variations in ro and y o ; thus by differentiation with respect to ro (19)

~ s p o ( y l , f ~ s o ~ Pdwo(s0) ~soY =0 9

where dw, is the Euclidean surface element. Putting f * ( 4 = f(X)PW,

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

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we have by (18) f * ( x o ) d x o = const. JB~o~yo,

so, differentiating with respect to yo, we get

J

(a,f*)(Yo

+ x 0 ) d x o = 0.

B,&)

By using the divergence theorem (26) in $2 on the vector field F ( x o ) = f * ( y o + xo) di defined in a neighborhood of Bro(o), the last equation implies f*(yo

+ s)si do,(s)

= 0,

S,,(O)

which in combination with (19) gives

Lv"(yo,

f*(s)si do,(s) = 0.

The Euclidean and the non-Euclidean Riemannian structures on Sro(yo) differ by the factor pz. It follows that do = p(s)"-' do,,so that (20) takes the form

j,.,,,f(s)p(s)si d o ( s )

(21)

= 0.

We have thus proved that the function x + f ( x ) p ( x ) x i assumptions of the theorem. By iteration we obtain (22)

s

satisfies the

f(s)p(s)ksi, . . . Sik dw(s) = 0.

SAY)

In particular, this holds for y = 0 and r > R . Then p(s) = constant and (22) gives f = 0 outside BR(o)by the Weierstrass approximation theorem. Thus Theorem 4.2 is proved. Now let L denote the Laplace-Beltrami operator on X. (See Chapter I1 $2, for the definition.) Because of formula (3) for the Riemannian structure of X,L is given by

a2

a

L = 7 + ( n - l)cothr-+(sinhr)-2L,, ar ar

where L, is the Laplace-Beltrami operator on the unit sphere in X,.

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4. Two-Point Homogeneous Spaces

159

(See also Chapter 11, Proposition 5.26.) We consider also for each the mean value operator M' defined by

I'

20

As we saw before this can also be written

if g E G is arbitrary and y E X is such that r = d(o, y). From Prop. 4.12 in Chapter I1 (which could easily be proved at the present stage) we have the commutativity

(25)

M'L = LM'

and the analog of the Darboux equation

(26)

Lx(F(x, Y ) ) = L,(F(x. Y ) ) for the function F(x, y ) = (Md(",Ylf)(x). For a fixed integer k (1 Ik I n - 1 ) let E denote the manifold of all k-dimensional totally geodesic submanifolds of X . If 4 is a continuous function on 5,we denote by r$ the point function

I < x ) = l E I Q ( C ) dP(<), where p is the unique measure on the (compact) space of 5 passing through x , invariant under all rotations around x and having total measure one. Theorem 4.5. (The Inversion Formula). For k even let Qk denote the polynomial

Qk(z)= [ z

+ ( k - l)(n - k ) ] [ z + (k - 3)(n - k + 2)]

+ .

[z + 1 (n - 2)]

of degree k/2. The k-dimensional Radon transform on X is then inverted by the formula cf = QdLXf)'-,

fEW X ) .

Here c is the constant

The formula holds also for f rapidly decreasing in the sense of condition (i) in Theorem 4.2.

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Proof. Fix ( E S passing through the origin o E X . If x E X , fix g E G such that g . 0 = x. As k runs through H , , g k . 5 runs through the set of all totally geodesic submanifolds of X passing through x and

Hence

-~

M r ) * o0) dm(y),

where r = d(o, y). But since ( is totally geodesic in X , it has also constant curvature -1 and two points in { have the same distance in as in X. Thus we have

<

(f)"(X)

= 0,[om(My)(x)(sinh

r)k- dr.

We apply L to both sides and use (23). Then (29)

( ~ ( j ' j ) ( x=) 0,Jom(sinh r > k - l L , ( ~ y ( x )dr, )

where L , is the "radial part"

a + (n - 1)coth r ar2 ar a2

-

of L. Putting now F(r) = ( M y ) ( x ) ,we have the following result. Lemma 4.6.

Let m be an integer, 0 < m < n = dim X . Then

Jomsinhmr L,F dr =

(m

+ 1 - n)[m Somsinhmr F(r) dr + ( m - 1)

;s

JOrn

1

sinhm-2r F(r)dr

.

If m = 1, the term (m - 1) sinhmn-2 r F(r) dr should be replaced by F(0). This follows by repeated integration by parts.

From this lemma combined with the Darboux equation (26) in the form (30)

L,(M'f(x)) = W W x ) )

041

4. Two-Point Homogeneous Spaces

161

we deduce

[L,

+ m(n - m - l)] =

Jo*

sinh" r ( M r f ) ( x )dr

-(n - m - l ) ( m - 1 )

so*

sinh"-' r ( M ' f ) ( p ) d r .

Applying this repeatedly to (29) we obtain Theorem 4.5. B. The Spheres and the Elliptic Spaces Now let X be the unit sphere S"(0) c R"" and Z the set of k-dimensional totally geodesic submanifolds of X . Each ~ E isEa k-sphere. We shall now invert the Radon transform

f(5)= / f ( x ) dm(x), f E &X), 5

where dm is the measure on 5 given by the Riemannian structure induced by that of X . In contrast to the hyperbolic space, each geodesic in X through a point x also passes through the antipodal point Ax. As a result, 3 = (,f 0 A ) - and our inversion formula will reflect this fact. Although we state our result for the sphere, it is really a result for the elliptic space, that is, the sphere with antipodal points identified. The functions on this space are naturally identified with symmetric functions on the sphere. Again let i ( x >=

lep(e)d A 5 )

denote the average of a continuous function on Z over the set of 5 passing through x. Theorem 4.7. Let k be an integer, 1 I k < n = dim X .

f-3

(i) T h e mapping ( f b~( X ) ) has a kernel consisting of the skew functions (the functions f satisfying f + f A = 0). (ii) Assume k even and let P, denote the polynomial 0

P,.(z) = [ z

-

( k - l ) ( n - k)][z - (k - 3 ) ( n - k

+ 2)] . - . [ z - l(n - 2)]

of degree k/2. The k-dimensional Radon transjorm on X is then inverted by the formulu c(.f + f A ) = P ~ ( L ) ( ( ~ ) " )f~ where c is the constant in (27).

~m),

Proof. We first prove (ii) in a similar way to that in the noncompact case, The Riemannian structure in (3) is now replaced by

ds' = dr'

+ sin'

r da',

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the Laplace-Beltrami operator is now given by L=

(31)

a2 a ar + ( n - I)cot r -ar + (sinr)-'L,

instead of (23), and (3)-(x> = ZZ, J o n ( ~ y ) ( xsink) 1 r dr.

For a fixed x we put F(r) = (M*f')(x). The analog of Lemma 4.6 now reads as follows. Lemma 4.8.

Let m be an integer, 0 < m < n = dim X . Then

Jonsinm r L,F dr

=(n-m-

I)

l)[mJ)in"'rF(r)dr-(m-

1

sin"-'rF(r)dr.

l j m = 1, the term ( m - 1) 1 ; r F(r) dr shoirld be replaced by F(0) F(n). Since (30) is still valid, the lemma implies

+

[L, - m(n - m - I)] =

Jon

sinmI' ( M r f ) ( x ) dr

- ( n - m - l)(m

-

1)

I

sinm-' r ( M r f ) ( x ) dr

and the desired inversion formula follows by iteration, since F ( 0 ) + F(n) = j(x)

+ f(Ax).

In the case when k is even, part (i) follows from (ii). Next suppose k = n - 1, n even. For each there are exactly two points x and A x at maximum distance, namely, n/2, from 5, and we write

<

P(X> =

3(A4 =

3co.

We have then (32)

3 w = WMZ'2f)(X).

Next we recall the decomposition a2

(33)

P(X)=

1E , 0

from 53 of the Introduction. Here the space E , consists of the restrictions to X of the homogeneous harmonic polynomials on R"+' of degree s. The space is the eigenspace of L for the eigenvalue -s(s + n - 1 ) (s 2 0). As proved there, E, contains a unique function 4, which is

4. Two-Point Homogeneous Spaces

a41

163

invariant under the group K of rotations around the vertical axis (the x,+ ,-axis in R"' I), and satisifies +,(o) = 1, o being the north pole. Since the mean value operator Mni2 is invariant under O(n + l), which in turn acts irreducibly on E,, we see from Schur's lemma that M n i 2 acts as a scalar c , ~on E,. The equations

Mn'24s= C,4sr

4s(o) = 1

then imply

(34)

c, = 4,(n/2).

Lemma 4.9.

The scalar 4,(n/2) is zero if" and only if" s is odd.

Proof. Let H , be the K-invariant homogeneous harmonic polynomial whose restriction to X equals &. Then H , is a polynomial in

+

(x:

+ xi)

and

x,+~.

Thus we see that if the degree s is odd. x , + ~ occurs in each term; hence 4s(n/2) = H,(I, 0 , . . . , O , 0) = 0. If s is even, say s = 2d, we write H , = u,,(x:

+ ...+

+ a l x I + 1 ( ~ :+

*.*

+ XI)"' + ... + u,,x,Z:~.

By using L,+l = L, + C?2/d~i+, and formula (31) in $2, the equation L,+ , H , = 0 gives the recursion formula

ai(2d - 2i)(2d - 2i

+ n - 2) + ai+,(2i + 2)(2i + 1) = 0

(0 Ii < d). Hence H,(l, 0,. . . ,O), which equals a,, is #O. This proves

the lemma. Now by Theorem 3.3 in the Introduction, each f ~ c f ( X )has a uniformly convergent expansion,

and by (32) m

3 = Q, c0 csh,. If 3 = 0. then by Lemma 4.9. h, = 0 for s even, so f is skew. Conversely. i f f is skew; thus Theorem 4.7 is proved for the case k = n - 1, n even. If k is odd, 0 < k < n - 1, the proof just carried out shows that f ( t ; ) = 0 for all g E 6 implies that f has integral 0 over every (k + 1)-dimensional sphere with radius 1 and center 0. Since k + 1 is even and < n , we conclude by (ii) that j + j o A = 0, so that the theorem is proved.

f =0

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INTEGRAL GEOMETRY A N D RADON TRANSFORMS

[Ch. I

2. Compact Two-Point Homogeneous Spaces We shall now extend the inversion formula in Theorem 4.7 to compact two-point homogeneous spaces X of dimension n > 1. By virtue of Wang’s classification [1952] these are also the compact symmetric spaces of rank one, so their geometry can be described very explicitly. Here we shall use some geometric and group-theoretic properties of these spaces [(i)-(vii) below] and refer to [DS], Chapter VII, $10, for their proofs. Let U denote the group I(X) of isometries of X. Fix an origin o E X and let K denote the isotropy subgroup U , . Let f and u be the Lie algebras of K and U , respectively. Then u is semisimple. Let p be the orthogonal complement of t in u with respect to the Killing form B of u. Changing the distance function on X by a constant factor, we may assume that the differential of the mapping u + u a 0 of U onto X gives an isometry of p (with the metric of - B ) onto the tangent space X , . This is the canonical metric on X which we shall use. Let L denote the diameter of X, that is, the maximal distance between any two points. If X E X ,let A,T denote the set of points in X of distance L from x. By the two-point homogeneity the isotropy subgroup U , acts transitively on A,; thus A, c X is a submanifold, the antipodal manifbld associated to x. (i) Each A, is a totally geodesic submanifold of X ; with the Riemannian structure induced by that of X it is another two-point homogeneous space. (ii) Let 5 denote the set of all antipodal manifolds in X;since U acts transitively on Z, the set E has a natural manifold structure. Then the mapping j : x + A , is a one-to-one di#eomorphism; also, x E A, i f and only i f YEA,. (iii) Each geodesic in X has period 2L. If X E X ,the mapping Exp,: X , + X gives a diffeomorphism of the ball B,(O) onto the open set X - A,.

Fix a vector H E p of length L [i.e., L2 = - B ( H , H ) ] . For 2 E p let T, denote the linear transformation Y + [ Z , [ Z , Y]] of p, [ , 3 denoting the Lie bracket in u. For simplicity we now write Exp instead of Exp,. A point Y E p is said to be conjugate to o if the differential d Exp is singular at Y . The line a = RH is a maximal abelian subspace of p. The eigenvalues of TH are 0, a(H)’, and possibly ( $ Y ( H ) ) ~ ,where + a (and possibly &;a) are the roots of u with respect to a. Let (35)

P

=a

+ P a + PW2).

§41

4. Two-Point Homogeneous Spaces

165

be the corresponding decomposition of p into eigenspaces; the dimensions q = dim(p,), p = dim(p(l/2)a)are called the multiplicities of a and *a, respectively. (iv) Suppose H is conjugate to o. Then Exp(a + pa), with the Riemannian structure induced by that of’ X , is a sphere, totally geodesic in X , having o and Exp H as antipodal points, and having curvature n2JL2. Moreover, AExpH =

(v) I f H is not conjugate t o

Exp(p(1/2)a)*

0, then

p(l12)a= 0 and

Exp Pa. (vi) The diferential at Y of’ Exp is given by AExpH

=

d Exp, = dz(exp Y )

a, 0

0

(TYIk (2k + l ) ! ’

where for u E U , T ( U ) is the isometry x -,u . x. (vii) By analogy with (23) the Laplace-Beltrami operator L on X has the expression

where Lsr is the Laplace-Beltrami operator on S,(o) and A(r) its area (Chapter 11, Proposition 5.26). Lemma 4.10. The surface area A(r) (0 < r < L ) is given by

A(r) = R,I-P(22)-q sinP(lr) sinq(21r), where p and q are the multiplicities above and I = 1 a(H)I J2L. Proof. Because of (iii) and (vi), the surface area of S,(o) is given by

where dw, is the surface element on the sphere I Y I = r in p . Because of the two-point homogeneity, the integrand depends on r only, so that we can evaluate it for Y = H , = (r/L)H. Since the nonzero eigenvalues of TH, are a(H,)’ with multiplicity q and (+a(H,))2 with multiplicity p ; a trivial computation gives the lemma. We consider now Problems A, B. and C from $3. No. 2. for the homogeneous spaces X and =, which are acted on transitively by the same group U . Fix an element 5, E S passing through the origin o E X. If 5, =

166

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

A,. then an element u E U leaves 5, invariant if and only if it lies in the isotropy subgroup K’ = U,,; we have the identifications

X

’

=

UJK,

=

U/K’,

and x E X and 5 E E are incident if and only if x E 5. On E we now choose a Riemannian structure such that the diffeomorphism j: x + A, from (ii) is an isometry. Let L and A denote the Laplacians on X and E, respectively. With 1 and [ defined as in 83, No. 1 we have

g = 5,

2 = M y ) : y Ej(X)l;

the first relation amounts to the incidence description above, and the second is a consequence of the property x E A, o y E A , listed under (ii). The sets 2 and [ will be given the measures d p and dm, respectively, induced by the Riemannian structures of E and X. The Radon transform and its dual are then given by

3<5> = J f ( x ) dm(x), c:

6 w = /$({I

dP(5).

But i < x >=

/

dp(5) =

i

/

#(j(Y)) dp(j(y)) =

YE j ( x )

J

(4 o j ) ( y )dm(y),

j(x)

so that

6 = (4 0j)-

(36)

0

j.

6,

Because of this correspondence between the transforms f + 3, 4 + D(X)denote the algebra of differential operators on X invariant under U. It can be shown (cf. Chapter 11, $4) that D(X)is generated by L. Similarly, D(S) is generated by A. it suffices to consider the first one. Let

Theorem 4.11. (i) The mapping f ping o f & ( X ) onto &(=) and

-,f

is a linear one-to-one map-

(Lf)-= A.! (ii) Except for the case when X is an even-dimensional elliptic space,

f

= P(LM3)- ),

f

E

&n

where P is a polynomial, independent off, explicitly given below, (44)-(50). I n all cases

degree(P) = 4 dimension of the antipodal manifold.

941

4. Two-Point Homogeneous Spaces

167

Proof. We first prove (ii). Let dk be the Haar measure on K such that j d k = 1 and let R, denote the total measure of an antipodal manifold in X . Then it($ = m(A,) = R, and if u E U ,

Hence

= Ox

( W ) ( u .o>dm(y),

where r is the distance d(o,y) in the space X between o and y. If d(o, y ) < L, there is a unique geodesic in X of length d(o, y) joining o t o y , and since to is totally geodesic, d(o, y ) is also the distance in to between o and y . Thus using geodesic polar coordinates in 5, in the last integral, we obtain (37)

where A , ( r ) is the area of a sphere of radius r in have

5,. By Lemma 4.10 we

A , ( r ) = C , sinP'(L,r) sinq'(2A,r),

(38)

where C , and L I are constants and p , , q 1 are the multiplicities for the antipodal manifold. In order to prove (ii) on the basis of (37), we need the following complete list of the compact symmetric spaces of rank one and their corresponding antipodal manifolds :

~

~~

Sphcres Real projective spaces Complex projective spaces Quaternion projcctive spaces Caylcy plane

= I , 2,. . .) P"(R) (n = 2 , 3 , . . .) P"(C) (n = 4, 6 , . . .) P"(H) (ti = 8, 12,. . .) P6(Cay)

S" (n

Point

P"-'(R) P"-'(C) P"-4(H) S8

Here the superscripts denote real dimension. For the lowest dimension, note that P'(R) = S',

P ( C ) = s2,

P4(H) = s4.

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

For the case S",(ii) is trivial, and the case X = P"(R) was already done in Theorem 4.7. For the remaining cases, a and +a are both roots, so by (v) H is conjugate to o; hence we have the properties in (iv). Since a closed geodesic in A, is a closed geodesic in X,we have L = diameter X = diameter A, = distance =

from o to the nearest conjugate point in X,

smallest number M such that lim A(r) = 0. r+M

The multiplicities p and q for the rank-one symmetric spaces were determined by Cartan [1927] (see also Araki [1962], or [DS], p. 532) and we can now derive the following list:

x = P"(C): p =n

-

2, q = 1, I = x/2L, sin'n-2)(Ar)sin(2Ar),

A(r) = 1R 2 n

A , ( r ) = 4Rn-2A-n+3 ~in'"-~'(Ar) sin(2Ar).

x = P"(H): p

=

n

3, A = n/2L,

- 4,q =

A(r) = $R,A-"+ ~in'"-~)(Ar) sin3(2Ar), A , ( r ) = &,-4A-n+5

sin"-*(Ir) sin3(21r).

x = P'6(Cay): p = 8, q = 7, I

=

71/2L,

A(r) = (1/27)R161l 5 sin*(Ir) sin7(2Ar), A , ( r ) = R, sin7(2Ar).

In all cases we have

1, L

(39)

R,

=

m(A,) =

A , @ ) dr.

Thus A(r) and A , @ ) are in all cases expressed in terms of n and L. But L can also be expressed in terms of n. In fact, the Killing form metric has the property

w, H ) c B(HY =

8

§41

4. Two-Point Homogeneous Spaces

169

as j? runs over the roots. Thus -L 2 =

- IH12

=

2p(+a(~))' + 2 q ( a ( ~ ) ) ~ .

But by (iii) H is the first conjugate point in the ray R'H, so that Icr(H)I = IC (see [DS], p. 294). Thus we get the formula L2 = p(n2/2)

(40)

+ 2qn2.

Lemma 4.12. I n the Killing form metric the diameters L of the projective spaces

P"(H), P' 6(Cay)

P"(C>, are given by, respectively,

(; +

(; +

y2n,

4)1'2n,

3JzIC.

As already used for spaces of constant curvature we have here the commutativity [cf. (25)]

M'L = LM'

(41) which implies [cf. (26)]

Lx((M'f)(x)) = Lr((M'f)(x)),

(42)

where by (vii) and Lemma 4.10 a2

+

L, = 7 l{p cot(lr) ar

a + 2q cot(2lr)) ar

(0 < r < L).

We now apply the Laplacian to (37) and use (42) in order to simplify the right-hand side. The result is reduced by repeated use of the following three lemmas. Lemma 4.13. Let X = P"(C), f~ Ca(X). If m is an even integer, 0 I m I n - 4, then

(L,

-

A2(n - m - 2)(m

=

+ 2))

-12(n - m - 2)m

sin"(Ar) sin(2lr) [ M ' f ] ( x )dr

1

sin"-2(lr) sin(2lr) [ M ' f ] ( x )dr.

For m = 0 the right-hand side should be replaced by -2l(n

-

2)f(x).

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

Lemma 4.14. Let X 0 c m I n - 8. Then

=

P"(H), f~ Cm(X). Let

( L , - A2(n - m - 4)(m + 6)) =

-A2(n - m

-

4)(m

sin"(Ar) sin3(2Ar)[M'f](x) dr

+ 2)

I

sin"-2(Ar) sin3(2Ar)[M'f](x) dr.

( L , - 6A2(n - 4))(L, - 4A2(n - 2))

16A3(n - 2)(n - 4)f(x).

Lemma 4.15. teger. Then

be an euen integer,

IoL

Also,

=

m

[Ch. I

I

sin3(2Ar)[M'f](x) dr

Let X = P16(Cay), j e Cm(X). Let m > 1 be an in-

( L , - 4A2m(1 1 - m))

I:

sin"(2Ar) [M'f](x) dr

L

=

-32A2(m - 1) Jo ~in"-~(21r) cos2(Ar)[Mlf](x)

+ 4A2(m - I)(m - 7) (L, - 4A2(m + 1)(10 - m))

s:

[M'f](x) dr;

6

sin"(2Ar) cos2(Ar)[M'f](x) dr

L

= 4A2(3m -

5) Jo sin"(2Ar) [Mlf'](x) dr

+ 4A2(m - l)(m - 15) Also,

(43)

( L , - 72A2) =

I

~in"-~(2Ar) cos2(Ar)[Mrf](x) dr.

s:

sin(2Ar) cos2(lr)(M'f)(x) dr

- 8A2 JoLsin(2Ar) (M'f)(r) - 28Af(x).

These lemmas are proved by means of long computations. Since the methods are similar for all cases, let us just verify the last formula (43). Here we have L,

a2

a + A { 8 cot(Ar) + 14 cot(2Ar)) ar

=7

ar

§41

4. Two-Point Homogeneous Spaces

171

Hence putting F(r) = ( M ‘ f ) ( x ) , we have by (42) L, IoLsin(21r)cos2(Ir)( M ‘ f ) ( x )dr

+ (441cos4(1r) - 141cos2(1r))F‘(r)] dr

=

IOL[sin(21r)cos2(1r) F”(r)

=

IoL[36I cos4(1r) - 81 cos’(Ir)]F’(r) dr

=

- 281F(O) - I’

I

F(r)[sin(Ur) (8 - 72 cos’(Ir))] dr,

which gives formula (43). We can now prove (ii) in Theorem 4.11. Consider first the case X = PI6(Cay). We have ( f ) ” ( x ) = R,R,

I

( M ‘ f ) ( x ) sin7(21r) dr.

Here

L’

=

R,

18n2,

= R,

[oLsin7(21r) dr,

I = 71/2L.

Taking m = 7 in Lemma 4.15, we get (L, - 1121’) =

IoL

(M‘j)(x) sin7(21r) dr

- 192A2IoL(My)(x)sin5(21r) cos’(Ir) dr,

and then taking m = 5 we get (L,

-

12012)(L, - 1121’)

IoL

( M ‘ f ) ( x )sin7(21r) dr

I0 L

= (- 1921’)(401’)[

-4

IoL

( M ‘ f ) ( x )sin5(21r) dr

1

(M‘f)(x) sin3(21r) cos2(Ir)dr .

172

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

Taking m = 5 again, we get ( L , - 12012)2(L, - 1 1212)

= (-

s:

( M ' f ) ( x )sin'(21r) dr

[

1921')(40A2) ( - 128A2)

JoL

sin3(21r) cos2(1r) ( M y ) ( x )dr

- 32A2 JoLsin3(21r)( M ' f ) ( x )dr

1

( M ' f ) ( x )sin3(21r) cos2(1r) dr .

Taking m

=

3, the last term is found to be

-4(L,

=

-

1121' - 8A2)

JoL

( M l f ) ( x ) sin3(21r) cos2(lr)dr

-64A2 JoL(M!f)(x)sin3(21r)dr

+ 384A2 JoL(M'f)(x)sin(21r) cos2(1r)dr

+ 321'

JoL(Mrf)(x)sin3(21r) cos2(1r)dr.

Hence ( L , - 12012)2(L, - 1 12A2)

=

192 4 0 . 9 6 .['A +

JoL

( M y ) ( x )sin'(21r) dr

JoL(My)(x)sin'(2Ar)

+ JoL(My)(x)sin3(21r) dr

1

( M ' f ) ( x ) sin(21r) cos2(1r)dr .

cos2(1r)dr

§41

4. Two-Point Homogeneous Spaces

Finally, we apply the operator ( L , in Lemma 4.15, we get

-

11212) to both sides. Taking m=3

I

(L, - 1 1212) (M'f)(x) sin3(21r)cos2(lr) dr L

=

1612Jo (Mlf)(x) sin3(21r)dr

-9612 JoL(Mlf)(x) sin(2lr) cos2(lr) dr,

(M'f)(x) sin3(21r)dr

=

- 1612JoL(M'f)(x) sin3(21r)dr - 6412 JbL(M'f)(x)sin(2lr) cos2(lr) dr - 3212 IOL(Mlf)(x)sin(2lr) dr

6

-4(L, - 112A2)

(M'f)(x) sin(2lr) cos2(lr) dr

= 16012 joL(M'f)(x) sin(2lr) cos2(lr) dr

+ 3212 IoL(M'()(x) sin(2lr) dr + 1 12lf(x). Fortunately, all terms except the last one cancel out and we obtain (L,

- 1 1212)2(L, -

=

173

I

12012)2 (M'f)(x) sin'(2lr) dr

192 * 40 * 96 * 112 * l'f(x).

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

If we now substitute the values

[Ch. I

72, we get

=

( L , - W 2 ( L , - Y)'(f>"<X> =

&(

/onsin7 s ds) 192 . 4 0 . 9 6 - 112A'j(x)

0;

Thus we have proved for X = P16(Cay),

f = P(L)(fL

f E Crn(X),

where

36 P(L) = -( L - $)2(L 2213

(44)

For X

=

Y)'.

P"(C) we find similarly from Lemma 4.13 the formula

f = P(L)((f)")' where, since A - 2 = 2(n

. f c $(XI

+ 2), 2(n

+ 2)

with

+ 2)]1-("'2).

c = [-8n2(n

(46)

For X = P"(H) we derive from Lemma 4.14 the formula

f = P(L)((f)"), where, since A- = 2(n

f e W),

+ 8), L--

*

2(n

+ 8)

with (48)

c = $[ -47t2(n

+ 8)]Z-("'2).

Finally, we determine P ( L ) for the case X = P"(R) because now the metric on X is normalized by means of the Killing form of U = I ( X ) rather than by the curvature + 1 condition in Theorem 4.7. Instead of the functions on P"(R) we shall deal with even functions f on the sphere

91

4. Two-Point Homogeneous Spaces

175

s" and define f(<)as the integral of J' over the totally geodesic ( n - 1)sphere 5. In order to preserve (36) we define &x) as an integral over an ( n - 1)-sphere (of unit normal vectors to hyperplanes through x). The Killing form metric on S" is obtained by multiplying the usual Riemannian structure (curvature + I ) by 2n; (see [DS], solution to Exercise 7, Chapter V). The new Laplacian is therefore 1/2n times the Laplacian in Theorem 4.7. Thus we have by Theorem 4.7, with k = n - 1, n odd,

P(LXf)" =

f ?

where

c a constant. To find c we apply the formula to the function ,f E 1. Since

1 ~ , ( 2 ~ ) ( 1 / 2 ) (1n),we have (50)

(IT = ~ , 3 2 n ) n - ' ,

= K-4n2n)(1/2)(I-n).

This concludes the proof of part (ii) of Theorem 4.1 1 . That f + f is injective as follows from (ii) except for the case X = Pn(R),

n even.

But in this exceptional case the injectivity follows from Theorem 4.7. For the surjectivity we use once more the fact that the mean value operator M' commutes with the Laplacian [cf., (41)]. We have

(51)

3O'(x)) = C(MLf)(X)?

where c is a constant. Thus by (36)

so

(52)

(3)" = C5W-MLf

Thus if X is not an even-dimensional projective space f is a constant multiple of MLP(L)MLf, which by (51) shows f + f surjective. For the case X = P"(R), n even, we use Theorem 4.7. If the map f + f were not surjective there would by (51) be a nonzero distribution T on X such that

(53)

T ( M L f ) = 0,

f E &(X).

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

Now take ,f to be an eigenfunction of L ; then, as used earlier, f is an eigenfunction of M L and, by the injectivity, with a nonzero eigenvalue. Thus (53) implies the contradiction T = 0. It remains to prove ( L f ) A = At. For this we use (36), (41), and (51). By the definition of A we have x E X,

(A$)(j(x)) = L ( 4 oj)(x),

4 EW).

Thus (Af)(.j(x))= ( U f o j M = cL(MLf)(x)= cML(Lf)(x) = (L!f)-cj(x)). This finishes the proof of Theorem 4.11. Corollary 4.16. Let B be an open set in R"'l, symmetric and star shaped with respect to 0, bounded by a hypersurface. Assume for a fixed k (1 I k c n ) (54)

for all (k

Area@ n P) = const.

+ 1)-planes P through 0. Then B is an open ball.

In fact, we know from Theorem 4.7 that if f is a symmetric function on X = S" with f ( S n n P) constant (for all P), then f is a constant. We apply this to the function

e E sn,

f(e) = p(elk+

1,

if p(0) is the distance from the origin to each of the two points of intersection of the boundary of B with the line through 0 and 8; ,f is welldefined, since B is symmetric. If 0 = (el,,. . ,6,) runs through the ksphere S" n P, then the point x = re

(0 Ir < p(8))

runs through the set B n P and Area(B n P) =

1

dw(8) JOp(")?

dr.

S"nP

It follows that Area(B n P) is a constant multiple of f(s" n P), so (54) implies that f is constant. This proves the corollary. Remark. R. Michel has in [1972] and [1973] used Theorem 1.11 to prove certain infinitesimal rigidity properties of the canonical metric on the real and complex projective spaces.

§41

177

4. Two-Point Homogeneous Spaces

3. Noncompact Two-Point Homogeneous Spaces Theorem 4.1 1 has an analog for noncompact two-point homogeneous spaces, which we shall now describe. By Tits's classification [1955] of isotropic homogeneous spaces (Riemannian or not) these spaces are the Euclidean spaces and the noncompact symmetric spaces X = G / K of rank one. (This is proved without classification in Nagano [I9593 and Helgason [1959].) Let d denote the distance in X and o the origin, that is, the coset {K}. Let g = f + p be the direct decomposition of the Lie algebra g of G into the Lie algebra f of K and its orthogonal complement p (with respect to the Killing form of g). Fix a one-dimensional subspace a c p and let

P = a + Pa + P(1/2)m be the decomposition of p into eigenspaces of TH [by analogy with (35)]. (55)

Let 5, denote the totally geodesic submanifold Exp(p{,,,,,); in the case p(l,2k = 0 we put 5, = Exp(p,). By the classification and duality for symmetric spaces we have the following complete list of the spaces G / K . X

€0

Real hyperbolic spaces Complex hyperbolic spaces Quaternion hyperbolic spaces Cayley hyperbolic space

H"(R) ( n = 2, 3 , . . .) H"(C)( n = 4,6,. . .) H"(H) ( n = 8, 12,. . .) H' 6(Cay)

H"-'(R) H"-'(C) H"-4(H) HEW

Here the superscript denotes the real dimension; for the lowest dimension note that

H'(R) = R,

H2(C)= H2(R),

H4(H) = H4(R).

Let E denote the set of submanifolds g . 5, of X as g runs through G; E is given the canonical differentiable structure of a homogeneous space. Each E E has a measure rn induced by the Riemannian structure of X , and the Radon transform on X is defined by

The dual transform

+

6 is defined by

178

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

<

where ,u is the invariant average on the set of passing through x . Let L denote the Laplace-Beltrami operator on X, the Riemannian structure being that given by the Killing form of g. Theorem 4.17. The Radon transform f + I is a one-to-one mappint of 9(X)into 9 ( Z ) and, except for the case X = H"(R), n even, is inverted by the ,formula f = Q(L)((3)")9

the polynomial Q being given as follow:

X

=

v)b

H"(R), n odd:

Q(L)= y ( L

+

7) (n - 4)3 ...

x = H"(C): ...

x = H"(H): (L

+

H).

x = H'6(Cay): Q ( L ) = y(L

+ Y)'(L+ y)'.

The constants y are obtained from the constants c in (50) by multiplication by rhe factor 0,in (39).

(a), (46), (48), and

We omit the proof since it is analogous to that of Theorem 4.1 1. The adjustment from the constants c to the constants y is caused by the difference in the normalizations of the dual Radon transform in the compact case and the noncompact case. 4. The X-Ray Transform on a Symmetric Space

Let X be a complete Riemannian manifold of dimension > 1 in which any two points can be joined by a unique geodesic. By analogy with R" we can define the X-ray transform f -+fon X by the formula (56)

3w = j-f (4ds(x), Y

y being any complete geodesic in

X , ds the element of arc length, and f

WI

4. Two-Point Homogeneous Spaces

179

any continuous function on X for which the integral converges. By analogy with the X-ray reconstruction problem on R" (52, No. 7b) one can consider the problem of inverting the X-ray transform f-3 on X . With d denoting the distance in X and o E X some fixed point, we now define two subspaces of C ( X ) . Let f E C ( X ) : sup d(o, x ) I~f (x)l < co for each k 2 0 X

i

I

9 ( X ) = f E C ( X ) : sup ekJ(o*x)l f (x)l < co for each k 2 0 . X

Because of the triangle inequality these spaces do not depend on the choice of 0. We can informally refer to F ( X ) as the space of continuous rapidly decreasing functions and to 9 ( X ) as the space of continuous exponentially decreasing functions. We shall now prove the analog of the support theorems (Theorems 2.6 and 4.2) for the X-ray transform on a symmetric space of the noncompact type. This general analog turns out to be a direct corollary of the Euclidean case and the hyperbolic case, already done. Corollary 4.10. Let X be a symmetric space of the noncompact type, B any ball in M .

(i) If a function f E ~ ( Xsatisfies ) (57)

f(y) = 0

whenever y n B

=@,

then (58)

f(x)=0

for x $ B.

In particular, the X-ray transjorm is one-to-one on .F(X). (ii) l j X has rank greater than one, statement (i) holds with 9 ( X ) replaced by F ( X ) . Proof. Let o be the center of B, r its radius, and let y be an arbitrary geodesic in X through 0. Assume first X has rank greater than one. By a standard conjugacy theorem for symmetric spaces y lies in a two-dimensional, flat, totally geodesic submanifold of X. Using Theorem 2.6, on this Euclidean plane we deduce f ( x ) = 0 if x E y, d(o, x) > r. Since y is arbitrary, (58) follows. Next suppose X has rank one. Identifying p with the tangent space X o , let a be the tangent line to y. We can then consider the eigenspace decomposition (55). If b c pa is a line through the origin, then

S = Exp(a

+ b)

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. 1

is a totally geodesic submanifold of X. To see this we verify that the subspace a + b c p is a Lie triple system. In fact, if H E a, Y E b are any nonzero vectors, we have for the Cartan involution 0 of g with respect to k, Y = Z - OZ, where [ H , Z ] = a(H)Z. If follows that [ Y , [ Y , HI1 = a,

which together with [ H , [ H , Y ] ] = a(H)'Y implies that a + b is a Lie triple system. Hence by [ D S ] , Theorem 7.2, Chapter IV, the set S = Exp(a

+ b)

is a totally geodesic submanifold of X . Being two-dimensional, nonflat, and simply connected, S is necessarily a hyperbolic space. From Theorem 4.2 we can therefore conclude that f ( x ) = 0 for x E y, d(o, x) > r. Now (58) follows since y is arbitrary. For a Riemannian manifold all of whose geodesics are closed the analog of (56) can be considered (with the integration taken over the finite length of the geodesic), and again we can consider the injectivity question for the transform f + 3. Corollary 4.19. Let X be a compact symmetric space of rank one. Let f be a continuous function on X satisfying

if (x) ds(x) = 0 for each (closed) geodesic y in X , ds being the element o f a r c length.

(i) If X is a sphere, f is skew. (ii) If X is not a sphere, f = 0. Taking a convolution with f we may assume f smooth. Part (i) is already contained in Theorem 4.7. For part (ii) we use the classification. For X = Pl6(Cuy) the antipodal manifolds are totally geodesic spheres; thus, using part (i), we conclude that 3 = 0, so that by Theorem 4.11, f = 0. For the remaining cases P"(C) (n = 4 , 6 , . . .) and P"(H) (n = 8, 12, . . .), (ii) follows similarly by induction because the initial antipodal manifolds, namely, P2(C)and P4(H), are totally geodesic spheres. 95. Integral Formulas In this section we derive some integral formulas for semisimple Lie groups; the formulas, which to a large extent are due to Harish-Chandra relate the Haar measures of the groups entering in the Iwasawa, Cartan,

051

5. Integral Formulas

181

and Bruhat decompositions. Sometimes we use the general tools developed in $1 : however, it is often better to use ad hoc invariance considerations.

1. Integral Formulas Related to the lwasawa Decomposition Let G be a connected semisimple Lie group, g its Lie algebra. Let 8 be a Cartan involution of g, g = f + p the corresponding Cartan decomposition, a c p a maximal abelian subspace, and Z the corresponding set of restricted roots (cf. [DS],Chapter IX). Fix a Weyl chamber a+ c a; let Z’ denote the corresponding set of positive roots and n = g, the nilpotent algebra spanned by the root subspaces for the positive restricted roots. Let m, = dim&) and put p = 9 CIEz+ (m,)a. If A and N denote the analytic subgroups of G with Lie algebras a and n, respectively, we have the Iwasawa decompositions

EXEX+

g =t

+a +n

(direct vector space sum),

G = KAN,

that is, the mapping (k, a, n ) + kan is a diffeomorphism of K x A x N onto G. If a E A , let log a denote the unique element H E a such that exp H = a. As usual, M will denote the centralizer of A in K, and m its Lie algebra. Proposition 5.1. Let G = K A N be an Iwasawa decomposition of a connected semisimple Lie group G. Let dk, da, and dn be left-invariant measures on K , A , and N , respectively. Then the left-invariant measure dg on G can be normalized so that

J G f ( g )d g =

1

f(kan)e2P(’ogcr) dk da dn

( f E Cc(G)).

KxAxN

Proof. While this can be derived from Proposition 1.12, we proceed more directly. Since the mapping (k, a, n ) + kan is a diffeomorphism, there exists a function D(k, a, n ) on K x A x N such that (1)

J G f ( g )d g =

S(kan)D(k, a, n) dk da dn,

f

E C,(G).

KxAxN

The groups G, K, A , N are all unimodular, (Proposition 1.6). In particular, the left-hand side of (1) does not change if we replace f ( g ) by

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

f(klgn,) (k, E K,n , E N). It follows that D(k; ' k , a, nn; ') = D(k, a, n), so D(k, a, n) is a function 6(a) of a alone. Next, let a , E A. Then

JGf(B) dg

=

f(kana,)W dk da dn

Jcf(gal) dg = JKAN

-

f(kaa,(a; 'na,)b(a)dk da dn L A N

LAN

which by (2) in 01 equals

f(ka(a; 'na,))b(aa; ') dk da dn,

f(kan)d(aa; ') dk da (Z(al)*(dn)). L A N

Here Z(al) denotes the automorphism n + a,na;' of N . It maps the Haar measure dn into another Haar measure. This new Haar measure is a multiple of dn by the factor det(dZ(a,),) = det(Ad(a,)l n) = e2p(10grrl).

(2)

Comparing with (I), we obtain 6(a) = 6(aa;

')e2p(loga1),

so that putting a = e, the proposition follows. Corollary 5.2.

Zf F E CJAN), we haue for a E A, J ~ ( n a )dn = eZp(lo*a)

This is a consequence of (2); equivalently we can proceed as follows: The left-hand side is

F

Corollary 5.3. For f~ C, ( G ) we haue

f ( k n a ) dk dn da =

f(g) dg = JG

JKNA

The first identity follows from Proposition 5.1 and Corollary 5.2. By replacing f by the function f :g + f(g-'), the second identity follows from the first one.

5. Integral Formulas

$51

183

Let a' c a denote the subset of regular elements, that is, elements H such that a ( H ) # 0 for a11 ct E Z. Let A' = exp a'. Lemma 5.4.

Let H E a' and h = exp H. Then the mapping

t: n -+

h-lnhn-'

is a diffeomorphism 0s N onto N. Proof. Clearly 5 maps N into N. Next we calculate the differential d t , at n E N. Let X ~ n Then . { ( n exp t X ) = h-'n exp t X hn-'n exp(-tX) n - l = h- 'nhn- exp(Ad(nh- ' ) t X ) exp Ad@) (- t X ) .

'

Thus

(3)

d{,(dL(n)X) = dL(h- 'nhn- ') Ad(n) (Ad(h- ') - 1) X.

Since H is regular, Ad(h-') has no nonzero fixed points on n, so { is everywhere regular. Thus there exist open neighborhoods U and I/ of e in N such that { is a diffeomorphism of U onto V . We have

U

= exp

U , , V = exp V,,

where U , and V, are neighborhoods of 0 in n. Fix H , exp t H , ( t E R). Then {(a, nu; ') = a, &)a;

(4)

E a+

and put a,

=

'.

Since the decomposition n = ~ I , o ga diagonalizes Ad(a,) (with diagonal elements erl(Ho)),it is obvious that

u

Ad(a,)V, = n.

I>,

u,,o

Since the map exp: ti + N is surjective, it follows that a, Va;' = N. But then (4) implies {(N) = N. Finally, 5 is one-to-one: if in fact { ( n , ) = l(n,), (4) implies {(a,n,a; ') = {(a,n,a; '). Choosing for t a negative number so large that both a,n,a;' and arn2aI-' belong to U,if follows that a,n,a;' = a,n,a;', so n , = n 2 . Corollary 5.5. Let H E a' and h = exp H . Then "

"

In fact, Lemma 1.11 and (3) above show that {*(dn) = det(Ad(n) (Ad(h- ') - 1)) dn, whence the result.

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

Proposition 5.6. Assume G above has finite center (so K is compact). For a = exp H, HE^, let

n

D(a) = _ _ (sinh(ja(H)))ma. aeZ+

Let a E A be such that D(a) # 0. Then for a suitable normalization of the invariant measure dgA on CIA we have *

*

for all f EC,(G). Proof. We must first verify that the integral on the left is defined. Let C denote the support off and put

CA = { g A E G / A :gag-' E C}.

By writing g E G as g = kna,, it is clear that gag-' E C - knan-lk-

' E C * nun-'

E

kCk.

Thus if gag-' EC, then nan-' lies in a compact subset of G, so by Lemma 5.4, n lies in a compact subset of N. Hence kn lies in a compact subset of G, so C, is compact. By Theorem 1.9 and the first relation in Corollary 5.3 we have for a suitable constant c "

(5)

J

I

F(gA) dgA = c J

F E C,(G/A).

F(knA) dk dn,

KXN

GIA

Hence by Corollary 5.5, JG,,f(gag-l) dgA = c

JKxN

=c

f ( k n a n - ' k - ' ) dk dn

n 11 -

U€X+

e-a(H)[-mm

f (kank-') dk dn, K xN

so the result follows by writing 1 - e-a(H)= 2 sinh(ja(H)) e-(1/2)u(H).

This proposition has an interesting functional equation as a consequence. Let M' denote the normalizer of A in K and W = M'/M the Weyl group. Theorem 5.7. (Harish-Chandra) Retaining the notation in Proposition 5.6, let W = W(g, a) denote the Weyl group of G / K and for s E H E a, a = exp H, put as = exp s(H).

w

5 . Integral Formulas

$51

Let f’ E C,(G) satisfy f ( k g k - ’ )

185

= f ( g ) ( k E K ) . Then the function

satisflies the functional equation a E A,

F f ( d ) = Ff(U), ,for each s E W. Proof. For g E G, let Proposition 5.6

= g a g - ’ . First assume D(a) # 0. Then by

a y ~

Ff(a)

(7)

=

1 D(a)I

1

f(u‘l”)

&A*

G/A

We shall now prove that the right-hand side of this relation is invariant under a + as. For this purpose f can be arbitrary in C,(G). First, since S E W permutes the roots, ID(a)l = ( D ( u ) ~ ) ’ is / ~ invariant under s. For the invariance of the integral select U E K such that Ad,(tr) coincides with s on a. Then u A u - ’ = A, so the mapping 4 : g A --* u g u - ’ A is a well-defined mapping of CIA onto itself. Put f’”(g) = f ( u g u - ’ ) , and as in Lemma 1.10,

and with d g suit-

Then the mapping ably normalized, r

r

r

p ( g A ) dgA*

On the other hand, p(gA)=

J f ( u g a u - ’ ) da = J A f ( u g u - ’ a S ) da = J”f ( u g u - ’ a ) da, A

so f” = (,f)4-’.Substituting this in (8), we deduce 4 * ( d g A ) = dg,. Thus, invoking (5), we have .f((as)lA)dgA = ~ G , A f ( ( a s ) ‘ u ’ ’A) ’ u ~dgA ’

= / G / A f ( u g a g - ’ u - l ) dgA

=

JKXN

f ( u k n a n - ’ k - ’ u - ’ ) dk dn.

186

[Ch. I

INTEGRAL GEOMETRY A N D RADON TRANSFORMS

Here u can be cancelled, so this equals

JC / Af(a”)

&A.

This proves F,-(as) = F,-(a) for all a in a dense subset of A ; by continuity it holds for all a E A . 2. Integral Formulas for the Cartan Decomposition

A. T h e Noncompact Case For the symmetric space X = G / K as above, consider as in [DS] (Chapter IX), the set A’ = exp a’ (a’ = the set of regular elements in a), G‘ = K A ‘ K , and the open dense subset X ’ = G’ o of X which is diffeomorphic to ( K I M ) x A + under the “polar coordinate map” 0:( k M , a) + kaK of ( K I M ) x A + onto X’. Here o is the origin { K } in X . Let d x denote the volume element on X corresponding to the canonical Riemannian structure ( , ) induced by the Killing form B of g (also denoted ( , )) restricted to p . Let 8 denote the Cartan involution of g corresponding to the Cartan decomposition g = f + p . Theorem 5.8. For a suitable constant c dkM

.fE

9

cc(x>,

where d k , is the K-invariant measure on K I M , normalized b y J d k M = 1, and G(exp H) = (sinh a(H))’”., H Ea+.

n

,ex+

Proof. Let I denote the orthogonal complement of m in f (with respect to B). Identifying I with tangent space of K I M , the restriction ( - B)II induces a K-invariant Riemannian structure on K I M , and d k , is a constant multiple of the corresponding volume element. According to Prop. 1.3 we just have to calculate det(d@,,,,,,) for the polar coordinate map CD above. We know from [DS] (Chapter VII, §11), that I is the direct sum

where

f, = { T E f : (ad H)’T = L X ( H ) ~for T all H

E

a}.

051

5 . Integral Formulas

187

Moreover, the mapping X -+ X + OX is a bijection of g, onto f,; in particular, dim €, = mu. Let TY, . . . , TkII be an orthonormal basis of t, (a E C ' ) and let H,,. .. , H , be an orthonormal basis of a. As usual, we denote by t(b0) the translation bC + bo bC on any homogeneous space B/C. Let n: G --* G / K be the natural mapping, so dn is the projection of g onto the tangent space (GIK), with kernel f. The curve t --* k exp(tT;) M in K I M has tangent vector d.r(k)T; at t = 0. Then for a = exp H ( H Ea'),

@(k exp(tT;) M , a ) =k

exp(tT;) a o a

d@(khf, a)(dT(k)T4,

ka exp(t Ad(a- l ) T;) . 0,

=

0)

= d ~ ( k adn(Ad(a) ')

T;) = d ~ ( k adn{KAd(a) ') T ; - 0 Ad(a- ') TP)} = d ~ ( k adn{.r(e-adHT; ) - eadHTy)} = d ~ ( k adn( ) - a(H)- '[H, TP] sinh a(H)),

where we used the relation (ad H)'Tq = a(H)'T;. Also, @(kM, a exp(tHj)) = ka exp(tHj). 0,

so dT(a)Hj) = d T ( k U ) dn(ffj).

d@(k~,

The maps

dn: P

+

(G/K),,

dT(ka): (GIK), + (G/K)ko.o

are isometries, and the vectors Hj,

a(H)-'[H, T9]

(1 < j II , a E X + , 1 Ii I mu)

form an orthonormal basis of p. The formulas above thus imply that (9)

1 det(d@(kM,a,) I =

fl (sinh

Q(H))"',

U€T+

and this proves the theorem. B. The Compact Case The case of a compact symmetric space can be handled by similar methods although the global properties of the "polar coordinate" decomposition now have a new but minor complication. With the Cartan decomposition as above consider the compact real form

u

=f

+ p*,

where p* = ip

188

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

of the complexification gc. Then the subspace a, = ia is a maximal abelian subspace of p,. Let ( U , K,) be a symmetric pair such that U and K , have Lie algebras u and H, respectively. Let M , denote the centralizer of a, in K,. The group A, = exp a, is a closed subgroup of U ([DS], 56, Chapter V), hence compact. Let du, dk, dm, and da denote the invariant measures on the compact groups U , K,, M,, and A, such that in each case the total measure is 1. Let duK, and dk,, be the corresponding invariant measures on UIK,, K,/M,, respectively. By analogy with the polar coordinate map @ above we consider the mapping Y:(K,/M,) x A ,

+U/K,

given by Y(kMl, a) = kaK,, which by [DS], Theorem 6.7, Chapter V, is surjective. By analogy with (9) we find 1ELf

if a = exp H (H E a,). Let 6,(a) denote the right-hand side of (10). Using Prop. 1.3 we have on each open subset on which Y is one-to-one and regular, (1 1)

v*(dUK,) = c6,(a) dk,, da,

where c is a constant. Let A; denote the set of points a E A, for which 6,(a) # 0 and let (U/Kl)r denote the complement in U/K, of the singular set as defined in [DS], 53, Chapter VII. The space J = K , n A, is compact and discrete, hence finite. Let j denote its cardinality and let w denote the cardinality of the Weyl group W = W(g,8). Lemma 5.9.

The mapping Y is a regular wj-to-one mapping of

( K , / M , ) x A;

onlo

(U/Kl)r*

Proof. The regularity and surjectivity of Y being known from Lemma 8.1, Chapter VII, in [DS), it only remains to prove that each point in (UIK,), has exactly wj preimages. Suppose a , , a2 E A ; , k,, kzEKl, such that Y(k,M,, a , ) = Y(k2M,, az). On putting k = k;'k,, we have ka, = a2k' for a suitable k' E K,. We apply the involutive automorphism 8 of U , eliminate k' and obtain ka:k-' = a:. This implies Ad(k) 2, = 2, if 2, (i = 1, 2) denotes the centralizer

Zi= {XE p, : Ad(a:) X

=

X}.

$51

5. Integral Formulas

Let H E a,, X

Ep

189

, . Writing

x = x, + C u,(x,- ex,), U€T+

where X ,

E a,,

X , E g,, a, E iR, we have

Ad(exp H) X = eadHX =

X, +

1 a,(eu(H)Xa- e-''H)BX,). ,EX+

If a ,

=

exp H , , X E Z , , we deduce that a, = a,e2u(H1)

(a E X+),

so that a(H,) E niZ if a, # 0. But the assumption d,(a,) # 0 then shows a, = 0 for all a E Z', that is, 2, = a,. Similarly 2, = a,. This proves that k lies in the normalizer M', of a, in K,. We have by the above (12)

a2 = (ka,k-')k(k')-'.

Now let S E W, a E A , . We recall ([DS], Chapter VII, $2) that W = M',/M,, even if K , is not necessarily connected. Thus if rn E Mi is any element such that s and Ad(m) coincide on a,, we can write mam-1 = a . It is also clear that 6,(as) = b,(a). Let b E J and select H E a, such that b = exp H . Since b2 = e, we have a ( H ) E niZ for all a E C + , so that J,(aob) = d,(ao) for all a,€ A,. Formula (12) shows that a2 = aib, where t~ W, b E J . Also, k2 M I = k , k - ' M , . O n the other hand, if b runs through the set J and k runs through a complete set of representatives of M', mod M I , then the elements ( k , k - ' M , , ( k a , k - ' ) b ) E ( K , / M , ) x A: (for ( k , M , , a , ) fixed) are all different. Hence they make up the complete inverse image Y - l ( k l a l K l ) .This proves the lemma. New A; fills up A, except for a set of measure 0 and ( U I K , ) , fills up U I K , except for a set of measure 0. Since the density d,, as noted, has the same value at all the points ka,k-'b as k E M i , b E J, we can derive the following result from (1 1) and Lemma 5.9.

190

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

Theorem 5.10. Let ( U , K,) be a Riemannian symmetric pair of the compact type. Let dUK,, dkM,, da be the invariant measures on UIK,, K,/M,, and A , normalized by JU/Ki f U K ,

=

I K I , M Idk,,

=

I*

da = 1.

Then for a suitable constant c,, ( 3,

f (ukl) dUK,

= c*

(

/A!(kaKl)B*(a)

KIIMI

IU/Kl

dkM,

for all f E C(U/K,), B,(a) denoting the right-hand side of (10).

In the case when U is simply connected, K, connected, we can put the result in a more precise form. As in [DS] (Chapter VII, 53), let D(w, 0) denote the diagram {HEa,:a(H)EniZfor s a m e a E Z + } ; a, is the complement a, - D(u, e), and Q , is any component of a, whose closure Qo contains the origin. We have then the following global polar coordinate representation of UIK

,.

Theorem 5.11. ( U is simply connected, K, connected.) The mapping

$: (kM,, H) + k(exp H)K, maps (K,/M,) x onto ( U / K ,),.

0,

onto UIK, and gives a bijection of ( K , / M , ) x Q,

Proof. That

*(Kl/Ml x

00) =

U/K,,

d4KllMl x Qo) = ( U / K , ) r is clear from Chapter VII, 58, in [DS]. For the remaining (injectivity) statement suppose H I , H, E Q o , k,, k, E K,,such that k,(exp H d K , = k,(exp H,)K,. By (12) we have (14)

exp H, = exp Ad(k) H,b,

where k = k; 'k, E M', and b E J . Since b = exp H, (H,E a,), it follows generated by W(u,0) that H, and H I are congruent under the group rKI and the lattice aKI= {H ~ a , : e x pH E K , } . But then Chapter VII, 58 in [DS] implies that H, = H,. Equation (14) then implies that (15)

Hi = sH1

+ H,,

551

5. Integral Formulas

191

where H * € a K Iand s is the Weyl group element induced by Ad(k). But since rKI permutes the components of a, simply transitively, (15) implies s = e. Thus k , M , = k , M , , proving the injectivity. Let us now assume U / K , in Theorem 5.11 to be irreducible. Let p,,.. . , p, denote the simple restricted roots, 6 the highest root, 6 = d i p i . Then the component Qo can be taken as the polyhedron

1:

Qo = { H E a, : ( I / i ) p j ( H ) > 0 (1 I j I I), ( l / i ) 6 ( H ) < n}.

We can then state Theorem 5.10 in a more precise form. Corollary 5.12. W i t h U , K , as in Theorem 5.11, U / K , irreducible, let d H be the Euclidean measure on a, induced by the Killing form. Then f(uKl) dUK,

/U/K

= cO

/K,MldkMl(/Q.f(k(exPH)Kl)6*(exp

dH

I

where c0' =

IQ0n

(sin a(i- lH))mm dH.

asT+

We now restate Theorem 5.11 for the case when the symmetric space is a group. Let I/ be a compact semisimple Lie group, T c U a maximal torus. As in [DS] (Chapter VII), 5547, let to and u denote the respective Lie algebras, t and g the complexifications. Let A denote the set of roots of g with respect to t. Let D(u) denote the diagram { H E to : a ( H ) E 2niZ for some a E A},

let t, = to - D(u) and fix a component P o of t, whose closure Po conwe , view U as the symtains the origin. As in [DS], Chapter VII, &I metric space U x U / U . Then 6* becomes a function on the set of pairs { ( t , t - ') : t E T } . Writing 6(exp 2 H ) = 6*(exp H, exp( - H ) ) ( H E to), we have (up to a constant factor) if

t =

exp H, H E to

asA

The product is taken over all of A, not just over the positive roots, because here each restricted root has multiplicity 2. Let du be the Haar measure on U normalized by j d u = 1 and let d H be the Euclidean measure on to induced by the Killing form of u. Then we have from Theorem 5.1 I and its corrollary the following result.

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

Theorem 5.13. Let U be a compact semisimple simply connected Lie group. The mapping t,h: (uT, H ) + u exp H u-'

maps ( U / T ) x Po onto U and gives a bijection of (U/T)x Po onto U,. Also,

Juf (4 du = c Ju,&

J P - f ( uexp H u-')&xp

H)dH,

where c is a constant. It is convenient to insert here a simple result about reduced root systems. Let E be a vector space over R, V the dual of E, and R a reduced root system in V ([DS], Chapter X, $3). Let a l r . . . , cq be a basis of R and R f the corresponding set of positive roots. Let ( , ) denote the Killing form on V ([DS], p. 523) and C the Weyl chamber corresponding to R', that is, C = { y E I/ : (a, y) > 0 for a E R +} = { y E V : ( a i , y ) > 0 (1 Ii II)}.

As in [DS], Chapter X, $3 let T ( R ) denote the subgroup of V generated by the vectors in R and, putting 6 = 2a/(a, a), let

T ( R ) = {oE V: (w,&) E Z for ct E R } . Let o,,. . . , o1be the "dual basis" of V given by ( w i , 6 j ) = d i j ; then (loc. cit.) 1

1

T(R)= It follows that (16)

Zwi,

T ( R )=

1

1 Za,. 1

+ -..+ m p , :m, > 0,. . . ,m, > 0).

T ( R ) n C = {rnlol

The element p = 3 3.1 1, Chapter X)

zsR+ a belongs to T ( R ) n C; in fact ([DS], p =01

+

Lemma

+ w,.

Consider now the algebra F of functions on E generated over R by the exponentials eo ( W E?(R)). The Weyl group W ( R ) operates on F by

s(eo) = esw. An element a € F is called invariant if sa = a (SE W ) and skew if sa = (det s)a (s E W ) , det denoting determinant. The alternation A : F + F given by A(e") = (det s)esw sE W ( R )

5. Integral Formulas

$51

193

maps F onto the set of skew elements in F (cf. [DS],Chapter I, $2, No. 3).

Lemma 5.14. The elements A(e"). w E f(R)n C,form a basis of the space of skew elements in F. Proof. Since W ( R ) permutes the Weyl chambers simply transitively (cf. the proof of Corollary 7.4 in [DSJ, Chapter VII), the elements sw (s E W(R), w E F(R) n C) are all different, so the linear independence of the elements A(e") follows. Next let f~ F be skew, f = f, em. Then if f, # 0, w is regular. In fact, suppose swo = wo for some wo and some reflection s E W(R). Since f i s skew,

c

sf=

(det s)f= -f,

I f m e s o = - Z foeo,

whence f,, = 0. But each regular element is W(R)-conjugate to an element in C. Thus f has a unique representation

f=

(17)

1

fA,sesA*

let(R)nC,s~W(R)

But if t E W(R), (det t ) f =

1 fweto,

tf=

o

so, by the linear independence of the esAin (17), = (det

fAVs

whence

f=

4-'fA= (det s)fA,

c

fn4eR).

AE t(R)nC

Proposition 5.1 5.

(i) The following identity holds in F : eP (1 - e - , ) = (18)

n

M€R+

1 (det s)esP.

seW(R)

(ii) The element given by (18) is skew and divides every skew element of F. Proof. The left-hand side of (18), say D, can be written in the form (e(1/2)a - e-(1/2)a),

fl

aeR+

If ai is a simple root, the corresponding reflection s,, permutes R f { a i } (cf. [DS],Chapter X, Lemma 3.11). It follows that s,,D = -D =

I 94

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

det(s,,) D, so that. since the s,, generate W ( R ) , D is skew. Thus, by Lemma 5.14, D is a linear combination D

=

c

+

apA(eP)

c,A(e"').

w e T(R)nC, w + p

On the other hand, by definition, D = ep + C d A e P - L

(19)

(d, E Z),

A

where A is a positive integral linear combination of the ai.Now if cco# 0, then by (1 6) w - p E T (closure) and, by (19), o = p - 1for some A. But this would give - 1 T~, contradicting Lemma 2.2O(iii) in [DS], (Chapter VII). (This lemma holds for abstract root systems because of Lemma 3.4, Chapter X.) Thus all c, vanish and u,, = 1, proving part (i). For part (ii) we consider w regular in T ( R ) and let a E R'. Then ((I),

5 ) E Z - {0}and ew - saew = ew - ew-(w.')a,

so ew - s,ew

1 - e' - z K w . 5) ) =

ew(l -

if ( w , 6 ) > 0 if (w, 6 ) < 0.

In t.: first case ew - s,ew is divisible by ( I - e-"), in the seconL case by (1 - e") = e"(e-" - 1). Thus in both cases ( I - e-") divides eo - s,e' and hence divides each A(e") ( = &A(e") - s,A(e"))). The factors (1 - e - I ) ( a E R + ) being relatively prime and F a unique factorization domain. D divides A(&") and, by Lemma 5.14, each skew element in F . Using Proposition 5.15, we can state the group case of Theorem 5.10 in an improved form. Corollary 5.16.

Let U be a compact semisimple Lie group, T c U dr and du uor-

a maximal torirs, b ( t ) given us above, arid the Haar measures malized by dr = j dii = 1. Then

1

where I W I is the order of the Weyl group W of U . It only remains to verify exp H , then d(t) =

n

jT 6 ( t )d t

=

I W I. For this we note that if

(e(lI2)a- e-(1/2)")(H) con,

l>O

n

(p(li2)" -

e-(l."")(H)),

(a>o

which by Proposition 5.15 equals

h(t) =

(det s)(det o)eSP-'p(H). s, u E W(R )

t =

951

5. Integral Formulas

195

If y is a sum of roots, we can define a character ey of T by ey(t)= ey(H) if t = exp H (HEto). In fact, if exp H = e, then ~ ( H ) 2niZ. E But sp - ap is a sum of roots and is #O unless s = a. Hence jT d(t) dt = I W I , as stated. C . The Lie Algebra Case We shall now state and prove the infinitesimal version of Theorem 5.8 corresponding to the decomposition of p into K-orbits. We retain the notation from the proof of Theorem 5.8. Consider the mapping (20)

4: (kM, H ) + Ad(k) H ,

k E K, H E a,

of KIM x a onto p. It is shown in [DS], Chapter VII, 53, that its differential at a point (k, M, H , ) E KIM x a is given by (21)

d6(kohf,Ho)(dz(kO)L, dz(HO) H) = d m K k , ) H,)Ad(k0) ([L HOI

+ H)

for L E I, H E a. As in the proof of (9) consider the orthonormal basis

H,,..., H , , Tq of a

+ I; then if H

(1 I i Im , , a E X + )

, E ~is regular,

H , ,..., H , , a ( H , ) - ' [ H , , Tq]

is an orthonormal basis of p. Expressing see that

(1 I i Im a , a E Z + )

d 4 in terms of these bases, we

Let p' denote the set of regular elements in p (elements whose centralizer in p is abelian) and let dX and dH denote the Euclidean measures on p and a corresponding to the Killing form metric. Theorem 5.17. The mapping of KIM x a onto p is a diffeomorphism of KIM x a' onto p'; moreover,

where c is a constant.

Only the injectivity on KIM x a+ remains to be verified. But suppose Ad(k,) H , = Ad(k,) H , , where k,, ~ , E K H , , , H , E ~ + Let . k = k;'kz, so that Ad(k) H , = H,.Then Ad(k) maps the centralizer of H, in p onto the centralizer of H, in p. These being equal to a, we have k E M'; but by the simple transitivity of the Weyl group on the set of Weyl chambers we conclude that k E M, as desired.

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INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

We now prove a simple result showing once again how a + behaves like a multidimensional radius vector. For X E p, let a(X) denote the unique point a(X) = ( K . X ) n

3

(cf. [DS], Chapter IX, $1). Let d denote the distance function on p and define distance between subsets as usual. Proposition 5.18. Let X, Y E p. Then

d(Ad(K) X,Ad(K) Y) = d(a(X), a( Y)). Proof. Let Ad&) Y be the point on the orbit Ad(K) Y which minimizes the square distance

F(k) = (X - Ad(k) Y, X The relation {iF(exprT-k,)

-

It=.

Ad(k) Y),

=0

(TEf)

= 0,

TE€,

gives

(X,[ T , Ad(k0) Y])

which implies [X, Ad&) Y] = 0. Thus we may assume that X and Ad@,) Y are in a. But the proof of Theorem 2.12 in [DS] (Chapter VII), then shows that X and Ad(k,) Y are in the same closed Weyl chamber, so the result follows.

3. Integral Formulas for the Bruhat Decomposition We retain the notation from the beginning of this section and put = ON. As proved in [DS] (Chapter IX, $l), the mapping

m

(23)

(ii, m, Q, n) + iiman

is a bijection of R x A4 x A x N onto an open submanifold NMAN c G whose complement has Haar measure 0. Furthermore, if we write g = k(g) exp If@) n@) (k(g) E K, H ( g ) E a, n(g) E N), according to the Iwasawa decomposition G = K A N the mapping (24)

ii + k(R)M

5 . Integral Formulas

$51

I97

m

is a diffeomorphism of onto an open subset of KIM whose complement is a null set for the invariant measure dk,. We shall now compute the Jacobian for these maps (23) and (24). As before, let p denote half the sum of the positive restricted roots (with multiplicity). Lemma 5.19. Let g E G. The mapping phism of K onto itselfand

F(k(g-'k)) dk =

q:k + k(gk) is

F(k)e-2p(H(8k)) dk,

a diffeomor-

F EC ( K ) .

Proof. Let X E G . By Proposition 5.1 we have

~ f ( k a n ) e 2 p ( ' " edk " ) da dn = But if g

=

kan, xg = xkan = k ( x k ) exp H ( x k ) n(xk)an = k ( x k ) exp H ( x k ) a(a-'n(xk)un) = k,a,n,,

so that our integral is / f ( k l a l n l ) e z p ( l o g adk) da dn. But the translation a

-,exp H ( x k ) a = a ,

preserves da, and the translation n + (a-'n(xk)a)n = n , preserves dn. Thus the integral equals ~ f ( k ( x k ) a l n l ) e 2 P ~ ' Znpe( H~( x1k )~) dk e ~d a , dn,. Taking j' of the form f ( k a n ) = F(k)F,(a)F,(n), we get (25)

F ( k ) dk = I K

F(k(xk))e-2p(H(xk)) dk, IK

and the lemma follows if we replace F with the function F T,We can also write the lemma in the form 0

(26)

(T,)*(dk) = e- 2 p ( H ( g k ) ) dk.

198

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

Theorem 5.20. The function ii ifdii and dk, are normalized by

+ e-2p(H(")is

integrable on

m and

1.

we have F E C(K/M). Proof. We know from (24) that there exists a function t,b that

E

&(N)such

n

J

(27)

F ( k M ) dkM =

KIM

Fix x E

m.By Lemma 5.19 we have

JK

F(k(x- ' k ) M ) dk =

J,

F(k(ii)M)IC/(ii)dii.

JK

F(kM)e-2p'H'xk))dk.

But k(x-'k(ii)) = k(x-'ii), so by (27) the left-hand side is J/(k(ii)MM(xii) dii, whereas the right-hand side is F(k(ii)M)e- 2p(H(xk(ii)))IC/(ii) dii. Thus IC/(xfi) =

e- Z / W ( x k ( W ) *m. )

and the theorem follows by putting ii = e. Proposition 5.21. Let dii, dm, da, and dn be Haar measures on N, M , A, and N , respectively. Then the Haar measure dg in G can be normalized so that

(28)

/J(g) dg =

JA X M X A X f(iiman)eZp(loga) dii dm da dn N

.for .f E CAG). Proof. Because of the properties stated for the map (23), formula (28) holds with e2"(Ioga)replaced by some unknown function D(ii, m, a, n). Then D can be determined by the method of Proposition 5.1. Replacing

461

6. Orbital Integrals

199

f(g) byf(ii,gn), we see that D is independent of ii and n. Replacingf(g) byf(m,g) and noting that, by the compactness of M . dfi is invariant under ii + m,iirn; we see that D is independent of m. Finally, D’s dependence on a is found just as in Proposition 5.1.

’,

56. Orbital Integrals

In this section we give examples of solutions to Problem D in 43, No. 3, which amounts to the determination of a function in terms of its integrals over generalized spheres. 1. Pseudo-Riemannian Manifolds of Constant Curvature

Let X be a manifold. We recall that a pseudo-Riemannian structure of signature ( p , q ) is a smooth assignment y + g,, where gy is a symmetric nondegenerate bilinear form on X , x X, of signature (p, q). This means that for a suitable basis Y,, ..., Yp+4 of X, we have P+4

g y ( Y . Y ) = y : + . . . + y ; - y p + 2,

-...-

Y2 p+q

( Y = FYiy).

If q = 0, we speak of a Riemannian structure, and if p = 1, we speak of a Lorentzian structure. Connected manifolds with such structures g are called pseudo-Riemannian (resp. Riemannian. Lorentzian) manifolds. An isometry of a pseudo-Riemannian manifold X is a diffeomorphism preserving g. Let I ( X ) denote the group of all isometries of X . For y E X let I ( X ) , denote the subgroup of I ( X ) fixing y (the isotropy subgroup at y) and let H , denote the group of linear transformations of the tangent space X, induced by the action of I ( X ) y . For each a E R let Z.(y) denote the “sphere” ZQ(Y)= (2E

x,: g,(X, X) = a, z # 0).

A pseudo-Riemannian manifold X is said to be isoDefinition. tropic if for each a E R and each y E X the group H , acts transitively on C,(y). (Note that the case a = 0 is included.) An isotropic pseudo-Riemannian manifold X is necessarily homogeneous. For this let y, z be any two points in X and join them by a curve consisting of finitely many geodesic segments y i . For each y i there exists an isometry of X fixing the midpoint of y i but reversing the direction of y i . Their product will map y to z.

200

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

Let X be a manifold with a pseudo-Riemannian structure g and curvature tensor R. Let EX, S c X , a two-dimensional subspace on which g y is nondegenerate. The curvature of X along the section S is defined by By(Ry(X,Y ) X , Y ) K(S) = g y ( X X)g,(Y, Y) - Qy(XYI2’ The denominator is in fact f O , and the expression is independent of the choice of X and Y (cf., [DS],p. 250). We shall now construct pseudo-Riemannian manifolds of signature (p, q) and constant curvature. Consider the space R p + q + l with the flat pseudo-Riemannian structure

+

+ yp’ - $ + I

- Y p + q + eyp+q+1 Let Q, denote the quadric in R p + q +given l by E,(Y) = y:

*

*

a

2

-

2

( e = rt 1).

B,(Y) = e.

(1)

+

+

The orthogonal group O(B,)[ =O(p. q 1) or O(p 1. q)] acts transitively on Q,; the isotropy subgroup at o = (0,. , . ,0, 1) is identified with O(p, 4). Theorem6.1. (i) The restriction of Be to the tangent spaces to Q, gives a pseudo-Riemannian structure ge on Q, of signature (p, 4). (ii) W e have

Q-

(2)

Z

O(p, q

+ l)/O(p, q )

and the pseudo-Riemannian structure g - 1. (iii) W e have

Q+

(3)

Z

O(p

(difleomorphism) on Q- has constant curvature

+ 1, q)/O(p,q )

(difleomorphism)

and the pseudo-Riemannian structure g + on Q+ has constant curvature

+ 1.

(iv)

The JIat space Rp+qwith the quadratic form go(Y) =

z:,”

y: -

yf and the spaces

O(P,4 + 1)/0(P, 41, o(P + 1, 4)/0(P, 4) are isotropic, and they exhaust the class of pseudo-Riemannian manifolds of constant curvature and signature ( p , q) except Jor local isometry and multiplication of the pseudo-Riemannian structure by a constant factor. Proof. If so denotes the linear transformation (Yll

*

’

* 9

Y p + q * Y p + q + 1)

+

(-Y19

*

-.

9

-Yp+q,

Y p + * + l),

063

6. Orbital Integrals

20 1

then the mapping 6 :g + s,gs, is an involutive automorphism of O(p,q + 1) whose differential do has fixed point set o ( p , q ) (the Lie algebra of O(p,4)). The (- 1)-eigenspace of do, say m, is spanned by the vectors

(4)

F = Ei,p+q+l + Ep+q+l,i

(1 I i I

(5)

(p+ 1 I j ~ p + q ) . y.J = E J. . p + q + ~ - E p + q + l , j Here Eij denotes a square matrix with entry 1 where the ith row and the jth column meet, all other entries being 0. The mapping I): gO(p, q) + g o has a differential d$ which maps rn bijectively onto the tangent plane y p + q + l= 1 to Q - l at o and d $ ( X ) = X o (X E m). Thus

-

d$(Yk)

= (81k,***,8p+q+l,k)

(l 5

5p

+ 4)-

Thus B - l ( d $ ( & ) ) = 1 if 1 I k ~p and -1 if p + 1 Ik < p + q , proving (i). Next, since the space (2) is symmetric, its curvature tensor satisfies ([DS], Chapter IV, Exercise Al),

R d X , Y)(Z) = - [ [ X , Y1, Zl, where [ , ] is the Lie bracket. A simple computation then shows K ( R & + RY,)= -1

(1 5 k, 1 I p

+ q),

and this implies (ii). Part (iii) is proved in the same way. For (iv) we first verify that the spaces listed are isotropic. Since the isotropy action of O(p, q + l), = O(p,q) on m is the ordinary action of O(p, q) on RP+q, it suffices to verify that R P + 4 with the quadratic form go is isotropic. But we know that O(p,q) is transitive on ge = + 1 and on ge = - 1, so it remains to show that O(p,q) transitive on the cone { Y # 0:ge(Y) = 0). By rotation in RP and in R4, it suffices to verify the statement for p = 4 = 1. But for this case it is obvious. The uniqueness in (iv) follows from the general fact that a symmetric space is determined locally by its pseudo-Riemannian structure and curvature tensor at a point (cf., [DS], pp. 200-201). This finishes the proof. The spaces (2) and (3) are the pseudo-Riemannian analogs of the spaces O(p, l)/O(p), O(p + l)/O(p) from $4, No. 1. For the pseudo-Riemannian analogs of the other two-point homogeneous spaces, see Wolf [1967]. We shall later need a lemma about the connectivity of the groups O(p,4). Let I,,q denote the diagonal matrix (diJ with dii = 1 (1 I i 5 p),

202

INTEGRAL GEOMETRY AND RADON TRANSFORMS

d, = -1 (p + 1 I j I p to O(p, q) if and only if

+ q), so a

[Ch. I

matrix g with transpose 'g belongs

'SIP49 = bJ.ll* If ~ E R ~ let+ yT ~ ,= ( y l,..., y , , O . . . O ), y" = (0, ..., 0, Y , + ~,..., y P + J , and for g E O(p,q), let gT and gs denote the matrices (6)

(gT)ij = gij

(1 Ii, j IP),

(gS)kl = gkl

(p

+ 1 Ik , 1 5 p + 4).

If gl, .. . , g p + q denote the column vectors of the matrix g, then (6) means for the scalar products that gT. I g?1 - 9:. g; = 1,

1 l i s p ,

gf.gT - g;.g;

p

=

-1,

+ 1I

j I p

+ q,

g[ = gy. gi, j # k. Lemma 6.2. W e have ,for each g

E O(p,q )

Idet(gT) I 2 1,

I det(gS) I 2

The components of O(p,q ) are obtained b y (identity component);

(7)

det gT 2 1,

det gs 2 1

(8)

det gT I - 1,

det gs 2 1 ;

(9) (10)

det gT 2 1,

det gs I - 1 ;

det

detg, I -1.

I- 1 ,

Thus O(p,q ) has 4 components q = 0.

if p

2 1, q 2 1, 2 components

if' p or

Proof, Consider the Gram determinant

which equals (det gT)2. By using the relations above it can also be written

963

6. Orbital Integrals

203

which equals 1 plus a sum of lower-order Gram determinants each of which is still positive. Thus (det gT)2 2 1, and similarly, (det gs)2 2 1. Assuming now p 2 1, q 2 1, consider the decomposition of O(p, q) into the four pieces (7)-(10). Each of these is # because (8) is obtained from (7) by multiplication by I l , p + q - etc. O n the other hand, since the functions g -P det(gT), g + det(g,) are continuous on O(p,q), the four pieces above belong to different components of O ( p , q ) . However, by [DS], Chapter X, Lemma 2.3, O ( p . 4 ) is homeomorphic to the product of O(p,4) n U(p + 4) with a Euclidean space. Since O(P9 4 ) n u(P + 4 ) = O(P9 4 ) n O(P + 4 )

is homeomorphic to O(p) x O(q), it just remains to remark that O(n) has two components. Now let X be a Lorentzian manifold of constant curvature. According to Theorem 6.1, X is (up to multiplication of the Lorentzian structure by a positive constant) locally isometric to one of the spaces

R'+, Q-1 = O(1.q Qi

(flat, signature (1.4))

+ l)/O(l,q):

= O(2, q)/O(1- 4):

y: - y i y ; - y; -

*

*

a

- Y,2+2 2

... - Y,+l

= -1

+ Y,z+2 = 1

the Lorentzian structure being induced (in the two last cases) by the form y: - Y; - ... - Y i + 1 T Y 2, + 2 .

2. Orbital Integrals for the Lorentzian Case We shall now define the Lorentzian analog of the spherical averaging operator M' from $2, No. 2, and $4. We start with some geometric preparation. For manifolds X with a Lorentzian structure g we adopt the following customary terminology: If J' E X , the cone

c, = {Y EXy:gy(Y,Y) = 0) is called the null cone (or the light cone) in X , with vertex y. A nonzero vector Y E X, is said to be rimelike, isotropic, or spacelike if gy( Y , Y ) is positive, 0, or negative, respectively. Similar designations apply to geodesics according to the type of their tangent vectors. While the geodesics in R'+" are just the straight lines, the geodesics in Q - l and Q + , can be found by the method of $4, No. 1 .

204

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

Proposition 6.3. The geodesics in the Lorentzian quadrics Q and Q + have the ,following properties:

,

(i) The geodesics are the nonempty intersections of the quadrics with two-planes in R2+, through the origin. (ii) For Q- the spacelike geodesics are closed; for Q + the timelike geodesics are closed. (iii) The isotropic geodesics are certain straight lines in R2+q. Proof. Part (i) follows by the symmetry considerations in §4, No. 1. For part (ii) consider the intersection of Q - , with the two-plane y , = y, = ... = Y , + ~ = 0. The intersection is the circle y 2 = cos t, y , = sin t whose tangent vector (0, -sin t , cos t, 0,. . . ,0) is clearly spacelike. Since O(1, q + 1) permutes the spacelike geodesics transitively, the first statement in (ii) follows. For Q + l we intersect similarly with the two-plane Y2 = ... = Y,+l = 0. For (iii) we note that the two-plane

,

,

R(1.0, ..., 0. 1) + R(0, 1 . . ..,O) intersects Q- in a pair of straight lines y, = t , v , = f l . y , = . . . = y , + ,

=0.yq+2=t.

which clearly are isotropic. The transitivity of O(l. q + 1 ) on the set of isotropic geodesics (Theorem 6.1) then implies that each of these is a straight line. The argument for Q+, is similar. Lemma 6.4.

,

,

The quadrics Q - and Q + ( q 2 1) are connected.

Proof. The q-sphere being connected, the point (y,, . . . ,yq+2) on can be moved continuously on Q to the point

QT

,

(Y1, (Y: + * . * + Y:+1)1’2, 0, * .. 0, y,+2), so the statement follows from the fact that the hyperboloids y: - y ; T y: = T 1 are connected. 3

Lemma 6.5. The identity components of O(1, q + 1) and O(2,q ) act transitively on Q - and Q+ respectively, and the isotropy subgroups are connected.

,

,,

Proof. The first statement comes from the general fact (cf., [DS], Chapter 11, &I that ) when a separable Lie group acts transitively on a connected manifold, then so does its identity component. For the isotropy groups we use the description (7) of the identity component. This shows quickly that

Oo(L q

+ 1) n O ( 1 , q ) = OO(1, 41,

Oo(2,q) n O ( 1 , q ) = OOU, q),

461

6 . Orbital Integrals

205

the subscript 0 denoting identity component. Thus we have

+ 1)/00(1,4),

Q-i

= oo(l,q

Q+ I

= Q o ( Z 9)/00(194).

proving the lemma. We now write our three spaces R’ + q , Q- Q + in the form X = G / H , where H = Oo(l, q) and G is either Go = R i + q .Oo(l, q ) (semi-direct product, R i + q normal), G - = Oo(l,q + 1) or G + = 00(2, q). Let o denote the origin {H}in X ; that is, o={

(0, . . ., 0) (0,..., 0, I )

if X = R i + q if X = Q - l orQ+l.

In the cases X = Q - X = Q+ the tangent space X , is the hyperplane {Y . . . Y q + 11 c R 2 + q . The timelike vectors at o fill up the “interior” of the cone C,. The consists of two components. The components which contain the set timelike vectors 9

1 9

13

e,

c,

0, =

(-1,o ,... O), )

( - 4 0 ,..., 0, 1),

( - l , O ,..., 0, 1)

in the cases G o / H , G - / H , G + / H , respectively, will be called the retrograde cone in X,. It will be denoted D o . The component of the hyperboloid q,(Y. Y) = r2 which lies in Do will be denoted S,(o). If y is any other point of X , we define C , , D , , S,(y) c X, by C , = g C,,

D,

=g

*

Do,

S,(Y) = g . S r ( 0 )

if g E G is chosen such that g . 0 = y. This is a valid definition because the connectedness of H implies that h . Do c Do. We also define B,(y) = { Y E D , : 0 < g,( Y, Y ) < r 2 } .

If Exp denotes the exponential mapping of X, into X, mapping rays through 0 onto geodesics through y, we put D , = Exp D,,,

C, = Exp C,,

SLY) = EXPSAY),

BAY) = EXPBAY).

Again C,,,and D , are called, respectively, the light cone and retrograde cone in X with vertex y. For the spaces X = Q + l we always assume r < n in order that Exp will be one-to-one on B,(y) in view of Part (ii) of Proposition 6.3. Lemma 6.6. The negative of’ the Lorentzian structure on X = G / H indiices on each S,(y) a Riemannian structiire of constant negative curvature ( q > 1).

206

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. 1

Proof. The manifold X being isotropic, the group H = Oo(l, 4 ) acts transitively on S,(o). The subgroup leaving fixed the geodesic from o with tangent vector v, is Oo(q).This implies the lemma.

Lemma 6.7. right angle.

The timelike geodesics j l o m 4’ intersect S,(y) tinder a

Proof. By the group invariance it suffices to prove this for y = o and the geodesic with tangent vector v,. For this case the statement is obvious.

Let r ( g ) denote the translation xH + gxH on G / H and for Y E m let T, denote the linear transformation Z + [ Y , [ Y , Z ] ] of m into itself. As usual, we identify m with (GIH),. Lemma 6.8. ential

The exponential mapping Exp: m

d Exp, = dr(exp Y) o

1 (2n T+; l ) !

+ G/H

has difer-

( Y Em).

Proof. If 11: G -+ G / H is the natural map. we have n(exp Y) = Exp Y ([DS], Chapter IV, Exercise Al), so Theorem 4.1 (loc. cit.) is still valid, giving the lemma.

Lemma 6.9.

The linear transformation

has determinant given by

for Y timelike. Proof. Consider the case of Q-l.Since det(A,) is invariant under H , it suffices to verify this for Y = cY, in (4), where C E R . We have c2 = g(Y, Y ) and T,,(?) = ? ( 2 < j Iq 1). Thus T, has the eigenvalues 0 and g(Y, Y); the latter is a q-tuple eigenvalue. This implies the formula for the determinant. The case Q + lis treated in the same way.

+

From this lemma and the description of the geodesics in Proposition 6.3 we can now conclude the following result.

461

6 . Orbital Integrals

207

Proposition 6.10. (i) The mapping Exp: m + Q - l is a diffeomorphism of' Do onto D o . (ii) The mapping Exp: m + Q + , is a difleomorphism of B,(o) onto B,(o).

Let dh denote a biinvariant measure on the unimodular group H . Let i r ~ 9 ( X ) Y, E X and r > 0. Select g E G such that y . o = y and select .Y E S,(o). Consider the integral

SHu(gh. x) dh. Since the subgroup K c H leaving x fixed is compact, it is easy to see that the set Cg,x= { h E H : gh x

E support(u)}

is compact; thus the integral above converges. By the biinvariance of dh it is independent of the choice of g (satisfying g . 0 = JJ) and of the choice of x ~ S , ( o ) . By analogy with the Riemannian case [-, (24)] we can thus define the operator M' (the orbital integral) by (M'u)(y) = JHu(gh. x) dh. If g and x run through suitable compact neighborhoods, the sets Cg,x are enclosed in a fixed compact subset of H , so (M'u)(y) depends smoothly on both r and y . It is also clear from (1 1) that the operator M' is invariant under the action of G : if I E G and ~ ( l denotes ) the transformation nH + InH of G/H onto itself, then M'(u 0 ~ ( 1 ) ) = (M'u) 0 ~ ( l ) . If dk denotes the normalized Haar measure on K , we have by Theorem 1.9

JHu(h . x) d h

=

u(hk . x) rik = JH,Ku(h . x) dh,

dh JH/K

JK

where dh is an H-invariant measure on H / K . But if do, is the Riemannian measure on S,(o) (cf. Lemma 6.6), we have by the uniqueness of H-invariant measures on the space H / K % S,(o) that

208

INTEGRAL GEOMETRY A N D RADON TRANSFORMS

[Ch. I

where A(r) is a positive scalar. But since g is an isometry, we deduce from (12) that

Now, we have to determine A(r). Lemma 6.11. For a suitable fixed normalization of the Haar measure dh on H we have

A(r) = r4,

(sinh r)4,

(sin r)q

for the cases

R'

fq,

O(1, q

+ 1)/0(1?d?

0(2,4)/0(1,4X

respectively. Proof. The relations above show that dh = A(r)-' d o , dk. The mapping Exp: D, + D , preserves length on the geodesics through o and maps Sr(o) onto S,(o). Thus if z E S'(o) and Z denotes the vector from 0 to z in X,, the ratio of the volume elements of S,(o) and S,(o) at z is given by det(d Exp,). Because of Lemmas 6.8 and 6.9 this equals

r+)q

(y)", sinh r

I,

for the three respective cases. But the volume element dw, on S,(o) equals r4 d o , . Thus we can write in the three respective cases r4 dh = -dw,

4)

dk,

sinhq r

-dw1

dk,

sin4 r A ( r ) d w , dk.

But we can once and for all normalize dh by cih = dw, dk, and for this choice our formulas for A(r) hold. Let 17 denote the wave operator on X = G / H , that is, the LaplaceBeltrami operator for the Lorentzian structure g on X as defined in Chapter 11, g2, No. 4. Lemma 6.12. Let y E X . On the retrograde cone D, the wave operator 0 can be written

where Lsr(,,,is the Laplace-Beltrami operator on S,(y).

461

6. Orbital Integrals

209

Proof. We can take J' = o. If (O1,. . . , 0,) are coordinates on the "sphere" Sl(o) in the flat space A', then ( r e l , .. . , r0,) are coordinates on S,(o). The Lorentzian structure on Do is therefore given by

dr2 - r' do2,

where do2 is the Riemannian structure of Sl(o). Since A, in Lemma 6.9 is a diagonal matrix with eigenvalues 1 and r-'A(r)''q (4 times), it follows from Lemma 6.8 that the image S,(o) = Exp(S,(o)) has Riemannian structure

r2 do2,

sinh2 r do',

sin' r do2

and

in the cases R1+(', Q - l , and Q+l, respectively. By the perpendicularity in Lemma 6.7 it follows that the Lorentzian structure on Do is given by dr2 - r2 do2,

dr2 - sinh' r do',

dr' - sin2 r do2

in the three respective cases. Now the lemma follows immediately. The operator M' is of course the Lorentzian analog to the spherical mean value operator for isotropic Riemannian manifolds. We shall now prove that by analogy with the Riemannian case, the operator M' commutes with the wave operator 0. Theorem 6.13. For each of' the isotropic Lorentz spaces X = G - J H , 0 and the orbital integral M' com-

G'JH, and G'JH, the wave operator mute :

OM'u

=

M' Ou

jor

u~9(X).

(For G'JH we assume r < n.)

Given a function u on GJH we define the function 12 on G by n(g) = u(g . o ) .

Lemma 6.14. There exists a dflerential operator under all left and all right translations such that

Bn = (nu)Proof.

.for

11 E

on G invariant

g(x).

We consider first the case X = G - / H . The bilinear form

K(Y, 2) = f Tr(YZ) on the Lie algebra o(1, 4 + 1) of G - is nondegenerate; in fact, K is nondegenerate on the complexification o(q + 2, C ) consisting of all complex skew symmetric matrices of order q + 2. A simple computation shows that in the notation of (4) and ( 5 ) K(Yl, Yl) = 1,

K ( 5 , 5) = - 1

(2 I j Iq

+ 1).

210

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

Since K is symmetric and nondegenerate, there exists a unique leftinvariant pseudo-Riemannian structure l? on G - such that Re = K . Moreover, since K is invariant under the conjugation Y + g Y g - ' of o(1, y + I), l? is also right invariant. Let denote the corresponding Laplace-Beltrami operator on G - . Then is invariant under all left and all right translations on G - . Let u ~ g ( X ) Since . a6 is invariant under all right translations from H , there is a unique function U E ~ ( X ) such that I% = ij. The mapping u + v is a differential operator which at the origin must coincide with 0 ; that is, l%(e) = Ou(o). Since, in addition, both 0 and the operator u + u are invariant under the action of G- on X, it follows that they coincide. This proves a6 = (nu)-. The case X = G + / H is handled in the same manner. For the flat case X = G O / H let

a a

I;. = (0,. . . , 1 , . . . ,O), the jth coordinate vector on Then 0 = Y : - Y ; - ... - Yi+l. Since R'+4 is naturally imbedded in the Lie algebra of Go, we can on GO. The operator 0 = P: extend r j to a left-invariant vector field - y ; - ... - 8:+, is then a left- and right-invariant differential operator on Go, and again we have 0 6 = (nu)-.This proves the lemma.

6

-

We can now prove Theorem 6.13. If g E G , let L ( g ) and R ( g ) , respectively, denote the left and right translations I + gl and I + Ig on G. If I . 0 = x, x E S,(o) (r > 0), and g - o = y, then

because of ( I 1). As g and I run through sufficiently small compact neighborhoods, the integration takes place within a fixed compact subset of H as remarked earlier. Denoting by subscript the argument on which a differential operator is to act, we shall prove the following result. Lemma 6.15.

n , ( / H i i ( g h l ) dh) = J i 8 6 ) ( g h I ) dh

=

ng

(1

6(ghI) dh

).

Proof. The first equality sign follows from the left invariance of In fact, the integral on the left is

a.

6. Orbital Integrals

561

21 1

so

The second equality in the lemma follows similarly from the right invariance of But this second equality is just the commutativity statement in Theorem 6.13.

a.

Lemma 6.15 also implies the following analog of the Darboux equation in Lemma 2.14. Corollary 6.16 . Let

11 E

9 ( X ) and

V(y, z ) = (M'u)(y)

piit

if' z E S'(0).

Then

O,(~Oi4)= OAWY, 4). Remark. In R" the solutions to the Laplace equation Lu = 0 are characterized by the spherical mean value theorem M'u = u (all r). This can be stated in this equivalent form: M'u is constant in r. In this latter form the mean value theorem holds for the solutions of the wave equation Ou = 0 in an isotropic Lorentzian manifold: If u satisfies Ou = 0 and [f u is suitably small ar m, then (M'u)(O) is constanf in r. For a precise statement and proof see Helgason [1959] (Chapter IV, 5). For R2 such a result was also noted by L. Asgeirsson.

3. Generalized Riesz Potentials

In this section we generalize part of the theory of Riesz potentials (§2, No. 8) to Lorentzian manifolds of constant curvature. Consider first the case X = Q- 1 = G - / H

=

Oo(1,n)/OO(l, II - 1)

of dimension n and let f E 9(X)and y E X. If z = Exp, Y ( Y ED,,),we put ry, = g(Y, Y)"' and consider the integral

212

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

where dz is the Riemannian measure on X,and

The integral converges for Re 1 2 n. We transfer the integral in (13) over to D, by the diffeomorphism Exp (= Exp,). Since

dz = dr d o , = dr

sinh r

and since dr dw, equals the volume element dZ on D,, we obtain

where r = g(Z, Z)l/'. This has the form (15)

where h(Z, A), as well as each of its partial derivatives with respect to the first argument, is holomorphic in 1 and h has compact support in the first variable. Because of Theorem 2.40 and the subsequent remark, the function 1+ ( I i ,f)(y), which by its definition is holomorphic for Re 1> n, admits a holomorphic continuation to the entire 1plane. Its value at 1 = 0 is h(0,O) = f ( y ) . Denoting the holomorphic continuation of (13) by (I!. f ) ( y ) we have thus obtained

(16)

I! f = .f.

We would now like to differentiate (13) with respect to y. For this we write the integral in the form j F f ( z ) K ( y , z ) dz over a bounded region F which properly contains the intersection of the support off with the closure of D,.The kernel K ( y , z ) is defined as sinhl-" ryr if z E D,,otherwise as 0. For Re 1 sufficiently large, K ( y , z ) is twice continuously differentiable in y, so we can deduce for such 1 that I l J 'is of class C2 and that

Moreover, given m E Z + ,we can find k such that I? f (and all f ) . Using Lemma 6.12 and the relation 1 dA - (n - 1) coth r, A(r) dr

E C"

for R e l > k

6 . Orbital Integrals

$61

213

we find O,(sinh"-" ryz) = O,(sinh"-" r,,) = ( A - n)(A - 1) sinhl-" ryr

+ (A - n)(A - 2) sinhA-"-2

ryz '

We also have K,(A)

=

(A - 2)(A - n)K,(A

-

2),

so by substituting into (17) we get

Of! f

= (A -

n)(A - 1)I.l 1'

+ 15-7.

Still assuming Re A large we can use Green's formula to express the integral (18)

s,

[ f ( z ) O,(sinh"-" ryz) - sinhl-" ryz (Of)(z)] dz

as a surface integral over a part of C, (on which sinhl-" ryz and its firstorder derivatives vanish) together with an integral over a surface inside D , (on which f and its derivatives vanish). Hence the expression (18) vanishes, so we have proved the relations (19)

O ( I . l f )= I ! ( O f ) ,

(20)

Z!(O/)

= (A - n)(A - 1)1!

j'

+ 1.1-7

for Re A > k , k being some number (independent off). Since both sides of (20) are holomorphic in A, this relation holds for all AEC. We shall now deduce that for each AEC, I t f ~ g ( Xand ) (19) holds. For this we observe by iterating (20) that for each p € Z + (21)

Itf

= I!+2p(Q,(0)j),

Q , being a certain pth-degree polynomial. Choosing p arbitrarily large, we deduce from the remark following (17) that I! f E &X); second, (19) implies for Re A + 2p > k that

OIt+2p(Qp(U)f) = I,l'2p(Q,(0) Of). By using (21) again, this means that (19) holds for all A. Putting A = 0 in (20), we get (22)

I12f

=

0.f - nf.

Remark. In Riesz's paper [1949] (p. 190), an analog I" of the potentials (82) in 42. is defined for any analytic Lorentzian manifold. These potentials I" are different, however, from our 15 and satisfy the equation 1-21. = U , j in contrast to (22).

214

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

We consider next the case

X

Q + I = G + / H = Oo(2, t~ - l)/Oo(l, n

=

and define for j

E

-

1)

9(X)

Again, &(A) is given by (14) and dz is the volume element. In order to bypass the difficulties caused by the fact that the function z -+ sin rYz vanishes on S,(y), we assume that .f’ has support disjoint from S,(o). Then the support of ,f is disjoint from S,(y) for all y in some neighborhood of o in X. We can then prove just as before that (24)

(1: ,f)(.Y)

=

f(Y),

(25)

(01: /‘)(.Y)= (1: O.~’)(.V).

(26)

(1; O f ) ( Y )=

-(A

- 17)(2 -

w: , m y ) +

(~:-2\’)(v)

for all 1E C. In particular, 1 ; ’1’ = Of

(27)

+

r7.f

Finally, we consider the flat case X

=

R ” = Go/H = R ” ~ O o ( l , n l ) / 0 0 ( I , j 7 - 1)

and define

These are the potentials defined by Riesz in [I9491 (p. 31), who proved (28)

1;f

=

.I;

01;

.1‘ = 1; 0.1= 1;-2f.

4. Determination of a Function from Its Integrals over

Lorentzian Spheres In a Riemannian manifold a function is determined in terms of its spherical mean values by the simple relation ,f = limr+oMrf’. We shall now solve the analogous problem for an even-dimensional Lorentzian manifold of constant curvature and express a function f in terms of its orbital integrals Mrf. Since the spheres S,(y) do not shrink to a point as I’ -+ 0 the formula (cf. Theorem 6.17) below is quite different.

961

6 . Orbital Integrals

215

For the solution of the problem we use the geometric description of the wave operator 0 developed in $6, No. 2, particularly its commutation with the orbital integral M', and combine this with the results about the generalized Riesz potentials established in $6, No. 3. We consider first the negatively curved space X = G - / H . Let n = dim X and assume n even. Let ./'E 9 ( X ) and put F(r) = (Mrj")(y). Since the volume element dz on D, is given by (12 = dr dw', we obtain from (12) and Lemma 6.9 I " sinh'.-' r F(r) dr. (15 . f ) ( y ) = Let Y,, . . . , Y, be a basis of X, such that the Lorentzian structure is given by n

g,(

Y . Y ) = y: - y 22

-

. . . - y,.2

Y =

c

yi yi.

1

If 01, . . . , we put

are geodesic polar coordinates on the unit sphere in R"-', y,

=

-r cosh (

(0 I ( < XI, 0 < r c co),

y2 = r sinh [ cos B,,

y,, = r sinh ( sin 8, ...sin

are coordinates on the retrograde cone D ,and Then ( r , [, 8,, . . . , the volume element on S,(y) is given by do), = r"-' sinh"-2 id i

is the volume element on the unit sphere in R"-'. It where follows that (writing sh for sinh, ch for cosh)

do,= sh"-' r shn-2 ( d( and therefore (30)

F(r) =

J's x

(fo

Exp)(-rch (. r sh (cos 8,. . . . , r sh ( sin 8, . . - s i n

sh"- id( do"-'.

+ +

Now select A such that j' 0 Exp vanishes outside the sphere y: in X,,. Then in the integral (30) the range of ( is contained in thc interval (0. io), where r2 cosh' + r2 sinh2 lo = A'. We see by

y i = A'

c0

216

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

the substitution t = r sinh ( that the integral expression (30) behaves for small r like

where

I$

is bounded. Therefore if n > 2, the limit a = lim sinh"-2 r F(r)

(31)

( n > 2)

r+O

exists. For n = 2 we find similarly by differentiating (30) that the limit b = lim(sinh r)F'(r)

(32)

( n = 2)

r-0

exists. Consider now the case n > 2. We can rewrite (29) in the form

where F ( A ) = 0. We now evaluate both sides for A = n - 2. Since &(A) has a simple pole for A = n - 2, the integral has at most a simple pole there and the residue of the integral is

I

lim (A - n + 2) sinh"-2 r F(r) sinha-"+' r dr. rl-"-2 Here we can take A real and greater than n - 2. This is convenient, since by (31) the integral is then absolutely convergent and we do not have to think of it as an implicitly given holomorphic extension. We split the integral into two terms: rA

(A

-

n

+ 2) J

(sinh"-2 r F(r) - a) sinha-"+' r dr 0

+ a(A - n + 2) JoAsinh""+'

r dr.

For the last term we use the relation A

lim p Jo sinhp-' r dr p-O+

sinh A

=

lim p Jo

tp-'(l

+ tz)-lz

rlr = 1

p+o+

by (71) in $2. For the first term we can for each that Isinh"-2 r F(r) - a1

-=

E

E

> 0 find a 6 > 0 such

for 0 < r < 6.

061

217

6. Orbital Integrals

If N = maxIsinhn-2 r F(r)l, we have for n - 2 < A < n mates

1

I

-

1 the esti-

+ 2) C(sinh"-' r F(r) - a) sinhl-"" r d r l 5 (A - n + 2 ) ( N + l a l ) ( A - 6)(sinh 6)'-"+';

(A - n

(A - n

+ 2 ) L ( ~ i n h ~r F(r) - ~ - a) sinh'-"+'

I &(A - n

+ 2)

Io6

r'-"+l

rdrl

dr = &61-n+2.

By taking A - (n - 2) small enough, the right-hand side of each of these inequalities is <2&. We have therefore proved lim ( A - n

+ 2)

1-11-2

=

SOm

sinhA- r F(r) dr

lirn sinh"-' r F(r). r+O

Taking into account the formula for integral (29)

&(A),

we have proved for the

On the other hand, using formula (20) recursively, we obtain for u E 9(X)

I Y Z ( Q ( 0 ) u )= U, where

Q ( O )= (0 + (n - 3 ) 2 ) ( 0

+ (n - 5 ) 4 ) .

* *

(0+ l(n - 2)).

We combine this with (33) and use the commutativity OMr = M'O. This gives

Here we can replace sinh r by r and can replace 0 by the operator

d d2 0, = - + (n - 1) coth r dr dr because of Lemma 6.12 and Corollary 6.16.

218

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

For the case n = 2 we have by (29) (1: f ) ( y ) =

(35)

1

K,o J,

sinh r F(r) dr.

This integral, which in effect only goes from 0 to A , is absolutely convergent because our estimate of (30) shows (for n = 2) that F(r) = O(1og r ) near r = 0. But using (20), (39, Lemma 6.12, Theorem 6.13, and Corollary 6.16, we obtain for u E 9(X) u = 12

:jom

Ou = -

sinh r M' Ou dr

dr d( M'u)

This is the substitute for (34) in the case n = 2. The spaces G + / H and Go/H can be treated in the same manner. We have thus proved the following result in response to Problem D (or D') in 83. Theorem 6.17. Let X be one of the isotropic Lorentzian manifolds G - / H , Go/H, G + / H . Let K denote the curvature of X ( K = -1,O, + 1) and assume n = dim X to be even. Put

Q ( 0 ) = (0- K(n - 3 ) 2 ) ( 0 - K ( n - 5)4)...(0 - K . l ( n - 2)). Then

(37)

if

u E 9(X),

1 d u = - - lim r - (M'u)

2,,,

dr

( n = 2).

Here 0 is the Laplace-Beltrami operator and Or its radial part: dZ I dA d 0 =-++-' dr2 A(r) dr dr' 5. Orbital Integrals on SL(2, R )

In this subsection we solve Problem D ($3, No. 3) in another special case. We show for the group G = SL(2, R) that a function ,f E 9 ( G )

$63

6. Orbital Integrals

219

is explicitly determined by its integrals over the conjugacy classes of elliptic elements (cf. [DS], Chapter IX, 57) and their translates. Consider the Iwasawa decomposition G = KAN, where

Since a,n,a;' = nxuZt.the positive root o! is given by a(1og a,) = 2t, so that, because of G = K CI(A+) K and Theorem 5.8, the Haar measure dg can be normalized so that

Denoting by dg the invariant measure on G / K [normalized by dg and dk = (1/(2n)) d o ] , we consider the orbital integral (39)

a,(o)= JGiK/(akeg-1)dg,

f WG),

off over the conjugacy class kf of k,. Theorem 6.18. The integral (39) converges absolutely, and

Proof. We have

=

(

e2'sin 8 cos 6 -e-" sin 8 cos 8

The last expression shows that kf is closed in G, so the integral (39) is absolutely convergent. Also, @,(8) can be written as jG f ( g k , g - ') dg and therefore equals @,.(fI), where f * ( g ) = (1/(2n)) J ,f(kegke I ) do. For the proof of Theorem 6.18 we can therefore assume ,f to be invariant under conjugation by K. Then writing g = k,a,K so by (38) dg = (1/(2n)) sinh(2t) dt d 4 ,

220

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

we have

@,-(O)

= /omj(a,k,a;

l)

sinh(2r) dt.

Here we use (40), replace 2t by t, and introduce the function

Then O@,-(O)

= ig(O),

where

In (37) we represented u(0,O) by means of the integral u( - r cosh

c, r sinh () dc,

which is the invariant integral of u over the semihyperbola x2 - y 2 = r2, x < 0. The integral (41) represents the integral of the function F(x9 Y)(X - Y ) over the quarter-hyperbola x y = r2, x 2 r. The following counterpart to (37) gives Theorem 6.18. Lemma 6.19. For F E .9(R2)dejine g by (41). Then

lim g'(r) =

- 2F(O, 0).

r+O

Proof. Let F,

=

/om(e'F

=

aF/dx, F , = dF/dy. Then

+ re2'F1)dt - /o"e-lF

dt

+ /0m(rF2 - r F , - re-2'F2) dr.

By the dominating convergence theorem, lim /ome-'F(rer, re-') dt = F(0, 0) r-0

and lim r /o"e-2'F2(re', re-') dt = 0. r-0

Exercises and Further Results

22 1

Since F i has support contained in a ball BA(0), F,(re', re-f) = 0 for t > log(Ar-'). Hence JFi(ref, re-') dt

=

O( I log r I),

so it remains to prove (43)

lim /om[e'F(re', re-')

+ re2'F1(re', re-')] dt = -F(O,

0).

r-0

The integrand, however, equals d dt

+ rF2(ref,re-'),

- (e'F(re', re-'))

so (43) follows from (42).

EXERCISES AND FURTHER RESULTS A. Invariant Measures

1. Let G be a Lie group and H a closed subgroup. Then (i) If H is compact, G / H has an invariant measure. (ii) If G is unimodular and H normal, then H is unimodular. (iii) If G / H has a finite invariant measure and if H is unimodular, then G is unimodular. 2. For the group O(2) the element g = (: A) satisfies Ad(g) = - I . 3. Let G be a connected Lie group.with Lie algebra 9, and H c G a closed analytic subgroup with Lie algebra E) c g. Let X , , . . . , X , be a basis of g such that X , , . . . X n span and put

,, .

RX, + Let cfj be determined by [ X i , Xj]= m

=

+ RX,.

xkc:X,.

(i) G is unimodular if and only if Tr,(ad X) = 0

for X E ~ ,

or, equivalently, n

C cfk= 0

for each i,

1 I i I n.

k= 1

(ii) The space G / H has an invariant measure if and only if Tr(ad,(T)) = Tr(ad,(T))

for

T E E),

222

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

or, equivalently, r

Ccya=O

for r + l < i < n

a= 1

(Chern [1942]). 4. Show that the group M(n) of isometries of R" is isomorphic to the group of matrices

where k E K = O(n) and x = (xl, . . . , x,) M(n) is then given by

s,f@)

dg =

s,

/er.

X I

E R".

A Haar measure dg on

fe C,(M(n)),

dk dx,

where dk is a Haar measure on K. 5. A biinvariant measure on the group G = GL(n, R) of nonsingular matrices X = ( x i j ) is given by n

f + J f(X)ldet XI-"

n dxij. i, i

G

6. A biinvariant measure on the unimodular group G = SL(n, R) is given by

f + S f(x)tdetX,,I-' G'

n

dxij.

(i.j)#(1. 1 )

Here X = (xij),Xij is the (i, j)-cofactor in X, and the xij (except for x,,) are taken as independent variables on the set G' given by det X,, # 0. 7. Let T(n, R) denote the group of all g E GL(n, R) which are upper triangular. A left-invariant measure on T(n, R) is given by

and a right-invariant measure by

-, JT(n,R )f(t)tftt;: ...t i , ifls j d t i j . G = SL(2, R) and let r be the modular group f

8. Let that

,4G/r)< 00

SL(2, Z). Prove

Exercises and Further Results

223

if p is an invariant measure on G/T. [ H i n t : Let G act on the upper halfplane H: y > 0 by z + (az + b)/(cz + d). Since r contains the translation T :z + z + 1 and the “inversion” S : z -P - l/z, show that each z E H can be mapped by r into a point in the set D = { z ~ H : I z 1 2l , I R e z l < + } , which in the G-invariant measure on H has finite area.] 9. Let G,,,,, = O(p + q)/O(p)x O(q) denote the Grassmann manifold of p-planes in R p + 4 . Let e l , . . . , ep and J;, . . . ,fb be the canonical bases of the orthogonal subspaces E , = 0 x R“.

El = RP x 0, Let

c;,,+,= I P E G,,,,: p n E2 = {O}I. (i) Let PEG;,,+,. For each i (1 5 i I p ) P contains a unique vector of the form ei

+ Pi111 + * . . + P i q f q .

The mapping P + (pij)lsisp,lsjsqis a bijection of C;sp+q onto the space MpJR) of p x q real matrices. is (ii) In the parametrization above an invariant measure on given by

f on

Gl.3

+

J(1

+ a;,>-’ dP11;

by

.1’

+

JU+ P:, +

~:,)-3/2

dfill dp12

and on G2,4 by j’+

J-{ 1 + (ad - b ~ +)u2~+ b2 + c2 + d2}- * da db dc dd,

where u = p l l , b = p12,c = Pzl, d 10. (i) The measure dp = ( d x l

=

P Z 2 . Generalization?

. . dx,-

I)/

IX, I

is a rotation-invariant measure on the sphere x; (ii) The measure dv = (dx1 .-dx,+,)/lxp+,+1I

+ ... + x i = 1.

224

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

is an invariant measure on the quadric

- ... - x;+,+1 = - 1

+ ... + x; -

+

viewed as a homogeneous space O(p,q l)/O(p, q) (cf. V, Exercise 7). 11. Let U =-SU(2) and o the involution 1

[Dsl, Chapter

-3

0

u + (0 - l ) . c

of U. Show that (i) The eigenspaces of the automorphism do of the Lie algebra of are given by

i = Re,, where

el

=

C e),

p*

=

Re,

11

+ Re2,

( '),

e2 = - 1O 0

e3 = (i0 -iO)).

(ii) The adjoint representation AdU of U is a double covering of SU(2) onto the group SO(3) of proper rotations of the Euclidean space

+ Re2 + Re3.

w = Re,

We have Adu(u)X = UXU-', u E U , X E W. (iii) Let

=

exp(iel)

=(

t

t

cos-

isin-

isin-

j:teR/. cos-

In terms of the basis (el, e2,e,) of u we have cos 0 -sin 0

Ad@,) =

(i

0

0

cos t si: t -sin t cos r

).

Exercises and Further Results

225

Thus the Cartan decomposition U = K A K ([DS], Chapter V, Theorem 6.7) gives for u E U u = kea,kb,

which under AdU gives the Euler angles decomposition of a rotation in R3. (iv) Since

where

define

Then (in the sense of exterior multiplication) dz d2 dw dw

=

i r 2 sin t dr dt d0 d#.

The group SU(2) is given by r = 1, i.e., the sphere x: = 1. Thus sin t dt dU d 4

+ xi + xi + x i

is an invariant measure on SU(2). This confirms Theorem 5.10 in this case. More precisely, the normalized Haar measure d c r on SU(2) is given by

s

j ( u ) du = -Jozsin t dt so2nd0 S f : ; f ( k O

SrJ(2)

1 6n2

, )d4.

ak

B. Radon Transforms

1. Derive Theorem 2.10 in this chapter from Theorem 2.10 in the Introduction. 2. (i) The range 9 ( R 2 ) ^ of 9 under the Radon transform on R 2 consists of the functions $ e 9 ( P 2 )which when expanded in a Fourier series

$(eie, p ) =

1 $.tp)eine neZ

have the following property: For each

where $,, E 9 ( R ) is even.

ti E Z

226

[Ch. I

INTEGRAL GEOMETRY AND RADON TRANSFORMS

(ii) Generalize (i) to the Radon transform on R". 3. Let f-3 denote the Radon transform on R" and 4 + its dual. Show that the mapping T + is one-to-one on &'(I"'). 4. Let J1' denote the kernel of the dual transform 4 + on &(Pn). Then Jlr is the closed subspace of &(P") generated by the spaces

6 6

k , I e Z + . k -I>Oandeven.

E,Qp',

(Here E , @ p' is the space of functions 4(m, p ) of the form $(w, p ) = t,h(w)p' where t,h belongs to the eigenspace E, from Introduction $3.) 5*. The range &'(I?")of &'(R") under the Radon transform consists of the distributions C E ~ ' ( P "with ) the following property: For each k E Z + the distribution

4 E 8(Sn- l)-+

J

Sn-IxR

4(m)pk d W , p )

is a homogeneous polynomial in m l , . . . ,on of degree k (cf. Helgason [1983al). The result can also be stated (see B4) b'(R"j = { C E b ' ( P ) :C ( N ) = O } . In particular, 8'(R")- is closed in &'(P").This implies by purely functionalanalytic results (Hertle [1984]) that the mapping C#J + d) of 8(Pn) into 8(R") is surjective. C. Spaces of Constant Curvature

1. (i) (Quadric Model) The quadric Q - l given by B-,(x) = x12

+ ... + x,

2

- x n2 + l

= -1,

x n + l > 0,

with Riemannian structure induced by B (cf. [DS], Chapter V, Exercise 7). (ii) (Unit Ball Model) The space

x = (XI,

. . . )Xn+l),

has constant curvature

n

B,(O) = {Y :

IY

1 < 117

Y = ( ~ 1 ,. .

. 3

YnX

I Y I'

with the Riemannian structure ds2 = 4(1 - ly12)-'(dy:

+ * . . + dy;),

has constant curvature - 1. (iii) (Half-Space Model) The space (21,

. . . , z,)

E R",

Z,

> 0,

=

1.Y?, 1

-1

Exercises and Further Results

227

with the Riemannian structure da2 = (dz:

+

. + ~z,Z)/Z,Z

* *

has constant curvature - 1. (iv) The mapping y = @(x) (x E Q-

given by

with inverse

is an isometry o f Q - l onto B,(O). given by

(v) The mapping z = Y(x) (x E Q(31,

. . ., z,) = (x,+

I

- xn>-I(x1, . .

. , xn- 1,

1)

with inverse

where

121

=

(z:

+ . . . -I-z:)~'~, is an isometry o f Q - l

onto H , .

2. On the unit sphere S 2 we have the formulas

_sin_ a_ _ _sin- b sin A

sin B

cos u = cos b cos c

sin c sin C"

--

+ sin b sin c cos A ,

for a geodesic triangle with angles A, B, and C and sides of length a, b, c. The duality between compact and noncompact symmetric spaces ([DS]. Chapter V.##2-3) suggests the analogous formulas sinh a sin A

-

sinh b sin B

cosh u = cosh b cosh c

(2)

-

-

sinh c sin C '

sinh b sinh c cos A

on the hyperbolic plane with curvature -1. (i) Verify (1) and (2). (ii) Generalize (1) and (2) to the hyperbolic plane with curvature -1;-.

7

(iii) What happens as

E -+

O?

228

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

D. Some Results in Analysis

1. Let k E Z+. Show that the polynomials x -,(x, w ) ~(oE S"- ') span the space of all homogeneous polynomials of degree k . 2. Show that the function

belongs to L2(R")but is not integrable over any plane of dimension 2 f n (Solmon [1976]). This example exhibits the difficulties in defining k-plane transforms directly on LP(R");while the distributional definition essentially requires compact support, it is well suited to nonsmooth behavior. 3. Let F E 9(R2)and as in Lemma 6.19 define g by g ( r ) = r JomF(re'. re-')(e'

- e-')

r # 0.

dt,

Prove that

iim g ( r ) = r-O+

JOm

lim g(r) = - Jom~(x.0) ilx.

~ ( x 0) , dx,

r-0-

Thus g may not extend to a continuous function around r = 0 although g' does (Harish-Chandra (1964~1). 4. Give a proof of the Poisson equation in R2, L(1og r ) = 2118, by use of the formula Lr" = a2rQ-2. 5. Let m , , . . . ,m, be distinct real numbers and put r

f(t)=

1cjeimJ',

t E R,

j= 1

c , , . . ., c, being complex numbers. Using lim ~ + +

I mT

T

- Jo If(t)12 dt =

i

deduce that

(*I

Iim supI.f(t)I 2 r-+m

C JciI2,

{1 i

r"

IciJ2

I

Now let k , , . .., k, be distinct complex numbers and p , , .. ., pr polynomials in t with complex coefficients such that all pi # 0. Suppose lim sup(p,(t) i++m

+ pl(t)ekl' + . . + pr(t)ekr.'(I a.

229

Notes

Using (*), deduce that po is a constant and lpol I a. Moreover, if Re k i 2 0 for each i, then each pi is a constant and each ki is purely imaginary (cf. Harish-Chandra [1958a], 51 5).

NOTES With differential forms viewed as "infinitesimal volumes," the notion of their integrals (integral invariants) becomes a natural one (Crofton [ 18681, Poincare [ 18871, Cartan [I8961 [1922], Lie [1897]). In Chevalley [I9461 differential forms appear as smooth families of multilinear functions on the tangent spaces. In $1 we follow his treatment of their integration. The Riemannian measure is usually presented under an unnecessary orientability condition. The invariant integral on a compact Lie group was used by Hurwitz [I897], Schur [1924], and Weyl [1925]. The Haar measure was constructed by Haar [1933] on locally compact groups. Invariant measures on coset spaces occur in special cases in classical integral geometry (Crofton, Czuber, Deltheil, Herglotz, Blaschke), but Theorem 1.9 (for locally compact coset spaces) was first proved by Weil [I9401 (see Chern [I9421 for the Lie group case, described in Exercise A3). Proposition 1.12 is from Harish-Chandra [1953]; see also Mostow [1952]. This $ 1 differs little from Chapter X, 91. Nos. 1, 2, in Helgason [1962a]. 92, Nos. I , 3, 4. The inversion formulas (1)

J ( ~= ) t(2rri)l - n L l l i Z ) l n -

1)

, J ( w((0,x)) dw

(n odd),

(ii) for a function , / E ~ ( R " )in terms of its plane integrals J(cu, p) go back to Radon [I9171 and John [1955]. According to Bockwinkel [1906], the case n = 3 had been proved already before 1906 by H. A. Lorentz. In John [I9551 proofs are given based on the Poisson equation Lu = f . Alternative proofs, using d!stributions, were given by Gelfand and Schilov [1960]. The dual transforms ,f + f , 4 + 4, the unified inversion formula, and its dual cf =

L l l i 2 ) f n - 1 ) ((,f)"),

c 4 = n"'2""-1)((4)-)

were given by the author in [1964b]. The proofs from Helgason [1959] (Chapter IV), and [1965a] are based on the Darboux equ'ation (Lemma 2.14) and therefore generalize to two-point homogeneous spaces. Formulas (17) and (55) were already given by Fuglede [1958]: according to Radon [1917], the first formula had even been observed by Herglotz. The modified inversion formula (Theorem 2.16) and Theorem 2.17 are proved in Ludwig [1966]; the latter result is attributed to Y. Reshetnyak in Gelfand et a/. [1966]. 92. No. 2. The support theorem and the Paley-Wiener theorem (Theorems 2.4, 2.6, and 2.10) were first given in Helgason [1964b. 1965a1. The example in Remark 2.9 was also found by D. J. Newman. cf. B. Weiss's paper [1967]. which gives another proof of the support theorem. The local result in Corollary 2.12 goes back to John [1935]; our derivation is suggested by the proof of a similar lemma in Flensted-Jensen [1977b]. p. 81. Another proof is in Ludwig [1966]. Corollary 2.8 was derived by Ludwig [I9661 in a different way. He proposes alternative proof of the Schwartz and Paley-Wiener theorems by expanding p ( o , p ) into spherical harmonics in OJ. However. a crucial point in the proof is not established and seems difficult

230

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

, f ( < )in (2.7), p. 57. at the point 0. (This is the function F in our proof of Theorem 2.4; here the smoothness of F at at 0 is the main point.) Since the inversion formula (Theorem 2.13) is rather easy to prove for odd n, it is natural to try to prove Theorem"2.4 for n odd by showing directly- that if ~ E . Y , , ( R " ) , then the function f = L'l'z)(n-l)(b)belongs to Y ( R " ) [in general, b#Y(R")]. This approach is taken in Gelfand et a/. [I9661 (pp. 16-17); however, this method seems to me to offer some unresolved technical difficulties. Corollary 2.5 is stated in Semyanistyi [1960]. In [I9841 Hertle has shown t h a t f + f i s not a homeomorphism of 9(R")onto its image inside 9(P"). $2. Nos. 5, 6. Thc approach to the Radon transform of distributions adopted hcrc is from the author's paper [1966a]. Other methods are proposed in Gelfand et a/. [1966], Ludwig [ 19663 [where formula (46) is also proved], Ambrose (unpublished), and Guillemin and Sternberg [19771. The d-plane transform, Theorem 2.27 and Corollary 2.28, characterizing the range are from Helgason [1959, 1980~1;an Lz version of Corollary 2.28 was given by Solmon [1976], p. 77. A different characterization of the image for d = 1, n = 3 was given by John [1938]. For further work in this direction see Gclfand and Graev [I9681 and Guillemin and Sternberg [1979]. $2, No. 7. The applications to differential equations given here are based on John [1955]. Other applications of the Radon transform to partial differential equations with constant coefficients can be found in Courant and Lax [1955], Gelfand and Shapiro [1955], John [195S]. Borovikov [IYSY], Garding [IY61]. Ludwig [1966], and Lax and Phillips [1967]. Applications of the Radon transform in medicine were proposed by Cormack [1963, 19641 and in radioastronomy by Bracewell and Riddle [1967]. See Shepp and Kruskal [1978], Smith et a/. [1977], and Shepp et a/. [I9831 for further account of applications. 52, No. 8. For a thorough treatment of distribution theory on R" and on manifolds see Schwartz [19663 or Trkves [19673. A concise, but nevertheless self-contained, treatment of the basics is given in Hormander [1963]. A more systematic study of the potentials 1" and the distributions r', x$ is given in Riesz [1949], Schwartz [1966], and Gelfand and Schilov [1960]. A more general version of Proposition 2.38 is given in Ortner [ 19801. $3. The notion of incidence in a pair of homogeneous spaces goes back to Chern [1942]. The duality between to settle in the context: the smoothness of the function

5

=

(i)

and

GIH,

GIH,,

the correspondences (ii)

x

+

i,

5 + i,

and the Radon transjbrrn for a double fibration of a homogeneous space were introduced in the author's paper [1966a], from which most of this section comes. A further generalization, replacing the homogeneity assumption by postulates about the compatibility of the entering measures, was given by Gelfand et al. [1969]. For the case G = U(4), H, = U(1) x U(3), H, = U(2) x U(2), the maps (ii) become Penrose correspondences (Penrose [ 19673) described in more detail in Example (iii) in No. I . The example where X is the set of p-planes in R", 2 the set of q-planes (n = p + q + 1) is worked out in Helgason [1965a] with solutions to Problems A, B, and C. However, because of the convergence problems encountered there, it may be more satisfactory

Notes

23 1

to adopt the group-theoretic incidence definition in the present section. If xo I to,xo n to # 0, then x and 5 are incident iff they intersect under a right angle. Here Problems Band C have been solved by Fulton Gonzales. $4. It was shown by Funk [I9161 that a function f on the two-sphere, symmetric with respect to the center, can be determined by the integrals off over the great circles. In [I9171 Radon discussed this problem and the analogous one of determining a function on the non-Euclidean plane from its integrals over all geodesics. The Radon transform on hyperbolic and on elliptic spaces corresponding to totally geodesic submanifolds was defined in Helgason [1959]. where the inversion formulas in Theorems 4.5 and 4.7 were proved. A generalization was given by Semyanistyi [1961]. An alternative definition, with corresponding inversion formulas, was given in Gelfand ef. al. [1966]. A support theorem and a Paley-Wiener theorem for this transform, valid for symmetric spaces, were proved by the author in [1973a] 58 (cf. specializations in [1983a, 1983b1). Some local support theorems and L2 Paley-Wiener theorems for the case of hyperbolic spaces were proved by Lax and Phillips [1979, 19821. The support theorem (Theorem 4.2) was proved by the author [1964b, 1980bl and its consequence, Corollary 4.18, in [198Od]. The theory of the Radon transform for antipodal manifolds in compact two-point homogeneous spaces (Theorem 4.1 1) is from Helgason [1965a]. Another proof of the inversion formula is given in Grinberg [1983]. R. Michel has in [I9721 and [I9731 used Theorem 4.11 in establishing certain infinitesimal rigidity properties of the canonical metrics on the real and complex projective spaces. Funk [1916], a paper noted earlier, showed that the standard metric on the projective plane cannot be deformed in such a way that the geodesics remain closed and of the same length. For S2, however, such a deformation is possible (cf. Guillemin [1976]).

55. While the volume element on the orthogonal group goes back to Schur [1924], the general formula in Corollary 5.16 is due to Weyl [1925]. Its generalization, Theorem 5.10, is from Cartan [1929] ($23). The other integral formulas are for the most part due to Harish-Chandra [1953], p. 239, [1954d] p. 507, [1956a], 512, and [1958a], pp. 261, 287. For the case of the complex classical groups many such integral formulas are proved in Gelfand and Naimark [1957]. The proof of Lemma 5.4 (from Harish-Chandra [1954d] p. 507) is due to Wallach [I9731 (p. 174), and so is Proposition 5.21. Proposition 5.18 is from Helgason [1980a], and the proofs of Harish-Chandra’s results 5.8, 5.19, and 5.20 are from some old unpublished lectures by the author. The presentation of Lemma 5.14 and Proposition 5.15 is much influenced by that of Stminaire Sophus Lie [I9551 (Exp. 19). $6, No. 1. The construction of the constant curvature spaces (Theorem 6.1) is from Helgason [1959, 19611. The proof of Lemma 6.2 on the connectivity is adapted from Boerner [1955]. For more information about these spaces as well as on isotropic manifolds see Wolf [1967]. $6. Nos. 2-4. The solution of the orbital integral problem on Lorentzian manifolds G / H of constant curvature given here is based on Helgason [1959]. For some related work for arbitrary signature see Faraut [1979]. $6, No. 5. The Plancherel formula for the Fourier transform on a complex semisimple Lie group G was given by Gelfand and Naimark [I9481 for G = SL(n, c) (see also [1957]) and by Harish-Chandra [1951b, 1954d] for general G. In [1955], Gelfand and Graev observed that the Plancherel formula for complex G results from an expression of the value of a function f E 9(G) at e in terms of the integral of f over each conjugacy class. As we noted in 53, No. 3, this amounts to the orbital integral problem for G x GIAG, which, in the quoted paper, Gelfand and Graev solved when G is a complex classical group.

232

INTEGRAL GEOMETRY AND RADON TRANSFORMS

[Ch. I

For G real and semisimple the solution of the orbital integral problem on G x G/AG was given by Harish-Chandra [1957b]. Theorem 6.18 in the text is the special case G = SL(2, R). In contrast to the case of complex G , the orbital integral problem is only a partial step towards the full Plancherel formula (Harish-Chandra [1970]). The orbital integrals considered in the above papers only involve orbits of semisimple gEG. However, orbital integrals exist for arbitrary g E G (cf., Rao [1972]) and have been studied by Harish-Chandra, Barbash [1979], Barbash and Vogan [1980, 19821; these offer new and interesting features. As a synthesis of the results above it seems to be a promising research project to develop an orbital integral theory (cf., $3, Problem D) for affine symmetric spaces C/H ( G semisimple) including a joint generalization of Theorem 6.17 and Theorem 6.18. For G / H of “rank one” this has been carried out by Jeremy Orloff.

CHAPTER II

INVARIANT DIFFERENTIAL OPERATORS While applications to differential equations served as a motivation for Lie's theory of transformation groups, the theory of Lie groups and Lie algebras has grown into a force in its own right, exerting ever-increasing influence on many other fields of mathematics. In particular, with Lie group theory so highly developed, it is reasonable to reverse Lie's original viewpoint, that is, consider the group as the given object and investigate differential operators invariant under its action. Such invariance conditions are very natural, and the literature abounds with examples; see $4, No. 1 for a few elementary ones. In $1 we review some structure theorems for CD(R") and a coordinate-free characterization of differential operators. In 92 we discuss for a manifold M the spaces 9 ( M ) = C:(M), I ( M ) = Cm(M), and their topologies, distributions, and differential operators on M with particular emphasis on the Laplace-Beltrami operator on pseudoRiemannian manifolds. In 93 we discuss projections of differential operators on submanifolds; also, when a group is acting on V we decompose an arbitrary differential operator on V into a "polynomial" in orbital operators with transversal operators as "coefficients." For the Laplacian this becomes a kind of a Pythagorean formula [see (21) in $31. When a cross section W is selected for the group action on V, each differential operator on V has a radial part on W. When the orbits intersect W orthogonally, the radial part of the Laplacian has a simple geometric expression which in $3, No. 4 is worked out for some examples. In $ 3 4 5 we consider the invariant differential operators on a homogeneous space G/H and describe these in terms of the Lie algebras of G and of H. The case of a symmetric space is studied in more detail: we relate the operators to Weyl group inyariants and for the rank-one case prove generalized forms of the Darboux equation and Asgeirsson's mean value theorem and give an explicit solution of Poisson's equation. Symmetric spaces G / K with G complex have some special features worked out in No. 8-9.

51. Differentiable Functions on Rn

Let R" denote the n-dimensional Euclidean space and x = (xl,. . . ,x,) an arbitrary point. Put 1x1 = (x: + + x;)l/'. If V c R" is an open subset, let a(V ) denote the space of complex-valued differentiable functions on V, and 9 ( V ) the space of functions in B ( V ) with compact support contained in I/. While the space &(V) is the oneof principal interest, 9 ( V ) is often more convenient to work with because 9 ( U ) c 233

234

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

9 ( V ) if U c V . Let ai denote the partial differentiation a / a x i , and if a (al,. . . , a,) is an n-tuple of integers ai 2 0, we put ~a

=

8 7 1 . .a?, .

l a [ = a1

xu

+ ... + an,

.

= x y . . ,:x

a! = a1!*.*an!.

If /3 = (pl,. , . ,fin) is another integral positive n-tuple and allj, we write p I a and put a-

=

fij I

aj for

p = (xi - Pi, . , a, - P n ) i

(2

a! = P!(a -

p)!.

We have then the generalized Leibniz rule for the differentiation of the product of two functions .f and 9 :

(Cf. Hormander [1963], Chapter I ; an identity of this type is in fact obvious and the coefficients can be determined by taking f and g to be exponent ials.) If S c V is any subset and m E Z + , we put (2)

Ilrll:

=

c SUPI(DY')(X)I.

lui<m x s s

When S = V we often drop the superscript S in the notation. Then we have for f , g E &(V),

1I.f + S l l i I II.rll: + Ilsl:. (3) Lemma 1.l. Let C c U c V c R", where U and V are open and C a closed subset of R". Then there exists a junction 4 E & V ) such that 4 = 1 on C , 4 = 0 on V - U,and 0 I 4 I 1 everywhere. For the proof see, e.g., [DS] (pp. 88 and 538), valid for V replaced by a paracompact manifold. Using the seminorms (2) we can prove a sharper version. Proposition 1.2. Let C c R" be any closed subset. Then there exists a ~ E & ( R "such ) that

C = { X E R" : 4 ( ~=) 0 ) . Proof. If V, is a (l/k)-neighborhood of C , then C is the intersection of the open sets V, (k 2 I). Let +k€b(R")such that 4k = 0 on C , 4k= 1 on R" - V,, 0 I 1. Let B,(O) denote the ball 1x1 < r and put Ilfll; for the seminorm (2) when S = B,(O).

911 Choose

1. Differentiable Functions on R" ck

235

> 0 such that m

and put

Then for a given k , if m' 2 m > r, k ,

and this tends to 0 as rn desired properties.

+

co. The limit

4 = limm+,,,(I,

clearly has the

Lemma 1.3. Let m > 0, and suppose f E d(R") has all derivatives Im Eanish at 0. Then given E > 0 there exists a g E &R") vanishing in a neighborhood o f 0 and satisfying

of' order

119 -

fllm

< E*

) that 4 ( x ) = 0 for 1x1 Ii, 4(x) = 1 for Proof. Let 4 ~ d ( R " such 1x1 2 I , and 0 I 4 5 I everywhere. For S > 0 put gab)

=

4(X/~)./'W

Then gaE b(R"), ga = 0 ncar x = 0, and gs(x) = f ( x ) for 1x1 2 S. Thus it suffices to prove that if IctJ I m,

(4)

sup I Daga(x)- D " f x ) I + 0

as 6 -+ 0.

1x1 5 6

But by our assumption about j ; (D"J'(0) sup I(Dy')(x)I + 0

(5)

=

0 for la1 I m, SO as 6 + 0.

IXlSd

Also, by (I),

so for a constant C,

(6)

IDag6(x)I I C

1S-IvII(D"f)(x)I,

X E

R".

p+v=a

But Dyf has all derivatives up to order m - Ipl vanishing at x = 0. Thus SUP(D~'(X)I= 0(6"-~"'

1 7

IXlS6

236

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

so by (6)

suPlD"g*(x)l = o( 1x15 6

1 gm-Irl-lvl p+v=a

1

= O(gm-ld

)7

and now (4)follows if we take ( 5 ) into account. Definition. Let V c R" be an open set. A diflerential operator on V is a linear mapping D : 9 ( V ) + .9(V) with the following property: For each relatively compact open set U c V such that B c V, there . . . ,a"), ai E Z+] exists a finite family of functions a, E b ( U ) [ a = (al, such that

(7)

D4

=

1aaDu4,

+E~(U).

a

It is clear that a differential operator D on V has the property

$ E 9(V ) , SUPP(D$) = SUPP($). supp denoting support. This property implies that D can be extended to a linear operator (also denoted D ) from b(V ) to &( V ) by means of the formula

(4f)W = (W)(X). Here x E V, f E b( V ) are arbitrary and 4 is any function in .9(V ) which

(8)

coincides with f in a neighborhood of x. The choice of material.

4

is clearly im-

Theorem 1.4. Let D : 9(V ) + 9(V ) be a linear mapping satisfying the condition (9)

SUPP(D4) = SUPP(4),

4 E %V).

Then D is a diflerential operator on V. Conversely, any diflerential operaror on V satisfies (9).

Remark. If D is assumed continuous [in the topology on 9 ( V ) defined in 82, No. 23, then for X E I/ the mapping 4 + ( D ~ ) ( x )is by (9) a distribution with point support. Hence (7) follows immediately from Schwartz's theorem that a distribution with point support is a linear combination of derivatives of the delta distribution. The main point of Theorem 1.4 is therefore that this continuity of D is automatic. Proof. Suppose D : 9 ( V ) -,.9(V) satisfies (9) and extend D to &(V) by (8). Next we shall prove that for each point U E V there exists an open

911

1. Differentiable Functions on R"

237

relatively compact neighborhood U of a, ( 0 c V ) , an integer m, and a constant C such that

U E % U - {a}). Suppose this were false for a point a. Let U o c V be an open relatively compact neighborhood of a, U o c V. Then there is a function

(10)

IlD~llo5 CIIUllm,

u1 E ~ ( U ' {a})

such that

> 2211u1II 1.

I P U l I10

If U , = { x : u , ( x ) # 0}, then U o - D, is an open neighborhood of a, so by our assumption there is a function u 2 ~ 9 ( U 0 Ul - { a } ) such that ll~u2Ilo> 2411u2112. Let U , = {x : u 2 ( x ) # 0). By induction we obtain a sequence U , , U , , . . . of open sets such that (1 1 )

u k

c

-

uo- { a } ,

DI= (ZI

Uk

for k # 1,

and functions (12) satisfying

ukEg(Uo u k

=

u1 -

-

"'

Uk-1

-

{ a } )c 9 ( V )

{ x :uk(x) # 0 ) and

(13)

~

~

> 22kllUkllk* ~ ~

k

~

~

0

Because of ( 1 1) and (12) the sum

is a well-defined function in 9 ( V ) , and on u k , u coincides with the function 2-kllukll; 'uk. Hence if I denotes restriction, (9) implies, DUl

Uk

=

2-kllUkll;1(DUk)I u k .

Thus, using (13), we conclude that there exists a point

XkE u k

such that

(OU)(xk) > 2k,

and this contradicts the boundedness of Du. This proves (10). Lemma 1.5. Let D : 9 ( V ) + 9 ( V ) satisfy (9). Let U c V be any open set. Assume there exist constants C > 0, m E Z' such that

(14)

IIDuII, I CIIu1Im

,fbr u ~ 9 ( U ) .

238

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

Then there exist ,functions a, E a(V ) such thal

1 a,(x)(Dau)(x>,

( D u ) ( x )=

(1 5 )

x E U, u E9(U).

l4sm

Proof. For each a E V let Q,Jx)

= (XI- a,)"' * . * ( x ,- a,)"".

Since this is a polynomial in (al,. . . , a,) with coefficients which are polynomials in ( x l , . . . , xn), the function

ba: a (DQa,a)(a) belongs to a(V ) . Now let u E 9(U ) and a E U.Then the function +

f

=u

-

1 lalsm

1

7 (D"u)(a)Qa,a a*

satisfies (D"f(a) = 0 for la1 5 m. Now we use Lemma 1.3 and its proof to approximate f in the seminorm 11 [I, by functions g v which coincide with .f outside some neighborhood N , of a, but vanish identically near a. From (9) [and (8)] we conclude that Dg, vanishes identically near a. Now using (14) on u, = f - g,, we conclude that Du,(a) -,0, so

(W(4= 0. This means

which gives the conclusion (15) with a ! a, = b,. We can now prove Theorem 1.4. Let U c V be open, 8 compact, B c V. Applying (10) to each point of 8, we get a finite covering of 8

by open relatively compact sets U 1 , . . . , U,, points a l 6 U1,. . . , a,€ U,, and constants C > 0, m E 2 ' such that for each i ( 1 I i _< r )

l l D ~ l5l ~CllUllm

(16)

1;"

for u e 9 ( U i - {ai}). Let 1 = + i be a partition of unity for the . . , U,, V - of V ([DS], p. 89) and let covering U1,. uE9(V

1;"

1;

- {a,}

- .* .

- {a,}).

Then u = +iu = +iu and (16) holds for each 4iu. Using (l), we deduce that (16) holds for u itself (with another C ) . The lemma then implies that (15) holds for X E U , x # a l , . . . , a,. But then (15) holds for all x E U , both sides being continuous. This proves Theorem 1.4.

423

2. Differential Operators on Manifolds

239

52. Differential Operators on Manifolds 1. Definition. The Spaces 9 ( M ) and S ( M )

Let M be a manifold. Motivated by Theorem 1.4, we define a diflerential operator D on M to be a linear mapping of C:(M) into itself which decreases supports: SUPP(Df) = SUPP(.f),

f

E CW).

If ( U , 4 ) is a local coordinate system on M , the mapping

4)) 4 - ' ,

(1)

D4: F

(2)

Of = 1 aa(Da(f

+ (D(F

F E Cp(4(U)), satisfies the assumption of Theorem 1.4. Using (7) in $1 we obtain for each open relatively compact set W , with W c U , a finite family of functions a, E Cm(W ) such that a

0

0

0

4-l)) 4, 0

f E C:(W>*

Thus the definition above of a differential operator coincides with the customary one. Just as for open sets in R" [cf. (8) in 511 we can extend a differential operator to Cm(M). We also adopt Schwartz's notation 9 ( M ) = C,m(M),

d(M) = Crn(M),

and if K c M is any compact subset, g K ( M )denotes the set of functions in 9 ( M ) with support in K . For convenience we now make the assumption that M has a countable base,for the open sets. We shall now give S ( M ) a topology. 2. Topology of the Spaces 9 ( M ) and S ( M ) . Distributions

If V c R" is an open set, d ( V ) is topologized by the seminorms I l f l l : [see (2) in 511 as C runs through the compact subsets of V and k runs through 2'. If ( U , 9) is a local coordinate system on M , this gives a topology of S(U) with the property that a sequence (f,) in b ( U ) converges to 0 if and only if for each differential operator D on U , D1;, + 0 uniformly on each compact subset of U . In particular, the topology of b ( U ) is independent of the coordinate system. The space b ( M ) is now provided with the weakest topology for which all the maps f + f I U (I denotes restriction), as ( U , 4) runs through all

240

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

local coordinate systems on M , are continuous. Here we can, by the countability assumption, restrict the charts ( U , 4) to a countable family ( U j , 4j),j = 1,2,. . . . Since each b ( U j ) is a FrCchet space, it follows that b ( M ) is also a FrCchet space, and again, a sequence (h) in b(M) converges to 0 if and only if for each differential operator D on M , Ofn + 0 uniformly on each compact subset of M . Writing M as the union of an increasing sequence of compact sets, we see that 9 ( M ) is dense in b ( M ) . We note also that a differential operator on M is automatically a continuous endomorphism of b ( M ) . For each compact set K c M , g K ( M ) is given the topology induced by b ( M ) . It is a closed subspace of b ( M ) , hence a FrCchet space. A linear functional T on 9 ( M ) is called a distribution if for any compact set K c M the restriction of T to g K ( M )is continuous. The set of all distributions on M is denoted 9 ' ( M ) . We often write jM f ( m ) dT(m) instead of T(f 1. We now give 9 ( M ) the inductive limit topology of the spaces Q K ( M ) by taking as a fundamental system of neighborhoods of 0 the convex sets W such that for each compact subset K c M , W n g K ( M ) is a neighborhood of 0 in g K ( M ) . With this topology of 9 ( M ) the continuous linear functionals T are precisely the distributions on M . In fact, if r > 0, T-'(B,(O)) is a convex set containing 0, so the continuity assumption amounts to the definition of a distribution. Thus 9 ' ( M ) is just the dual space of 9 ( M ) . A similar argument shows that a differential operator gives a continuous endomorphism of 9 ( M ) . A distribution T is said to vanish on an open set V c M if T(f) = 0 for each f' E 9 ( V ) ( c9 ( M ) ) . Let {U,}aeAdenote the family of all open sets on which T vanishes and U their union. Then T vanishes on U . In fact, if f E ~ ( U ) supp(f) , can be covered by finitely many U , , say U,,. . . , U , . Then { U , , . . . , U , , M - supp(f)} is a covering of M . If 1 = 4i is a. corresponding partition of unity, we have ,f = 4 i f , so T(f) = 0. The complement M - U is called the support of T. Let b'(M) denote the set of distributions on M of compact support. We shall now prove that this set can be identified with the dual space of b ( M ) .

1;"

c{

Proposition 2.1. The restriction of a functional from b ( M ) to 9 ( M ) gives a bijection of' the dual of b ( M ) onto b'(M). Proof, Let t be a continuous linear functional on d ( M ) , T its restriction to 9 ( M ) . Since T is continuous on each g K ( M ) ,it is a distribution. Also, t l # t2 implies Tl # T2 because 9 ( M ) is dense in b ( M ) . Next we prove that T has compact support. For this let K , c K 2 c be a sequence of compact subsets of M such that each compact set K c M is contained in some K , . If supp(T) were noncompact, we could find 4 i

~

921

2. Differential Operators on Manifolds

24 1

6 ( M ) such that 4, = 0 on K , but T(4,)= 1 ( i = 1,2,.. .). Then 4i --t 0 in b ( M ) , which is a contradiction. To see that the mapping t + T is surjective let S be a distribution of ) that 4 = 1 on a neighcompact support. Fix a function + E ~ ( M such borhood of supp(S). The mapping f + 4f is a continuous mapping of b ( M ) into itself; hence we can define a continuous linear functional D on & M ) by

41)= S(4!), the choice of 4

(3)

f

It is independent of neighborhood of supp(S), then

S(41.f) - S(4f) =

E

mw.

because if

E ~ ( M )41 , = 1 on a

w41 - 4)f) = 0,

since supp((4, - 4)s) c M - supp(S). The same argument shows that S coincides with the restriction of D to 9 ( M ) .

3. Effect of Mappings. The Adjoint

Let E ( M ) denote the set of all differential operators on M . Iff E b ( M ) , D E E ( M ) , the value ( D f ) ( p ) of Df at p will sometimes be denoted LIP(!@)). The composition of two differential operators D 1 and D z will often be denoted D1 D 2 . Let M and N be manifolds and 4: M + N a differentiable mapping. The differential of 4 at p E M which maps M, into N 4 ( p )is denoted d4p. If 4 is a diffeomorphism of M onto N and if f ~ 9 ( N ) g, E d ( N ) , T E 9 ' ( M ) , D E E(M), we put 0

(4)

9"'

=g

0

4,

T ' ( f ) = T ( f '- ') ,

D'(g) = (D(g'-'))'.

Then g'-' E b ( M ) , T' E 9 ' ( N ) , and D' is a differential operator on N , the image of D under 4. Suppose 4 is a diffeomorphism of M onto itself; D is said to be invariant under 4 if D' = D, that is, if Dg = (

Wo4))04-'

for all g E b ( M ) . A measure p on the manifold M is said to be equivalent to Lebesgue measure if on each coordinate neighborhood on M it is the multiple of the Lebesgue measure by a nowhere vanishing C"-function. In this case the adjoint D* of D E E ( M ) is defined as the operator on 9 ' ( M ) which is the transpose of D : (5)

( D * T ) ( f )= T (D f ),

f

E9(M),

T E 9YW.

242

[Ch. I1

INVARIANT DIFFERENTIAL-OPERATORS

The space b ( M ) is imbedded in 9 ’ ( M ) by means of p if to g E b ( M ) we associate the distribution

f

+

J fg

f

44

M

EWM),

on M . Thus the restriction D * l b ( M ) is well-defined. Also, D*, as an operator on 9 ’ ( M ) , decreases supports, so if { x ~ , . ,. x m } is a coordinate system on an open set U and g E 9 ( U ) , the distribution D*g is determined by its restriction to 9 ( U ) . But if in terms of the local coordinates we calculate the usual adjoint D‘ of D on U , we have (6)

J;DMx).f(x) 4 4 x ) =

J g ( x ) ( D f ) ( x ) 444, U

f E 9W),

so the distribution D*g equals the function D’g on U . This shows that D* maps d(M) into itself and is a differential operator. We remark that while this definition of the adjoint D* depends on the measure p, it does not require orientation of M. Now if E E E(M), T E 9’(M), we define ET E 9 ’ ( M ) by ( E T ) ( . f )= T(E*f), If T is a function g . i.e.,

.f E W M ) .

T(f) = J f g 4 4 then formula (6) shows that ET is the function Eg. If morphism of M leaving p invariant, then (E*)’ = (E‘)*,

T

is a diffeo-

(ET)’ = E‘T‘

for E E E ( M ) and T E 9 ’ ( M ) . 4. The Laplace-Beltrarni Operator A pseudo-Riemannian manifold M always possesses a differential operator of particular interest, the so-called Laplace-Beltrarni operator which we shall now define. Let g denote the pseudo-Riemannian structure on M and let 4: q -P ( x I ( q ) ,. . . , xm(q)) be a coordinate system valid on an open set U c M. As customary we define the functions g i j , g i j , g on U by

623

2. Differential Operators on Manifolds

243

We often write ( , ) in place of g and extend it bilinearly to complex vector fields. Each C"-function f on M gives rise to a vector field grad f (gradient of .f) defined by (grad f ,x > = X f

(7)

for each vector field X . In terms of the coordinates on U , we have on U grad f = 19'' i. j

a aif -, axj

where aif = (a/dxi)(f'). On the other hand, if X is a vector field on M, the divergence of X is the function on M which on U is given by

c

1 div X = - ai(&Xi)

(9)

& i

if X = Ci X i ( d / a x i ) on U . To see that div X is well-defined (independent of the coordinate system), we interpret it in terms of the Riemannian measure. Let O ( X ) denote the Lie derivative by X as defined in [DS, Exercise B, Chapter I]. We shall use some of its properties established there. Lemma 2.2. Suppose U is connected and let wu denote the differentia1,form on U given by wu =

&dx,

A

- A dx,.

O ( X ) w , = (div X)wu. Proof. Since O ( X ) commutes with the exterior differentiation d, we have

O(X) dxi = d e ( X ) x i = d X x i = dXi =

1( d j X i )d x j . j

Since O(X) is a derivation of the Grassmann algebra of U , we obtain

O(X)(& d x , =

A

... A dx,)

X(,/gj) d x ,

A

- .A

dx,

244

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

where the parenthesis on the right occurs at the ith factor. It follows that the right-hand side equals

which, by comparison with (9), proves the lemma. Now let { y l , . . . , y,} be another coordinate system on U and put

h

=

Idet(hij)l.

Then

so if J denotes the Jacobi matrix J=

a(x,, a(y 1,

. . . ,x m ) * * 9

Ym)’

we have

h

JI2g On the other hand, we have (cf. [DS],Chapter I, 82, No. 3) dx,

A

A

= ldet

dx, = det J dy,

A

A

dy,.

Hence, if det J > 0,

h1I2 d y ,

A

..

A

dy, = g1I2d x ,

A

. . . A dx,.

Thus (11) shows that the right-hand side of (9) is the same for all coordinate systems on U which satisfy the positivity condition det J > 0 relative to the original {x,, .. ., x,}. But the right-hand side of (9) is obviously also invariant under the coordinate change ( x , , x 2 , . . . , x,) + (x2, x l , . . . ,x,); thus it is invariant under all coordinate changes. Hence divX is a well-defined function on M (without the customary assumption of orientability). Remark.

Another justification of the definition of div X comes from

Lemma 2.5. The Laplace-Beltrami operator L on M is now defined by (12)

Lf

= div

grad f,

f~6(M).

2. Differential Operators on Manifolds

$21

245

In terms of local coordinates we have

so L is a differential operator on M . Proposition 2.3. Let M be a pseudo-Riemannian mangold, L the Laplace-Beltrami operator on M. Then L is symmetric; that is,

JM

u ( x ) ( L u ) ( xdx ) =

JM

u E Q(M), u E b ( M )

(Lu)(x)u(x)dx,

ifdx is the Riemannian measure on

M.

Proof. Let X be any vector field on M . Then we have from the definitions (8) and (9),

div(uX) = u div X

(14)

(15)

+ Xu,

grad u (0) = grad u ( u ) = (grad u, grad

0).

Consequently, uLu - uLu = div(u grad u ) - grad u (u) - div(u grad u ) + grad u (u), so uLu - uLu = div(u grad u

- u grad u).

It suffices therefore to prove

X ) dx

J;div

=

0

for any vector field X on M vanishing outside a compact subset. While this could be derived from (1 1) above [see also (2) in $1 of Chapter I and [DS], Exercise B3, Chapter I], we can proceed more directly as follows. Using partition of unity, we may assume that X vanishes outside a coordinate neighborhood U . Writing X = Xi a/Bx, on U , we have

xi

SM

1 ai(&Xi)

(div X) dx = S

d x , . . . d x , = 0,

i

proving (16) and the proposition.

In connection with (14) and (15) we mention the simple identity (17)

L(uu) = u(Lu) + 2(grad u, grad u )

+ ~(Lu),

246

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

whose proof is left to the reader. Next we prove a simple invariance property of the Laplacian. If (V, $) is a local chart on M and f E d ( M ) , we often write f* for the composite function f 40

'.

Proposition 2.4. Let # be a difleomorphism of the pseudo-Riemannian manijold M . Then @ leaves the Laplace-Beltrami operator L invariant fi and only if it is an isometry. Proof. Let p E M and let (V, $) be a local chart around p . Then (#( V ) , I,$ 0 0-') is a local chart around #(p). For x E V, let y = W x ) and 44x1 = ( X I , * * * 3 xm), ($o@-')(Y) = (Yl,***,Ym),

x E V,

YEWV

Then

For each function f E &(M),

(19)

(Lj*-')(x) =

1

Because of the choice of coordinates we have aj* - a ( f O q * _ aYi

axi

ay* - a 2 ( f 0 @ ) * -, ayi ay,

axi axj

1 Ii , j Im.

Now if # is an isometry, then giJ(x) = gi,(y) for all i, j . Thus the righthand sides of (18) and (19) coincide and L' = L . On the other hand, if (18) and (19) agree, then we obtain, by equating coefficients, gi,(x) = gi,(y), which shows that # is an isometry. The Laplace-Beltrami operator was defined above in terms of the pseudo-Riemannian structure on M. We shall now describe it more directly in terms of the Riemannian connection V ($9, Chapter I in [DS]). Lemma 2.5.

If X is a vector field on M and p E M , then

(div X), = trace of the endomorphism v + V,(X) of M ,

§21

2. Differential Operators on Manifolds

247

Proof. Let {xl, ..., x,} be a coordinate system as in Lemma 2.2 around p. Denoting the endomorphism u + V , ( X ) of M , by Dx,we have by 55, Chapter I in CDSl

i

if X = X i Xi(d/axi). On the other hand, div X =

1 d i X i + 1 dj(log &)Xj, i

j

so it suffices to prove

The formula for V in [DS] (Chapter I, §9), can be written

(22)

2

1gi1r:k

= ajgik

1

+ &gij

-

digjk.

On the other hand, if G i j is the cofactor of gji in the determinant go = det(gij), we have G I.J . = g ogij'

(23)

Applying to g o the rule for differentiating determinants, we have

Now (22) implies (25)

adgij)

=

c (rfkgj1 + r:,cgiA 1

whereby the right-hand side of (24) reduces to

Since g = [ g o [ ,this proves (21) and the lemma. Proposition 2.6. I n an arbitrary coordinate system, the LaplaceBeltrami operator is given by

248 Proof.

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

Because of Lemma 2.5 and (20), we have

But differentiating

g'jgjk

(28)

a,(gir) = -

=

and using (25), we derive

dik

1 (rS1gij+ r;rgjr). i

Carrying out the differentiation in (27) and using (28), we get the result We consider now the case of a Riemannian manifold (i.e., g positive definite). Then (grad f,grad 3) =

1

gij

d i f djf 2 0.

Let us call a Riemannian manifold M simply convex if any two points in M can be joined by a unique geodesic. Every point in any Riemannian manifold has an open neighborhood, which with the induced Riemannian structure is simply convex (cf., [DS], Chapter I, Theorem 9.9). Theorem 2.7. Let M be an analytic simply convex Riemannian manifold and p E M . Given c E C there exists a sphere SR(p) around p such that

if Lu = CU, u E b ( M ) , and u

=0

on SR(p),

then u=O

on M .

Remark. The example u(t) = sin I t in R which satisfies Lu = -A2u shows that the radius R will depend on c. The proof below will show, however, that for a given eigenvalue c all sufficiently small R will do. Proof. Let N o be a normal neighborhood of the origin in M , ([DS], Chapter I, @), let SR(0)c N o be a sphere inside N o with center 0, BR(0) the corresponding open ball, and put SR

=

Expp(SR(o))? B R

=

Expp(BR(o)).

For u E b ( B R )we put llull = [JBRlu(x)12dx

+ JBR(grd

ti,

grad Q ( x ) dx

I

. 1'2

Let u ~ d ( M such ) that u = 0 on S R . We shall prove that given there exists a 4 E 9 ( B R ) such that

(29)

IIU

- $11 < E.

E

>0

$21

2. Differential Operators on Manifolds

249

For this let ( O l , . . . ,&), where 8, = r, be geodesic polar coordinates around p (cf. [DS], p. 543). For 6 > 0 small, let ad be an even function in b ( R ) satisfying 1. a6 = 1 on [0, R - 461, 2. a6 = 0 on [ R - 6, R ] , 3. lab] 5 d-’, 4. Ia6JI 1.

Let a E b(BR) be defined by a(x) = a,(d(p, x)), d denoting the distance. We shall prove that 4 = au satisfies (29) provided 6 is small enough. Since (r = 8,) grad u = gij(aiU)aj i. j and since a au aa -(au - u ) = (a - 1) - + - u,

aei

aei

aoi

it would be sufficient to prove

But if g(r, e l , .. . , 8m-1)= Idet(gij)l, we have

Using now u

E

0 on S R , we have

Hence

Thus if D

=

maxIdu/ap(2 on B R , we obtain

IBR I2

u 1’dx

I 160 Volume(&),

where Ed is the “shell” R - 46 I d(p, x ) I R - 6. This proves (30) and therefore also (29). Now we choose an orthonormal basis of the tangent space M, and let {x,,. . . ,x,} be a system of normal coordinates with respect to this basis.

250

and then integrating over

(3 Now let

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

BR,

I4(x)I2 dxl

we derive the Poincare inequality,

dx, I 4R2

S.. x(aj4)’

dxl

s . 9

dx,.

i

/.BR

cR = inf gii(x),

dR =

i,xEBn

eR =

sup lgij(x)l i# j.xEBn

SUP~”~(X),

fR

XEBR

inf g ” 2 ( ~ ) .

=

XEBR

Then

We assume R to be so small that cR - (n - l)dR > 0. Then, by (31),

c

I~ R R ’ ( C R- (n - l ) d ~ ) - ’J

C g i ’ ( a i 4 ) ( a j 4 ) dX1 ...d x ,

BR i . j

I 4 e R f i 1 R 2 ( c R - (n - 1)dR)-l

Thus we have for small R > 0,

(32)

s1 BR i,i

g i j ( a i 4 ) ( d j 4 )dx.

4 E 9(B,) real,

IBR14(x)12d x 5 gR4R2 IBR(grad 4, grad

4 ) dx,

where g R is a constant depending only on the Riemannian structure of M and g R + 1 as R + 0. To finish the proof of Theorem 2.7 suppose Lu = cu, where C E Cand u = 0 on S R . Let 4, $ E 9 ( B R ) .By (17) we have

L(4$) = 4W

+ 2(grad

4, grad $) + $W.

$31

3. Geometric Operations on Differential Operators

25 1

Integrating this over BR and using (16) together with Proposition 2.3, we obtain IE:L$

dx = - IER(grad4, grad $) dx.

Using (29), we deduce by approximation

I,

uL$ dx =

(grad u, grad $) dx,

IB.

whence by Proposition 2.3 and Lu = cu c

S..

u$ dx =

-

(grad u, grad $) dx. JER

Again, by (29) this implies, letting $ + a,

s..

S..

1uI2 dx = (grad u, grad ii) dx. (33) Thus c is real (or u = 0 on BR), so we may assume u to be real. Again, by (29) we may replace 4 by u in (32). But if gR(4R2)(-c) < 1, we can conclude from (33) that u = 0 on B,. But since the operator L has analytic coefficients and is elliptic, its eigenfunctions are analytic (see, for example, John [1955], p. 57). Hence u = 0 and the theorem is proved.

Remark.

The proof shows that if c = 0, we can take R arbitrary.

$3. Geometric Operations on Differential Operators 1. Projections of Differential Operators

Let V be a Riemannian manifold, D a differential operator on V. If S c V is any submanifold, we shall now define the projection of D on S . For each S E Sconsider the geodesics in V starting at s perpendicular to S . If we take sufficiently short pieces of these geodesics, their union is a submanifold S,L of V. Putting this more precisely in terms of the exponential mapping Exp, for V ([DS],Chapter I, $6), we let V(0) be a spherical normal neighborhood of 0 in the tangent space V,, let V , = S, 0 S: be the orthogonal decomposition, and put S,l = Exp,(V(O) n SL). Fix so ES. Shrinking the S,l further, we can assure that as s runs through a suitable neighborhood So of so in S the manifolds S: are

252

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

disjoint and fill up a neighborhood Vo (a “tubular neighborhood”) of so in V :

u s:.

v, =

ssso

Given F E ~ ( S )we , define a function F on Vo by making it constant on each S,’ and equal to F on S o . Given the differential operator D on V, we define an operator D’ on 9(S) by

(2) (D’F)(so) = ( D m S o ) , Since F is C”, the right-hand side makes sense, and since D decreases supports, the right-hand side is independent of the choice of So and V, as long as we have the smooth disjoint decomposition (1). This D’F is a well-defined function on S . If we vary so slightly, decomposition (1) can still be used and the right-hand side of (2) varies smoothly with s o . Thus D’F is a C“-function on S. It is also clear that the linear map F + D’F decreases support, so D’ is a differential operator on S . Summarizing, we have proved the following result. Proposition 3.1. I f D is a digerential operator on V, ,formulas (1) und ( 2 ) dejine u rliJyi.rentia1 operator D’ on the submanifold S. Example. Consider the Laplacian L on R3 and let S denote the unit sphere x: + x: + xi = 1. If we express L in the spherical polar coordinates (r, 8,4), we obtain the standard formula

(3)

L

+ r ara + r2

= 7 --

-

+ cote-a + sin-2 8-).a2

w2

ar az ( aae2 2 ae The operator in the parentheses is clearly the projection L‘ of L on S . It is a familiar.fact, which we generalize below, that L‘ is the LaplaceBeltrami operator on S.

Theorem 3.2. Let V be a Riemannian manifold, S c V a submani.fold, and Lv and Ls the corresponding Laplace- Beltrami operators. Then Ls equals the projection of Lv on S :

Proof. We use the coordinate representation of Lv given by Proposition 2.6. Now let so E S be arbitrary and choose the local coordinates (xl, ..., x,) on a neighborhood of so in V in accordance with (1) as follows :

(i) The mapping s

+ (XI(*$,

. . . ,x,(s),

0,.. . 10)

is a system of local coordinates near so on S.

3. Geometric Operations on Differential Operators

431

a,,

253

(ii) For each S E S sufficiently close to so and any constants . . . ,a,, not all 0, the curve

(4)

t

--+

(x1(s), *

* *

9

x,(s), a,, I t , . * . 9 ant)

is a geodesic in V, starting at s, perpendicular to S . When writing out (2) in this coordinate system it is convenient to adopt the following range of indices: r+l
l
The geodesic (4)satisfies the differential equations

At the point s this reduces to

o + C rhpa,as + o = 0, a. B

so that, since by (22) $2 rksis symmetric in a and

and a,, as arbitrary,

rLs(s)= 0.

(5)

Since the geodesic is perpendicular to S at s, we have

(6)

gia(s) = 0,

With F and

(7)

g'"(s) = 0.

F" related as above we have

( G O ( s 0 )

=

( L F " ) ( s o )=

c gp4 a,a,F

P. 4

(

-

1 a,F)(so). t

By definition, F is constant in the n - r last variables; because of this, (5) and (6), the right-hand side of (7) reduces to

But F = F on S and by (22) in $2 the Christoffel symbol rfjon S is the restriction to S of the corresponding symbol for V. Thus expression (8) equals (L,F)(s,), so the theorem is proved. 2. Transversal Parts and Separation of Variables for Differential Operators

Let V be a manifold satisfying the second axiom of countability and suppose H is a Lie transformation group of V . If u E V , let H"denote the subgroup of H leaving u fixed. As usual, the image of u under h E H will

254

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

be denoted h - u. Let t, denote the Lie algebra of H . For X E t, let X + denote the vector field on V induced by X ; that is,

A C”-function f on an open subset of V is said to be locally invariant if X’f = 0 for each X E $ . We recall that a submanifold B c H is called

a local cross section over an open set U c H / H ” if the natural map II: H + H / H ” gives by restriction a diffeomorphism of B onto U. Lemma 3.3. Suppose W is a submanifold of V such that for each W the tangent spaces at w satisfy the following “transuersality”

W E

condition: (9)

V, = ( H e w),

+ Ww

(direct sum).

Fix w o e W. Then there exists an open, relatively compact neighborhood Wo of wo in W and a relatively compact submanifold B of H forming a local cross section through e ouer an open neighborhood U o of eHWo in H/Hwo such that the mapping q : (b, W ) + b * w is a diffeomorphism of B x Wo onto an open neighborhood of wo in V. Proof. Let I)’ denote the Lie algebra of Hwo and n c t, any subspace such that t, = t,’ 0 n (direct sum). If no is a sufficiently small neighborhood of 0 in n, exp no is a local cross section in H over a neighborhood of eHwo in H/Hwo (cf. [DS], Lemma 4.1 in Chapter 11). Therefore, it suffices to prove that the mapping 4: (X,w ) + exp X . w of n x W into V is regular at (0, wo). Since ( ~ 4 ) ( O , W 0 d O ~W W , )

=

ww,,

it remains to verify that (10)

4 0 .w , , h

0) = (H* wo)wo.

The mapping h + h . w , maps H onto the orbit H . w o ; its differential maps t, onto the tangent space (H.W,,),,,~, has kernel $O, and therefore gives a bijection of n onto (H - w ~ ) ~ This , . proves (10) and the lemma. Now we assume V has a Riemannian structure g invariant under the action of H . We assume furthermore that all the orbits of H have the same dimension. [Even if this assumption is not satisfied for V, it may be satisfied for a suitable open submanifold; cf. O(n) acting on R”.] Let D be a differential operator on V. We shall associate to D a new differential operator DT on V whose action is “transversal to the orbits.”

$31

3. Geometric Operations on Differential Operators

255

Let so E V and let S denote the orbit H . s o . We construct the manifolds S; as before with SES. Shrinking S,l, if necessary, we may assume that it satisfies the transversality condition (9) above for W. Put wo = so and choose Wo c S;, as in the lemma. Its conclusion means that the disjoint union B . w is a neighborhood Vo of so in V. Iff E &‘(V), we consider its restriction to W,, which we then extend to a function f,, on Vo by

uwcwo

f,,(b*w) = .f(w),

(1 1)

bEB,

W E

Wo.

We then define D T by Since fs, is smooth near so, the right-hand side makes sense; also, since B w is a neighborhood of w in the orbit H . w (for reasons of dimensionality), the method for constructing B is immaterial. Thus D Tf is a welldefined C”-function on V. If f vanishes on an open subset U of V, then so does DT f. Hence the mapping f --t DT f decreases supports, so DT is a differential operator. The operator DT is called the transversal part of D. If D = DT,then D is said to be transversal. Observing now that (j.io)pg).o = tio,we have for 4 E & ( V ) ( 12’)

(4ET)T

= 4ET.

In fact, ((4ET)T

f )(SO)

= ($ET)(fs0)(~0)=

4(sO)(ET

= 4 ( ~ o ) ( ~ f s 0 ) ( S o= ) W O ) ( E T f )(so) = (4ETfhO).

We shall now prove a global “separation of variables” formula for an arbitrary differential operator E on V. Theorem 3.4. Let H be a Lie transformation group of isometries of a Riemannian manifold V ; all orbits are assumed to have the same dimension. Let X , , . . . , X , be a basis of the Lie algebra f) of H and let = X,? (1 _< i I I ) denote the induced vector fields on V. Then each differential operator E on V can be written as a locallyfinite sum

where ET is the transversal part of E and each is transversal.

((i) = (i,, .. . , i,))

Remark. Here “locally finite” means that each point of V has a neighborhood inside which all but a finite number of the terms in (13) vanish identically.

256 Example.

( x , y) -,( x , y

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

Let V be the plane R2 and H the group of translations 7;:

+ t), t E R. In this case, the operators of the form

1bi(x, ~)
are the transversal operators, d/ay the vector field induced by H,and (13) reduces to the representation

of an arbitrary differential operator on RZ. Proof. Let so E V and let S , S,l, B, W,, and Vo be as above. Let b + (yl (b), . . . ,yr(b)) be any coordinate system on B, yl(e) = . . . = yr(e) = 0, and let w + ( z r +l(w),. . . , zn(w)) be a coordinate system on Wo such that the geodesics through so correspond to the straight lines through 0. Then we define a coordinate system { x l ,. . . ,x , } on V, by * * xr(b * w), xr+ 1(b * w), * * * xn(b * w)) (YI(b),- * * yr(b), zr+ 1(w), * * * z ~ ( w ) ) .

( x , ( b * w), * =

9

9

7

9

For each U E V, the vector (d/dxi), (1 I i I r ) is tangential to the orbit H . u, hence is a linear combination of the vectors (Yl),,, . . . , (q),,.Pick r linearly independent among these vectors, say (I;.,),, .. . ,(I& and write (a/axi), as a linear combination

where aij(u)= 0 if j # {jl, .. . , j r } . Since the vectors (q,), (1 < k < r ) remain linearly independent for u near v , (14) extends to a relation for vector fields,

valid on a relatively compact neighborhood N u of u, the aij," being C"functions and a i j , += 0 if j .$ { j l , .., j r } . The neighborhoods N u form a N , of Vo which covering of V,. We can find a countable covering is locally finite and such that for each m there exists an Nu(,) such that c Nu(,, (cf. [DS], Chapter I, #l, 15). Let 4,,, be a corresponding partition of unity. Then we have

urn"=

m,

I:=,

931

3. Geometric Operations on Differential Operators

257

on Nu(,,,,,hence on all of V,. Summing over m, we obtain on Vo a

1

-=

axi

cbijT

(1 I i l r ) ,

j=l

where bij = Ema i j ,u ( m ) + m lies in b( Vo), because of the local finiteness. By definition, we have (16)

( E - ET)($)(SO) = 0 whenever the function $ satisfies (17)

$(hew) = $(w),

bEB,

WE

Wo.

However, condition (17) just amounts to local invariance of $ on Vo (dim B = dimension of the H-orbits), so (16) holds not just for so but (18)

(E

-

E,)($)(s) = 0

for s near so in V,.

In terms of our coordinate system (17) amounts to $(XI,.

*

., X")

=

$@,.. ., 0, % . + I , . . ., X"),

so since E - E , annihilates all such functions, each term in its coordinate expression must include some ai (1 I i I r), whence by (15) 1

E- ET=zEix

(19)

1

in a neighborhood N o of so, each Ei being a differential operator on N o . Again, we can form a locally finite covering {U,} of I/ such that (19) holds on each member of the covering. Let 1 = 4, be a partition of unity subordinate to this covering. Then by (19)

c

1

+aE

- (+aE),

=

c Eqx, 1

with EPEE(U'). However, by (12'), (+.E), = + . E T , by summation over a,

SO

we obtain

E - E , E E( V)b', (20) where, as before, E ( V ) denotes the set of all differential operators on V and Ij+ denotes the set {X' : X E 5 ) of vector fields induced by H . Now the theorem follows from (20) by iteration. We shall now work out a more explicit form of (13) in the case when E is the Laplace-Beltrami operator on V. Theorem 3.5. Let V be a Riemannian manifold, H a Lie transformation group of isometries of V ; all orbits are assumed to have the same

258

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

dimension. Let S be any H-orbit and

3 the

restriction to S of a function

f E b ( V ) . Then the Laplace-Beltrami operators Ls on S, Lv on V satisfy (21) ( b f1- = L S ( 3 ) + ( ( L Y ) T f Remark. This formula generalizes decomposition (3) for the Laplacian on R3. Here I/ = R3 - {0}, H is the rotation group 0(3), the operator (a2/&') + (2/r)(a/ar) is the transversal part of LR3,and the operator

is the Laplacian on the sphere S,(O). For the proof of (21) let S , E S and let the notation be as in the last proof. Since H permutes the various S,l ( s E S ) , the coordinate system { x l , .. . ,x,} in question has properties (i) and (ii) from the proof of Proposition 3.1. We saw before that if $ E 8(Vo) satisfies (22) then

$(XI, * *

(23) so

-

3

xn)

= $40, . . ., 0, x,+

ij

$ = $,

4 ( ~ 1 ,*

* 3

Xn)

=

. . ., Xn),

= const.,

(24) ((LV)T$)(SO) = (&$)(SO), On the other hand, if 4 E 8(V,) satisfies (25)

1,

.

=

(&$)(SO)

* *

9

0,

XI, *

9

0.

O),

then 4 ( b . w) = 4 ( b . so) (w E Wo),and since H permutes the various S,l (sES),we have in a neighborhood of so

4so= const., 4 = ($I-, where is defined in connection with Proposition 3.1. From Proposition

(26)

3.1 we therefore deduce (27) (Lv+)(so)= ( G $ ) ( s o ) = (Ls$)(so), Now, by (17) in 42,

( ( L " ) T ~ ) (= s ~0.)

L"(4$) = 4LYlc/ + 2g(grad 4, grad $1 + @"4. (28) We have d,$=O ( l i i i r ) , a,4=0 (r+l
so using the orthogonality (6) and the formula (8) in 42 for grad, we see that the middle term in (28) - is 0 at the point so. But by (23) and (27) the

$31

3. Geometric Operations on Differential Operators

259

last term at so equals Ls(4$)(so). Similarly, using (24), (26), and the definition of we see that the first term on the right in (28) equals ( L V ) T ( 4 $ ) ( ~ato )the point so. Consequently, iff = 4$, (L" f)(so) = Ls(f><so>+ ( ( L " ) T f ) ( S O ) * It is an elementary fact (cf., Schwartz [1966], Chapter IV, $3) that the finite linear combinations of functions 4$ [where $ satisfies (25) and $ satisfies (22)] is dense in b(V,). The theorem therefore follows by continuity. Remark. Theorem 3.5 remains true with the same proof if V is a manifold with a pseudo-Riemannian structure g, provided g is nonsingular on S.

3. Radial Parts of a Differential Operator. General Theory

Again let V be a manifold satisfying the second axiom of countability, H a Lie transformation group of V, and l~ its Lie algebra. As in Lemma 3.3 we assume W c V is a submanifold satisfying this condition: (29)

Vw

= (H . w),

+ W,

(direct sum)

for each w E W.

Theorem 3.6. Assume the transversality condition (29). Let D be a differential operator on V. Then there exists a unique differential operator A(D) on W such that

(Df)- = A(m.7

(30)

for each locally invariant function f on an open subset of V, the bar denoting restriction to w. Definition. The operator A(D) is called the radial part of D . It has an obvious connection to the transversal part defined earlier. Proof.

Let w, E W and select W,, B, V,, and q as in Lemma 3.3. Let V, by

4 E &( W,) and define f on

b E B, w E Wo. f ( b * W ) = +(w) Since (f 0 q)(b, w ) = +(w), we have f~ b(V,). Let X E b. For reasons of dimensionality the tangent spaces q(B x w ) ~, . and (H w ) ~, . coincide, whence ( X + f ) ( b .w ) = 0, so f is locally invariant. Consider now the linear mapping Aw~,U'~,B~b(wO)

b(wO)

260

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

given by 4 +(Of)-. Since this mapping decreases supports, it is a differential operator on W,. By the remarks above, the set B . w is an open submanifold of the orbit H - w (w E Wo);hence if B’ is another local cross section with the same properties as B in Lemma 3.3, we have A w ~W , O ,B

= A w ~W, O .8’.

Thus we can, for each W , E W and each open relatively compact neighborhood W, of w , in W for which a B with the properties of Lemma 3.3 exists, define Awo,wo- Awo,wo,B.It is obvious that if W’ and W “ are two such neighborhoods, then

Awe, w , = Awo,

on d’(W’ n W”).

Thus we can define the linear form

Awe: 4

wo(4))(wo), 4 E &(W), the choice of W, being immaterial. Finally, we define the mapping A(D) by (A(D)4)(w,) = Awo(4), wo E w. -+

@WO,

Then (A(D)41)(wo)= (A(D)&)(w,) if 41 and 42 coincide near w,; also, on the subspace 9(W,) c b ( W ) , A(D) coincides with the differential operator Awo,wo,B.It follows that A(D) is a differential operator on W, and its construction shows that it has the properties stated in the theorem. The operator A(D) is called the radial part of D. We shall now give a formula for the radial part A(L,) of the Laplacian for a suitable transversal submanifold W for an action by isometries. Let V be a Riemannian manifold, I ( V ) its group of isornetries in the compact open topology. We shall now assume the result of Myers and Steenrod [1939] (cf., also Kobayashi and Nomizu [1963], Vol. I, Chapter VI) that I ( V ) has an analytic structure compatible with this topology in which it is a Lie transformation group of V. (The case when V is symmetric is the one of principal interest for us; for this case a direct proof is given in [DS], Chapter IV, $3.) Let H c I ( V ) be a closed subgroup. With the induced topology H is a Lie transformation group of V ([DS], Chapter 11, $2). If u E V, the orbit H . u is a closed submanifold of V, the isotropy group H“ is compact, and the mapping hH” + h . u is a diffeomorphism of H / H ’onto H . u (with the topology induced by V ) . For proofs see [DS], Chapter IV, $2, Chapter 11, Theorem 3.2, and Exercise C5. We shall now introduce a function 6 on V, the density jirnction, which in a certain manner measures the relative “size” of the orbits of H. Let u E V. The orbit H v inherits a Riemannian structure from that of V.

631

3. Geometric Operations on Differential Operators

26 1

The corresponding Riemannian measure do, is, of course, invariant under H. On the other hand, if we fix a left-invariant Haar measure dh on H and let d k denote the unique Haar measure on the compact group H" normalized by dk = 1, we obtain an H-invariant measure d h on H/H" by the formula

i

f ( h ) dh =

1

H / H " (JJ(hk)

dk) d k

f E

CAH)

(cf. Chapter I, Theorem 1.9, and the remark following). Under the identification H av = H/Hu the measures do, and d h must, by the uniqueness, be proportional. Hence there exists a function 6(v) on V, which we call the density junction, such that

do, = 6(v) dh,

(3 1) Example.

v E V.

For the usual action of O(n) on R", 6(x) = c ( x ~ " - ' ,

x

E

R",

where c is a constant. We shall now determine the radial part of the Laplacian on V assuming the H orbits intersect the transversal submanifold orthogonally and in just one point. Theorem 3.7. Let V be a Riemannian mangold, H a closed unimodular subgroup of the isometry group l ( V ) . Assume that a submanijold W c V satisjes the following transversality condition: For each w E W,

( H . w ) n W = {w}, V, = ( H . w), 0 W, (32) (orthogonal direct sum). Then the radial part of Lv is given by A(Lv) = 6 - '/'L, o a'/' - 6 - 1/2Lw(61/2), (33) where denotes composition of differential operators and 6 is the density firnction. 0

We shall see that (33) can also be written

+

A(Lv) = Lw grad,(log 61, (33') where grad, refers to the submanifold W and the vector field gradw(log 6 ) is viewed as a differential operator on W.

262

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

Proof. Put V* = H - W. The mapping onto V* has a differential satisfying ~Z(/I, w)(Hh

ww)

= dZ(h, ~ ) = ( H * W)h.

(

~

A:

(h, w) + h . w of H x W

io)) h + dn(h, w ) ( i o }

ww)

+ (dh),(WW)= (dh),((H

*

W),

+

WW).

Because of (32) or (29), this image equals Vh.w.This implies that V* is open in V ([DS], Chapter I, Proposition 3.1). Next we prove the formula

(34)

Syw do = sw6(w)( J-,,

F(h . w) dh) dw,

F

E 9(V*),

where du and dw are the Riemannian measures on V and W, respectively. For this let wo E W and select B, W,, V,, and q with the properties of Lemma 3.3. Assuming, further that b + ( y l , . . . ,yr(b)) is a coordinate system on B, with yl(e) = ... = yr(e) = 0, and the mapping w

+

(zr+ 1(w),.

. Zn(w>) . ?

a coordinate system on W,, we have a coordinate system on V, by putting

(35)

7

=

{xl,.. .,x,}

( x , ( b * w), . . .,x,(b * w), xr+ l ( b * w), . . .,xn(b w ) ) = (Y 1 (b),* * * yr(b), zr + 1(w), . * - zn(w))* +

7

9

Once more we adopt the following range of indices: 1 s i, j , k I r, r + 1 I a, fl, y 5 n. We have the following coordinate expressions for the Riemannian measures: dv=g1'2dX1.-.dx,,

dw=~1'2dXr+1-..dXn,

where

(36)

S = Idet(gpq)l
7 = ldet gapl.

Because of the orthogonality condition in (32) we have

(37)

gia(W) =

Let b E B. Since the mapping w expression ( x r + 1, *

it follows that

* * 7

--t

0,

w E W,.

b w of W, onto b . W, has coordinate

xn)

+

(xr+ 1, * * * Xn), 9

GI, La.W.

db-

= -

931

3. Geometric Operations on Differential Operators

263

On the other hand, the submanifold of V, given by fixing the xu lies on an orbit of H . It follows that (39) where aij E R. By (37) and (38) Sup(’

*

W)

= gu&w),

gidb . W ) = 0.

Hence, using (39), we obtain

e ( b . w ) = Idet(gij(b . w))IY(w).

(40)

But the Riemannian measure dow on the orbit H . w has coordinate expression do, = Idet(gij)(b. w

)I~’~

d x l . . .dxr(b . w),

so by (40) we obtain for F E 9(Vo),

By using now the definition (31) of 6, formula (34) follows for F E 9(Vo). But then it holds also for F E 9 ( h - V,) for each h E H. But as wo runs through W the sets h Vo ( h E H ) form a covering of V*. Passing to a locally finite refinement and a corresponding partition of unity, (34) follows in general. If f~ 9(V*), let f~ 9 ( W ) be determined by

The mapping f f is surjective. In fact, let F E 9 ( W ) and let C c W be a compact subset outside which F vanishes. Let c; be a compact subset of V * such that ( H . w ) n c = 521 for w $ C and such that for each w E C, ( H . w ) n c has measure > O (for do,). For example, take c = CH C, where CH is a compact neighborhood of e in H . Let fl E 9 ( V * ) be such that 1; 2 0 on V* and f i > 0 on c. Then fi > 0 on C and the function f;(u)(F(w))/j l ( w )

f ( 4 = {o

if u E H w, w E C otherwise

belongs to 9 ( V * ) and satisfies f = F. This proves the surjectivity

(42)

9(V*)’ = 9(W ) .

264

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

Since the vector fields d/axi (1 I i I r) are tangential to the H-orbits, we have on Vo, just as in (15), > U

(43)

I

-

axi

=

1bij x ,

bij E g(V0).

1

Now we use the coordinate expression of L y as given in Proposition 2.6. Combining it with (37) and (43), we deduce that (44)

A&)

= L,

+ lower-order terms.

On the other hand, we have by the symmetry (Proposition 2.3) (45)

j"Wl)(U)f2(U)

du =sy*f1(u)(L"f2)(u) do

for 1;,f 2 E 9 ( V * ) . But then the relation holds for all f2 E&'(V*). In particular, let us takef, invariant under H. Then, by (34), the left-hand side of (45) is

However, H" is compact for each u E written

so the inner integral can be

(47) and for the same reason, since Lv is H-invariant, (48)

(W"( JpV) d h ) = SH(L"l;)(h . 4 dh. *

By the unimodularity of H the function u + jHfi(h . u ) dh is H-invariant, so by (48) expression (47) equals (A(L,)f,)(w), and the left-hand side of (45) reduces to Sw(A(L")l;XW)h(W)a(W) dw. Now, by using (34), the H-invariance of L v f 2 , and the definition of A&), the right-hand side of (45) reduces to

J

W

fi(W)(A(L")f2)(W)a(W) dw.

By (42) the functions fl (and also j2)fill up 9 ( W ) , so the equality of the last expressions means that A(L,) is symmetric with respect to S(w)dw. But since L , is symmetric with respect to dw (Proposition 2.3), the oper-

$31

3. Geometric Operations on Differential Operators

265

ator b-II2LW 0 S'12 is symmetric with respect to 6(w) dw, and, of course, it agrees with L , up to lower-order terms. Thus by (44) the difference A(L,)

-

6- "*LW 6'", 0

which is symmetric (with respect to S(w) dw) has order 51. But no firstorder differential operator can be symmetric (cf. Exercise A7), so this difference is a function; the desired formula now follows by applying the operators to the constant 1. Remark 1.

By using (7) and (17) in 52 and the formula

26-

grad(6'") = 6- grad(@ = grad(1og S),

the formula for the radial part can be written

+

A(Lv) = L , gradw(log S), (49) the vector field grad,(logS) being viewed as a differential operator on W. Remark 2. Theorem 3.7 remains true if V is a manifold with a pseudo-Riemannian structure 9, provided the following two conditions are satisfied :

(i) For each w E W the orbit H w is closed and g is nondegenerate on it. (ii) For each w E W the group H"' is compact. 1

In fact, by (i) the orbit H . w is locally compact and thus homeomorphic to HIH"' ([DS], Chapter I, Theorem 3.2). By (ii), H" is compact for each u E V * , so no change is necessary in the proof. The general formula (33) for the radial part is proved in Helgason [1972a,b] where a further variation in the assumptions (weakening of the compactness in (ii)) is also given. 4. Examples of Radial Parts

We shall now use Theorem 3.7 to determine the radial part of the Laplace-Beltrami operator for various examples. (i) O(n) Acting on the Euclidean Space R". Let V = R", H = O(n) the orthogonal group, acting on R". Then the submanifold W = R + - (0) (the positive real axis) satisfies condition (32) in the theorem; hence we conclude [since 6(x) = cIxI"-']

266

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

This can be reduced to the familiar expression [cf., (49)] d2 n - l d A ( L p ) = - + -dr2 r dr Note that exactly in the case n = 3 does the last term in (50) disappear, and then the formula reduces to

A(LR3) = r -

d2 r. dr 0

In this case, a radial function f ( x ) = F( Ix I) is an eigenfunction of LRn if and only if the function rF(r) is an eigenfunction of d2/dr2. Thus f ( x ) is an elementary function. More generally the function

where ( x , w ) is the inner product and do the Riemannian measure, is a radial eigenfunction of L p on R" for eigenvalue - I 2 . If I E R, the function ( 5 1 ) is a constant multiple of

where J , is the Bessel function of order v . Formula (50) therefore explains the classical fact that J 1 , 2 ( r )is an elementary function. (ii) The Group N Acting on the Symmetric Space X = G J K . As usual, let G = K A N be an Iwasawa decomposition of a connected semisimple Lie group G with finite center. We take I/ = G / K and H = N in Theorem 3.7, and adopt the notation from 85 in Chapter I. Then the submanifold A o c V satisfies assumption (32) because of the uniqueness in the Iwasawa decomposition ([DS], Theorem 5.1, Chapter VI) and because of the orthogonality N . o IA o (loc. cit., Exercise B2), which implies the orthogonality N . (a . 0) IA so. Using Corollaries 5.2 and 5.3 in Chapter I, we obtain a

a

I,,,

f ' ( g K ) dg,

=

J;-2p('oea)(JNj'(naK)d n ) da,

so comparing with (34), (52)

6(a 0) = e-2p('0e0), a E A.

Now Theorem 3.7 implies the following result.

$31

3. Geometric Operations on Differential Operators

267

Proposition 3.8. When N acts on the symmetric space X = G / K , the radial part jor the transversal submanifold A . o is given by

(53)

A(Lx) = ePL, e - p - ( p , p ) , 0

where LA is the Laplacian on A o and where ep denotes the function a o + eP(loga)on A . 0. +

Remark. One can compare this formula with the radial part of LRn when R"-kacts on R" by translation. Then the orthogonal complement Rk serves as a transversal submanifold and from (33) or directly we see that A(LR") = LRr. This corresponds to taking p = 0 in (53).

(iii) The Group K Acting on X = G / K . Again, we follow the notation of 55 in Chapter I for the symmetric space X = G/K. We shall verify that the manifold A+ o satisfies the two transversality conditions in (32) for the group K playing the role of H. The first condition is guaranteed by the properties of the polar coordinate map ( k M , a) + kaK. To check the orthogonality condition (54)

( G / K ) o . o= ( A + . o ) ~CD. ~(Ka.o),..,

a E A+,

let T E f and consider the curve t + exp rTa . o in the orbit Ka . 0. Its tangent vector for t = 0 equals, since exp tTa . o = a exp Ad(a- ') t T . 0, the vector dz(a) dn(Ad(a- ') T). [Here z(a) is the translation gK + agK and n: G + G / K the natural mapping.] But if H E a (the tangent space to A' . o), we have (dz(a) dn(Ad(a-') T),dz(a) H ) = (f(Ad(a- ' ) T - 0 Ad(a-

l)

T), H) = 0,

verifying (54). Proposition 3.9. For K acting on X = G / K with A+ . o as a transversal manijold, the radial part of'L x is given by (55)

A(Lx) = L A +

m,(coth cr)A,. ,€I+

Here the vector A, E a is, as usual, determined by (A,, H) = a ( H ) (H E a), and in ( 5 5 ) it is considered as a differential operator on A+ 0. Using Theorem 5.8 in Chapter I, we find for each H E a that (56)

H(log 6) =

1 m,(coth ,EX+

a)cr(H).

268

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

If H I , .. .,H Iis an orthonormal basis of a, we obtain from (49) and (56) A(Lx) = L A

+ 1 Hi(log 8)Hi i

=

But

xi a(Hi)Hi

LA +

C mu coth a C a(Hi ) H i . i

U€Z+

=

Xi ( A a , H J H , = A,, so (55) is proved.

For the case when G is complex, the radial part can be put in a better form. Proposition 3.10. Suppose G is a complex semisimple Lie group. For K acting on X = G / K with A' . o as a transversal manifold, the radial part of' Lx is given by

(57)

A(L,) = 8 - l i 2 ( L A - ( p , p ) ) 0 d112,

where hli2(u)=

1w (det

s)eSp('Og").

SE

This is an immediate consequence of Theorem 5.8 in Chapter I if we recall that each c1 E C' has multiplicity 2 and then use the formula

n

(eu - e-') =

1 (det s)esp, sew

UEZ+

which follows from Proposition 5.15 in Chapter I for the root system (2a: a E C}. Remark. Consider, as in $4 in Chapter I, the hyperbolic space X = H" with the Riemannian structure

(58)

dsz = dr2

+ (sinh r)'

do2,

where do2 is the Riemannian structure on the unit sphere in the tangent space X o . Under the action of the isotropy group at o the radial part of the Laplacian is given by [cf. (23) in $4, Chapter I] (59)

dZ d A(Lx) = 7 (n - 1)coth r -. dr dr

+

Viewing H" as the symmetric space O,(n, l ) / O o ( n ) (subscript indicating identity components), we can give it the canonical Killing form Riemannian structure, which by irreducibility is a constant multiple of (58). Thus (59) follows also from the general formula (55). We can also write (59) in the form

931

3. Geometric Operations on Differential Operators

269

By analogy with what happened for R" [cf., (i)], the case n = 3 is one of special simplicity; the zero-order term in (60) then reduces to a constant, and in fact A(L,) = sinh-' r (:r:- - 1)

o

sinh r.

This again implies that for dimension 3 the radial eigenfunctions of L x are elementary functions. While for R3 we did not have a conceptual explanation of this fact (see, however, Proposition 3.13 for G = SL(2, C), K = SU(2)), Proposition 3.10 gives such an explanation for H3:If n = 3, the Lorentz group Oo(n, 1) has a complex structure. In fact, it is locally isomorphic to SL(2, C ) ([DS],Chapter V, $2). (iv) The Group K , Acting on the Compact Symmetric Space U I K , .

,

Consider now the simply connected compact symmetric space UIK from Theorem 5.11 in Chapter I. Under the action of K 1 the submanifold Exp Qo = exp Qo. o serves as an orthogonal transversal submanifold for which Theorem 3.7 applies. [The orthogonality is proved in the same way as (54).] Now the Riemannian structure of U I K , is induced by B , = - B, where B is the Killing form of 9' restricted to w. Let C , = ia', and for a root a E Z put a, = ( l / i ) a and C, = { c t , : a ~ C } , C i = { a , : a ~ Z + } .If a,EZ,, A , E a , be determined by B , ( A , , H ) = a,(H). We call the a, the roots of w with respect to a,. The roots in Z i are the ones which are positive on the Weyl chamber C,. We may assume the polyhedron Qo selected inside C,. On Exp Qo the density function is up to a factor given by (61)

G,(Exp H )

=

n (sin a,(H))"a.

a. E X,+

Proposition 3.11. For K , acting on the compact simply connected symmetric space UIK with transversal manijold Exp Q o , the radial part of' the Laplaciun LUlKlis given by

(62)

A(LU/K~)

= LAt

+

C

ms cot a* Aat*

a. E X,+

where LA. is the Laplacian on the jlat manijdd A , .o.

The proof is entirely analogous to that of (55). In fact, if H E a,, H(log 8,) =

ma(cot a*)a*(H) a,

E X,+

and A,, = y, a,(Hi)Hi if H , , . .. , H , is a basis of a, orthonormal with respect to B,.

270

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

(v) A Compact, Semisimple, Simply Connected Lie Group U Acting on Itself' by Conjugacy. This situation is a special case of Proposition 3.11, so that the radial part can be found from (62); however, it is easier to use Theorem 5.13 in Chapter I. We use the notation prior to that theorem. Let the Riemannian structure of U be given by B,, the negative of the Killing form of w. Again, we put a,(H) = (l/i)a(H)for H E to, a E A. Let C o c to denote the Weyl chamber containing Po and A' = {a,: a,(Co) c R'}.

Let p, = we have

a, and let Ip*l denote its norm in the metric B,. Here

if t = exp H,H E P o . But using Proposition 5.15, Chapter I, we find that this reduces to (63)

B(t)"' =

1 (det s)eisp*(H). SE

w

Proposition 3.12. For U acting on itself' by conjugacy with transversal manijold exp P o , the radial part of' the Laplacian L , is given by

A(L") = 6- "'LT

0

B"'

+ I p, I',

where LT denotes the Laplacian on T.

This follows from Theorem 3.7 if we just take (63) into account. (vi) The Adjoint Action of K on p. This is the situation of Theorem 5.17 in Chapter I for the symmetric space X = G/K. The Weyl chamber a+ can serve as an orthogonal transversal manifold. In fact, if H , E a + , the tangent space to the K-orbit K . H , is, by (21) in 45, Chapter I, given by [I, H O ] = [f, HO]which is orthogonal to a under the Killing form on p. Using the quoted theorem we thus derive the following result. Proposition 3.13. For the udjoint action of K on p with transversal manijold a' the radial part of the Laplacian L, is given by

where Lo is the Laplacian on a.

$31

3. Geometric Operations on Differential Operators

27 1

For the case when G is complex thejormula can also be written where n

=

nasZ+

A(L,) = 7c-'L0 0 n,

a.

In fact, the density is now proportional to flcrrZ+ a(H)mm,so the first formula above follows from (49). The second formula is a consequence of (33) and the equation (64)

L,(n) = 0.

This equation is established as follows. If c( is a simple root, the reflection s, permutes the set Z + - { a } and consequently IF-= - n ; it follows that n is a skew polynomial, i.e., ns = (det s)n for s E W. Since Lo is invariant under the Weyl group, it follows that L,(n) is skew. For each re' ) the relation (L,n)sII = - L a x implies that L o n vanishes flection s, (a E E on the plane c( = 0. Thus the polynomial Lon is divisible by a, and since the positive roots are prime to each other (as polynomials), Lax is divisible by n. This proves (64). (vii) The Adjoint Action of a Complex Semisimple Lie Group G on Its Lie Algebra g. Let t j be an arbitrary Cartan subalgebra of g and tj' c tj the subset of regular elements. Let A denote the set of roots of g with respect to Ij and g = I, + Cas6 9" the corresponding root space decomposition ([DS], Chapter 111). If X E g, the orbit G . X has tangent space [g, X] at X ; the root space decomposition therefore shows that the submanifold 6' c g satisfies the transversality condition (29). The Killing form ( , ) on g is nondegenerate on both g and on tj, so that we can use it to identify g and b with their duals g* and tj*, respectively. If p is a polynomial function on g (resp. t j ) let d(p) denote the constant coefficient differential operator on g (resp. b) which corresponds to p by means of this identification. If f' is a function on g, let 1: denote its restriction to fj (or tj'). Let o ( X ) = (X,X) (XE 9). We shall now determine the radial part of the "complex Laplacian" d(w). For c( E A we denote by H a the vector in b satisfying (H", H) = a ( H ) ( H E tj). Then the space tjR = R H , is the subset of on which all the roots are real-valued. Fixing a Weyl chamber tj' c t j R , a root tl is said to be positive if c ( ( t j + ) c R + .

xaeA

Proposition 3.14. For the adjoint action of'G on g with transversal manfold tj', the radial part of'the operator d ( o ) is given by

A(d(w))= 7 ~ -' d ( 1 5 )0 TC, where n is the product of' the positive roots.

272

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

Proof. Since we are dealing with a situation over C and since the isotropy groups are noncompact, Theorem 3.7 does not apply without some adjustment. We therefore adopt a more direct approach. Choose elements X , E g" (a E A) such that ( X z , X - , ) = 1. Then [ X , , X - , ] = Ha( [ D S ] ,Chapter 111, $4(3)). Since ( X , , X , ) = 0 if a, fi E A, a + fi # 0, we have a(w) = a@)

+ c x,x-,. aeA

Now let H E b' and suppose J' is analytic and locally invariant in a neighborhood of H in g. Then if tl E A, f(ead(sXm

+ 1.X - d ~ = )f ( ~ )

for s and t sufficiently small. Writing this as J'(H

+ X(S, t ) ) = f ( H X

we have by Taylor's formula

Here we collect the coefficients of X ( s , 1 ) = [SX,

st.

Since

+ tX-,, H] + g s x , + tx-,,

csx,

+ t X - , , HI1 + . .

only terms with n I 2 in (66) give a contribution: X ( s , t ) gives coefficient + [ X - , , [ X , , H ] ]

+ ;[X,,

[X-,, HI];

+X(s, t ) 2 gives coefficient [ x , , H ] [ x - , , H I . It follows that [ ( O ~ ( H ) ~ X ,-X ~- (~H ) H , ) ~ ' ] ( = H )0. Thus by (65) (67)

A(a(w))= d(3)

+2

a- ' H a .

a>O

On the other hand, by (15) and (17) in 52 (which are also valid in the pseudo-Riemannian case), we have (68)

n-' 13(3)(nJ')= a(G)f

But grad(1og R) =

+ n-'(d(G) n ) f + 2(grad(log n ) ) ( f ) .

c grad(1og a)

a>O

=

1a - l H , . as0

931

3. Geometric Operations on Differential Operators

273

Finally, we have by analogy with (64) d(c?i~)n = 0, and now the proposition follows from (67) and (68). (viii) The Laplacian on X and on p . Continuing Example (iii) above, consider the diffeomorphism Exp: Y -P (exp Y ) K of the tangent space p onto the symmetric space G / K and let J denote the corresponding volume element ratio; i.e.,

L

f ( x ) dx =

I

f(Exp X ) J ( X ) d X ,

j~ C,(G/K).

Then by [IDS],Chapter IV, Theorem 4.1,

J ( X ) = det(

(

sinh ad X ad

)J

when the subscript denotes restriction to p. Theorem 3.15. Let L x and L , denote the Laplace-Beltrami operators on X and Q , respectively. Then the image ( L X ) E X Pof- lLx under Exp-' (cf. 92) satisfies the relation

(70)

LFP-' F = ( L ,

+ grad(1og J ) ) F

f o r each K-invariant C" junction F on

(71)

LFP-' F

=

Q.

The jormula can also be written

(J-'i2LPo J1I2)F- J-1/2L,(J'/2)F.

Proof. Let 6 denote the density function for K acting on X and do the density function for K acting on p. It suffices to verify the relation above on a. The functions 6 and do can be taken as

6(Exp H ) =

n (sinh a(H))"=,

6,(H) =

aEE+

n a(H)"=

acE+

for H E a and J = 6/6,. Now if j = F 0 Exp-', (LpP-' F ) ( H ) = (L,f')(Exp H ) = (A(L,)f)(EXP HI, which by Theorem 3.7 equals (6- 112L" p ) f-'6 - l/2LA(61/2)f' 0

at Exp H. On the other hand, the right-hand side of (71) has restriction to a given by J - "2[(6, 1/2La0 ( g / 2 ) ( J ' / 2 F ) - (6 0 La(SA'2))J 'ZF] -.I"2[(6, 1/2La0 i p ) ( J " 2 ) F - 6, 1/2La(6y)(J1/2F)].

Since the second and fourth terms cancel, this proves (71). Formula (70) now follows by (17), 92.

274

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

§4. Invariant Differential Operators on Lie Groups and Homogeneous Spaces 1. Introductory Remarks. Examples. Problems

Let M be a manifold and itself. As before, we put

4: M

p = 1'. 4-1,

+M

f

a diffeomorphism of M onto E

&w,

and if D is a differential operator on M , we define D@ by

D':J+

(DJ'"')'

=

(D(j'c'$))0 4 - ' ,

f'~d?(M).

By the definition in $2, D@ is another differential operator. The operator D is said to be invariant under 4 if D@ = D,i.e., if D ( f 0 4) = ( O f ) 4 for all 1'.Note that (Of')@= DY@, justifying the notation. If T is a distribution on M , we write T' for the distribution T ' ( f ) = T ( j qf E 9 ( M ) . Since we want to take advantage of the invariance concept we shall now discuss differential operators invariant under a transitive group of diffeomorphisms. Let G be a Lie group, H c G a closed subgroup, G / H the manifold of left cosets gH ( g E G ) and D ( G / H ) the algebra of all differential operators on G / H which are invariant under all the transformations z(g): x H + g x H of G / H onto itself. We write D ( G ) instead of D(G/{e}).The algebra D ( G / H ) will play a central role in the remainder of this book. We recall briefly the concept of a representation. Let L be a locally compact group and V a topological vector space, Aut(V) the group of linear homeomorphisms of V onto itself. A representation 7c of L on V is a homomorphism of L into Aut(V) such that the mapping ( I , u) + n(l)u of L x V into V is continuous; 7c is said to be irreducible (or topologically irreducible) if ( 0 ) and V are the only closed subspaces of V invariant under x(L). Now we list some problems which arise naturally in connection with the invariant differential operators on G / H ; these were stated at the International Congress in Nice 1970. Considerable progress has been made on these problems since then; in particular, for symmetric G / K , Problems A-D are completely solved (see description below). 0

A. The Algebra D(G/H). Describe the algebra D(G/H) in terms of the Lie algebras of G and H . See Proposition 4.11, Exercises A3, C1, C2, C3, and C5 for information about this problem.

W

4. Homogeneous Spaces

275

B. Solvability. Given D E D(G/H), is the differential equation Du = f, for .f E b ( G / H ) arbitrary, globally solvable (respectively, locally solvable)? In this case we say that D is globally solvable (respectively, locally solvable). C . Joint Eigenfunctions. Determine the functions on G / H which are eigenfunctions of each D E D(G/H). Similar problem can be considered for eigendistributions. D. Eigenspace Representations. Let p : D(G/H) -, C be a homomorphism and let E , denote the corresponding joint eigenspace, i.e., E , = { f ' b~( G / H ) :Of' = p(D)f' for all D E D ( G / H ) } . Let T, denote the natural representation of G on this eigenspace, i.e., ( T , ( g ) f ' ) W ) = J'(g-'xH),

g, x E G

for J'E E , . For which p is this "eigenspace representation" T, irreducible and what representations of G are so obtained? Here the closed subspace E , c & ( G / H ) is given the relative topology. The definition of the topology of &(G/H) (52, No. 2) shows easily that T, is indeed a representation. E. Global Properties of Solutions. What geometric properties (functional equations, mean value properties, behavior at infinity) do solutions of the invariant differential equations have? It is to be expected that the local character of a D E D ( G / H ) (as a differential operator) together with its G-invariance will result in global limitations on the solutions. These problems will be discussed in this book as well as in another volume. But now we shall illustrate the problems in terms of some simple examples. Here we quote some results from elsewhere and anticipate special cases of results to be proved later. Examples

(i) The Euclidean Space. Consider the space R" as a homogeneous space under all translations. Here D(R") consists of all differential operators of constant coefficients in the standard coordinate system on R". In fact, d/dxi is invariant under all translations, so the invariance condition amounts to all the coefficients being invariant under the translations, hence constant. Problem B has a positive solution in this case; each D E D(R") maps &R") onto itself (Ehrenpreis [1954]; Malgrange [1955]). For the case R" we therefore have the following result. Theorem 4.1. (a) The algebra D(R") consists of' the diflerential operators on R" with constant coeficients.

276

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

( b ) Each D E D(R”) is globally solvable. (c) The joint eigenjunctions are the exponential functions; the eigenspace representations are one dimensional, hence irreducible. For R2 viewed as the homogeneous space M ( 2 ) / 0 ( 2 ) see the Introduction, Theorems 2.1 and 2.6. (ii) The PoincarC Group. Consider the space R4 as a homogeneous space G / H , where H is the identity component Oo(l, 3) of the Lorentz group and G is generated by H and the translations (the Poincare group). Here D ( G / H ) is generated by the d’Alembertian

0= a: - a; - a; - a: (cf. Exercise A3). This implicit physical significance of the Lorentz group so far as electromagnetic phenomena are concerned is made explicit in Einstein‘s special theory of relativity. Here the Lorentz group is given an interpretation in terms of pure mechanics. (iii) The Heisenberg Group N . Let N be the group of matrices B =

[t ‘b’ :’I 1

0

~

2

,

XER~,

with the obvious Lie group structure. The differential operator

a

a

a

E = - + iax, ax, + ix 1- 8x3 on N is invariant under all left translations La: B + au on N . To see this note that under the correspondence B -, ( x l ,x 2 , x3), a -, ( a l ,a 2 , a3), we have au + (al + xl, a, x2, a3 x3 a1x2),so

+

CE(.f0 L@)l(0) =

f)(aO)

+ +

+ i(a2f’Xafl)+ ia1(a3f)(aa)+ ixl(a3f>(a.)

and

+

+

[Efl(au>= ( a J ’ ) ( a d + $4f’)(aa) i(al x l ) ( a 3 f ) ( a 4 . This proves the stated invariance. Although E by its left invariance seems a natural analog of constant coefficient operators on R”, we shall see that it is not locally solvable on any open subset 0 c N . Theorem 4.2. For each open subset 0 c N we have

E b ( 0 ) $ %a).

+

This nonsolvability of E [or rather of the operator dl id2 2i(x, ix2)a3]was proved by Lewy [1957]. Later Hormander found a

+

277

4. Homogeneous Spaces

§41

more general necessary condition for local solvability which for firstorder operators can be described as follows. Replacing di in the expression of a differential operator by the variable ti,we obtain a polynomial in t,, . . . ,(,, called the symbol of the operator. The symbol of E, denoted E(x, t), equals t l it2 ix153. According to Hormander’s theorem, [19631, Chapter VI, specialized to first-order operators, we have the following: The solvability Eb(R) 13 9(0) implies

+ +

[E, El (x, 5 ) = 0

(2)

x E 0, 5 E R3.

if E(x, 5 ) = 0,

Here ?!, is the operator obtained by conjugating the coefficients in E. A simple computation shows [E, E](x, t) = 2it3, so taking 5 , = 0, t2 = -xl, = 1, we violate (2). Thus the solvability fails. It turns out that for the solvability question one should look at N as a homogeneous space under the action of the product group N x N by

c3

(Sl,

9 2 ) . 9 = 9199;

l,

g

E

N.

With this action all invariant differential operators are globally solvable (cf. Rais [1971] and Wigner [1977] for, respectively, local and global solvability on any simply connected nilpotent Lie group N ) . In the Introduction, Problems A-D have been discussed for the cases R2, S2, and H 2 . Now we shall briefly describe their solutions for the case of a symmetric space X = G / K of the noncompact type; refer to the original papers for detailed proofs. A. The algebra D ( G / K ) is a (commutative) polynomial ring in 1 algebraically independent generators D , , . . . ,D, whose degrees d , , . . .,d, are canonically determined by G (Chapter 11, Theorem 5.18, and Chapter 111, 53, No. 1). Here 1 = rank(G/K).

B. Each D # 0 in D ( G / K ) is globally solvable. This result was proved by Helgason [1973a]. In order to describe the solution to Problem C we consider the Iwasawa decomposition G = K A N and if g E G, let g = k exp H(g)n (k E K , H(g) E a, n E N ) . Let a,* denote the space of complex-valued linear functions on a. The following result of Harish-Chandra will be proved in Chapter IV, §4 [Theorem 4.4 and (7)J C(i). The K-invariant joint eigenfiinctions of D ( G / K ) are the constant mirltiples of the Jirnctions

4,(gK) =

jK

e(i,-P)(H(f7k))

dk,

g

E

G.

278

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

Here 3, E a:, p as in Chapter I, $5, No. 1, and dk the normalized Haar measure on K . Let M denote the centralizer of A in K and B = K / M . If x = g K in X = G / K , b = kM in B = K / M we put

A(x, b) = - H ( g - ’ k )

E

a.

This “vector-valued inner product” generalizes the inner product (2, b ) considered in the Introduction, $4. No. 1. If p E a,*, b E B, the function is a joint eigenfunction of D ( G / K ) (see Chapter IV, Exercise B15). Each joint eigenfunction f of D ( G / K ) can be expanded as f = xdetfd, where .fa is a K-finite joint eigenfunction of type 6 (Corollary 3.4 in Chapter V). We have now the following generalization of C(i) (cf. Helgason [19761). C(ii). The K-Jinitejoint eigenjirnctions of D(G/K) are the functions

f(x)=

bl)F(b)db,

where F is a K-Jinite continuous firnction on B and p E a:.

If f ’ is any joint eigenfunction of D(G/K), the expansion f’ = xdfi above together with C(ii) imply a formula

f(x) =

J;fl(A(x.b))

dT(b),

cdsi

where T is a formal Fourier series Fd on B = K / M . How can these T be characterized? Theorem 4.3 of the Introduction suggests an answer. The space . d ( B ) of analytic functions on the manifold B = K / M has a natural topology generalizing that explained in the Introduction, $4, No. 1, for the circle. The members of the dual space d ‘ ( B ) are called analytic functionals (or hyperjirnctions). Then we have the following result generalizing Theorem 4.3 of the Introduction. C(iii). The joint eigenjunctions of’ D ( G / K ) are the .functions f(x) =

J;plr(r.s,)

dT(b),

x

EX,

where p E a: and T E d ’ ( B ) (cf. Kashiwara et al. [1978]). A different and a more general proof based on C(ii) above has been given by W. Schmid.

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4. Homogeneous Spaces

279

In order to describe the solution to Problem D we consider again the function 4, above, [see (C(i)]. Let c,(D), D E D ( G / K ) , denote the corresponding system of eigenvalues, i.e.

04, = cA(D)4L3

D E D(G/K).

For 1 E a: let € , ( X ) denote the joint eigenspace

g A ( X )= {IE € ( X ) : Df' = c,(D),f' for D E D ( G / K ) } . Each joint eigenspace has this form for a suitable 1E a,*; in fact, each joint eigenspace contains a K-invariant function which by C(i) has the form +A ( 1 ~ a r ) . For Problem D we have the following answer (Helgason [1970a], Chapter IV, [1976]).

D. For 1 E a: let TAdenote the eigenspace representation of'G on €,(x). Then

T, is irredricible e l/rx(l)# 0. Here the function T x is defined by

rx(1)=

n

r(+t(fma

aeZo

+ 1 + (U, ao>))r(+(ma + mza + ( i A ~ 0 ) ) ) ~

where r is the usual gamma function, C, runs over the set of indivisible restricted roots for G / K , mC denotes the multiplicity of a root p, and a,= a/(a, a), where ( , ) is the bilinear form on a t induced by the Killing form of 9. The function Tx is the denominator of the function c(L)c(-1), where c(1) is the c-function from Harish-Chandra's expansion of the spherical function 4, (cf. Chapter IV, s5-6). Since the symmetric space G / K is determined by the triple (a, C, m), where C is the set of restricted roots and m the multiplicity function (cf., [DS], Chapter X, Exercise F9), it is reasonable that, in principle, Problem D should be answerable in terms of these data. The criterion rx(l)-# 0 is a confirmation of this. With this notation we can state the solution to Problem C in a more precise form. Let ri(1) denote the product above with a only running over the positive elements in C,. Let P , denote the Poisson transform ( P AT ) ( x ) =

J~e(id+p"a(x-b)) dT(b),

x E X , T E d'(B).

B

Then the following refinement of C(iii) holds. C(iv). Let 1E a:. Then the following conditions are equivalent: (a) i/T&I) # 0. (b) The Poisson transjorm P A :d ' ( B ) 4 g A ( X )is injectioe. (c) The Poisson transjorm P,: d ' ( B ) -+ b , ( X ) is surjective.

280

INVARIANT DIFFERENTIAL OPERATORS

[Ch. 11

The equivalence (a)-(b) is proved in Helgason [1970a] and [1976] and the implication (c) * (a) is a simple corollary of this and the proof of C(ii) above. The implication (a) =-(c) is proved in Kashiwara et al. [1978]. See Schlichtkrull [1984b1 for further information.

2. The Algebra D ( G / H ) Given a coset space G / H , our aim is now to describe the operators in D(G/H). First we consider the case when H = ( e ) and write D(G) for D(G/(e)),the set of left-invariant differential operators on G. If V is a finite-dimensional vector space over R, the symmetric algebra S ( V ) over V is defined as the algebra of complex-valued polynomial functions on the dual space V*. If X,, . . .,X , is a basis of V, S ( V ) can be identified with the (commutative) algebra of polynomials ak,...k.X:l

' '

x?.

(k)

Let g denote the Lie algebra of G (the tangent space to G at e ) and exp: g -+ G the exponential mapping which maps a line RX through O_ in g onto a one-parameter subgroup t -+ exp t X of G. If X E g, let X denote the vector field on G given by

where L, denotes the left translation x + gx of G onto itself. Then a differential operator on G, and if h E G, then

(XL")(s)= ( so

E D(G). Moreover,

m

o

is

L m h - ' 9 ) = (X/"dl

the bracket on g is by definition given by

[ X , Y]-

=

x7- 7r7, x,Y E g ,

the multiplication on the right-hand side being composition of operators. The following result, which connects S(g) and D(G), shows in particular that D(G) is generated by the (X E 9). Thus it coincides with the algebra introduced in [DS], (Chapter 11. $1, No. 4), with the same notation, except that now we are allowing complex scalars. Theorem 4.3. Let G be any Lie group with algebra g . Let S(g) denote the symmetric algebra over the vector space g. Then there exists a unique linear bijection I : S(g) -+ D(G)

§41

28 1

4. Homogeneous Spaces

such that I(X") = Rm(X E 9, m E 2'). If' XI, . . . , X, is any basis of g and P E S(g), then

(4)

(Wf)(g) =

{W1,. .., a,)f(sexp(tlX1 +

where j ' ~ b ( G )di, = a/&, und t = ( r l , .

0

.

.

+ tnXn))}t=o,

. . , t,,),

Proof. Fix a basis X I , . . . ,X, of g. Then the mapping

gexp(r,X,

+ + rnX,J+(tl, ..., r,,)

is a coordinate system on a neighborhood of g in G, so formula (4) defines a differential operator I ( P ) on G. Clearly A(P) is left invariant, and by (3) I(Xi) = Xi, so by linearity I(X) = 3 for X E g. Also,

1:x i x i , equals C xixj(I(Xi xj>.f)(g)= (Jix2)J)(g)*

which, writing X =

By the same argument (5)

I(x") = X",

X E ~ ,m E Z +

For a fixed m E Z + , the powers X" ( X E ~ span ) the subspace S"(g) c S(g) of homogeneous elements of degree m (cf. Chapter I, Exercise Dl). Thus ( 5 ) shows that although I is defined by means of a basis, it is actually independent of this basis. Next we prove that 1 is one-to-one. In fact suppose I ( P ) = 0 where P # 0. With respect to a "lexicographic ordering," let ax:' . . . X:n be the leading term in P. Let f ' be a smooth function on a neighborhood of e in G such that

+ t,,X,)) = t;"' ... t r n f'(exp(tlX1 + for small t. Then ( I ( P ) f ) ( e )# 0, contradicting 1(P) = 0. Finally, I maps S(g) onto D(G). In fact, if U E D ( G ) ,there exists a polynomial P such that (uf')(e) = {P(dl,.. . , d,)f(exp(tlXl

Then by the left invariance of u,

II

=

+ ... + tnXn)lr=O.

1(P) so I is surjective.

282

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

The mapping A is usually called symmetrization.

Definition.

The mapping

A has the following property. If Y,, . . . , Y, E g, then

where 6, is the symmetric group on p letters. This follows from (5) used on ( t , Y, . . . t , Y,), by equating the coefficients to f I t,. We now recall some facts concerning the adjoint representation Ad of G (or AdG) and the adjoint representation ad of g (or ad,). If g E G, the mapping x -,gxg-' is an automorphism of G; the corresponding automorphism of g is denoted Ad(g). Thus

+

(6)

+

exp Ad(g) X = g exp Xg-

',

X E g, g E G.

Then the mapping g + Ad(g) is a representation of G on g. By general theory it induces a representation of g on 9, denoted ad (cf., [DS], Chapter 11, $5). Thus by definition (7)

Ad(exp X)

X E g,

= eadX,

where for a linear transformation A, eA denotes and (7) one can deduce (cf., loc. cit.)

(8)

c; ( l / n ! ) A " .From ( 6 )

X, Y E g.

ad X( Y) = [X, Y],

These operations can now be extended to differential operators. Let us calculate (Ad(g)X)-. Recalling the translations

L,: x + gx,

R,: x

-P

(Ad(g) X)- = X R g - I Thus we define for D E D(G) (9)

Ad(g)D = DRg-'.

Then Ad(g) is an automorphism of D(G).

xg,

441

4. Homogeneous Spaces

283

Next we observe that (ad(X)(Y))-

=

xi?- i?x;

hence we defne for D E D(G) (ad X ) ( D ) = X D - D x ,

(10)

and then ad X is a derivation of the algebra D(G). We can also define (1 1)

ead”(D)=

“ 1 1 I(ad X)”(D), n*

D E D(G),

0

because (ad X ) ” ( D ) by (10) is a differential operator of order
F z Obi,z+j=n

so ead‘ ( D , D 2 ) = ead‘(D,)ead‘(0,).

Thus Ad(exp X ) and eadXare automorphisms of D(G); they coincide on g, hence on all of D(G), since by Theorem 4.3, g generates D(G). Consequently, (12) Lemma 4.4.

Ad(exp X ) D = eadX(D),

D E D(G).

Let X E g, D E D(G). Then X D = D x $and only iJ

DRerptX = D

for t~ R.

In fact we have by (9)-(12) lim -1 ( ~ R = p ( - t x )- D ) = X D 1-0

-

D X* ,

t

so the “if” part is immediate. On the other hand, if X D = O x , we have D R C X p f=X D by (9) and (12). Corollary 4.5. Assume G is connected. Let Z ( G ) denote the center of’ D(G) and I ( g ) c S(g) the set of’Ad(G)-inuariants. Then

(13)

W9))

=

Z(G).

284

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

Moreover Z ( G ) consists o j the right-invariant differential operators in D(G), in other words, ojthe bi-invariant diflerential operators on G. The last statement is immediate from the lemma (since G is connected). Since 4Ad(g) P ) = Ad(g) &P),

P E S(9)

statement (13) follows immediately. Now suppose G is a connected Lie group and H c G a closed subgroup. Let g 3 b be their respective Lie algebras and m a complementary subspace, g = m @ t, (direct sum). We now use m to introduce coordinates on G/H (cf. [DS],Chapter 11, $4). Let (XI, ... , X,) and (X,+ . . , X,) be bases of m and t,, respectively, and n: G + GJH the natural projection. Then, if g E G, the mapping (xl,. .., x,)

(14)

-, n(g

exp(x,X,

+ + x,X,))

is a diffeomorphism of a neighborhood of 0 in m onto a neighborhood of n(g) in G / H . The inverse of (14) is a local coordinate system near n(g), turning GJH into a manifold. The mapping n: G + G/H has a differential dn which maps g onto the tangent space (GJH), to G / H at the origin o = {H}. The kernel of dn is t,. The translation r(g): xH + gxH satisfies 7t

and since ferentials,

n o

0

L, = r(g) 0 x ,

Rh = n, Ad,(g)X = dR,-l odL,(X), we have for the difdn 0 AdG(h) X = dz(h), 0 dn(X),

X E 9.

Thus under the isomorphism the linear transformation Ad,(h) of g/t) corresponds to the linear transformation dr(h), of (GJH),. The coset space GJH is called reductive if the subspace m c g can be chosen such that 9 = m @ @, Ad,(h)m c m

(hE H).

If H is compact [or if just Ad,(H) is compact], then G/H is reductive. In fact, g will then have a positive definite quadratic form invariant under Ad,(H) and we can take for m the orthogonal complement of t, in 9. Let

DH(G)= {DE D(G) :DRh= D

for all h E H},

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4. Homogeneous Spaces

285

and if .f’ is a function on G / H , we put f = J ’ o 7c. We have now a Liealgebraic description of D(G/H) for a reductive coset space. Theorem 4.6. Assume G / H reductive. Then the mapping p : u -+ D,, where

(D,,j)- = ~

f , . f b~ (G/H),

is a homomorphism of’D,(G) onto D(G/H). The kernel is DH(G)

D(G)h

so we have the isomorphism

DH(G)/DH(G) D(G)t)

D(G/H)* Proof. Let UED,(G> and ~ ’ E ~ ( G / HThen ) . uf is right invariant under H, thus of the form fl,1; E ~ ( G / H ) D, ; is the map f+f,. It decreases supports, so D, is a differential operator. It is G-invariant because (DL(B1J’)-= ((DUf.r(8)-1)r(B))- = ( ( D , j r b - ’ ) ) - ) L g =(u(fT(q-)k7 = u3 = (D,f)-,

so that D;(e) = D,, whence D, E D(G/H). Also, u -+ D, is a homomorphism Next we prove that the mapping is surjective. Let E E D ( G / H ) . We express E at a in terms of x,, . . . ,x,; there exists a polynomial P such that

By the G-invariance, ( E f ’ ) ( g .0) = # 5 f r ( g - ’ ) ( o )

In particular, take g = h E H. Then

hence we conclude that P is Ad(H)-invariant. Put u = A(P)ED(G).Then l , R ~ , -‘

=

Ad(h) u

=

A(Ad(h) P) = A(P) = U,

286 SO

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

u E DH(G). Also,

=

(Ef’)-(g),

so D, = E. Thus our map is surjective. It remains to prove D, = 0 o u E DH(G) n D(G)t). For this we insert the following lemma. For d 2 0 let Dd(G)= A ( x e < d Se(g)). Lemma 4.7. D(G) = D(G)t, 0 A(S(m)) (direct sum). Moreover, if D c D d ( G )decomposes D = D , + D , , then D,, D , €Dd(G). Proof. Given P E S(g), we claim that there exists a Q E S(m) of degree Ideg(P) such that I(P - Q) E D(G)t,. This is clear if P has degree 1. We now assume that the claim is true for P e S ( g ) of degree < d . We must prove it holds for P of degree d. We may assume P = X;l X:“ in terms of the bases X,, . . . , X, of m, X r + l , . . . , Xflof 5. If e,, ... + efl = 0, we can take Q = P. If e , + , + + e, > 0, A(P) is a linear combination of terms where X , , E t, for some i. Then

+

xxl x,,, e

e

e

s,, . . f,,- sm1 . . . zai,xai , .. . x,,xUiE Dd-’(G), +

so that A(P) - D E D(G)t)

for some D E Dd-’(G).

By the induction hypothesis, there exists a Q E S(m) of degree I d such that

-

1

A(Q) - D E N G ) ~ ) , whence I(P - Q)E D(G)t,. This gives the decomposition; it remains to prove the directness. Let PeS(m), P # 0. Then there exists a function j ’ * ( x , , . . .,x,) such that

aax3

(

)

. . ., - j’*

P -,

(0) # 0.

Choose f ’ Cm(G/H) ~ such that f’(n(exp(x,X,

+ . + x,X,)))

= j’*(xl,

. . . , x,)

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4. Homogeneous Spaces

287

Since both summands are stable under Ad,(H), we deduce Corollary 4.8. Lei I(m) denote the set of' Ad,(H)-invariants in S(m).

Then = (DdG)

D(G)t))@ We can now finish the proof of Theorem 4.6. Let U E D , ( G ) such that D, = 0. Let DH(G)

11

= u,

+ u,,

as ,in Corollary 4.8. Then D,, = 0, so that D,, = 0. But u2 = A(P2), P , E I(m). We claim u2 = 0. If not, then as we saw above, there exists f~ b ( G / H ) with u2f # 0, so that D,, # 0, which is a contradiction. Thus u2 = 0, so that u E D,(G) n D(G)t). Combining Theorem 4.6 and Corollary 4.8, we obtain the following result. Theorem 4.9. Let GIH be a reductive homogeneous space. The mapping Q + Da(Q,is a linear bijection of I(m) onto D(G/H). Explicitly, if' Q E I(m), rhen

While the mapping Q + D,,,, is not in general multiplicative [even when D ( G / H ) is commutative], we have -

-

D d ( P ~ P ~ ) D d ( P ~ D) d ( P z )

where Q E I(m) has degree <deg P , By induction we obtain,

+

Dd(Q)9

+ deg P , .

Corollary 4.10. I f I(m) has a j n i t e system of' generators P I , .. . ,P , , each D E D ( G / H ) can be written and we put Di = D a ( p i )then

3. The Case of a Two-Point Homogeneous Space. The Generalized Darboux Equation

We consider now the case in which M is a two-point homogeneous Here ). the Laplace-Beltrami operator can be space (Chapter I, &I characterized in terms of its invariance.

288

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

Proposition 4.11. Let M be a two-point homogeneous space, M = G / K , where G = Z(M). Then D(G/K) consists of’ the polynomials in the Laplace-Beltrami operator. Proof. The two-point homogeneity implies that Ad,(K) acts transitively on the unit sphere in the tangent space (GIK),; thus it is clear that I(m) is generated by X : X : if (Xi) is an orthonormal basis of m. This implies the result.

+

+

Remark. It is useful to observe that the result holds also for D(G’/K‘) if G’ is a subgroup of Z(M) doubly transitive on M = (G’/K’). Proposition 4.12. Let G/K be a symmetric space of’rank one, L the Laplacian, and M’ the mean value operator 1

”

where dw is the area elbment o f S r ( x ) and A(r) the total area. Then (16) (17)

M‘L

=

LM’,

L , ( m Y ) ) = L , ( m Y)),

where F(x, y ) = ( M d ‘ ” s y ) f ’ ) ( ~ ) . Remark. This result is the analog to Lemma 2.14 in Chapter I, and (17) is the generalized “Darboux equation.” Proof. Let SZ denote the Laplacian on the semisimple Lie group G , the pseudo-Riemannian structure defined by means of the Killing form. Let R: G + G / K be natural projection and f = f R. Then the operator ~ ( $ 2 ) from Theorem 4.6 is G-invariant, and by Proposition 4.11 (or directly) p(Q) is proportional to L. If h E G is such that r = d(o, h . 0). then 0

( M ‘ f ) ( g * 0) =

1

f ( g k h ) dk.

K

By the bi-invariance of $2,

and this proves both (16) and (17).

$51

289

5. Differential Geometry on Symmetric Spaces

$5. Invariant Differential Operators on Symmetric Spaces 1. The Action on Distributions and Commutativity

Let G be a separable unimodular Lie group, d y a fixed Haar measure on G, and D -+ D* the corresponding adjoint operation on E(G), the set of all differential operators on G . The convolution

(1' * g)(x) =

(1)

JG

m y -

l)

g ( y ) dY =

J f ( Y ) d y - ' 4d y G

is well-defined for 1; g E C ( G ) if at least one of them has compact support. More generally, we define the convolution t , * t2 of two distributions on G , at least one of compact support, as the distribution (2)

0 1

* t2Hd) =

J J W Y )d t , ( x ) dt,(Y),

4 E Q(G)*

G G

Then we have the associativity (tl * t 2 ) * t 3 = t l * ( t 2 * t 3 ) if at least two of the t i have compact support (cf. Chapter I, 42, No. 8). A locally integrable function F on G can be viewed as a distribution C#J + J +F on G. With this identification we have for f' E &G), t E Q'(G), at least one of compact support,

( f * t)(x) =

(3) If

T

s

f ( W - 9 dt(Y),

(t *f ) ( x ) =

J f W1x1 dt(Y).

is a diffeomorphism of G leaving dg invariant, we have for D E E(G)

(4)

(D*)' = (D')*,

The mapping -: x functional

-+

(DT)' = D'(T'),

TE~'(G).

x - ' is a diffeomorphism and, given D E D(G), the ED:

4

-+

(D$)(e)

d

Q(G)

is a distribution with support { e } . We have v

(5) whence

( t , * t2)" = I ,

v

* t,,

Dt = t * E D ,

& D , D ~= ED^ * E D , . (6) Let X E g, the Lie algebra of G, and let (resp. denote the left=X invariant (resp. right-invariant) vector field on G such that (resp. X, = X).Since

x

s;Cxf)(x exp t X ) dx =

x)

dt

x,

exp t X ) dx = 0,

290

INVARIANT DIFFERENTIAL OPERATORS

we have

jG xf’= 0. Using this on a

[Ch. I1

product f’g,we obtain

(X)*= 4, (X)”= -x.

(7)

Also, if D,,D ,

E E(G), we

have

(DlD2)= DYD;.

(D1D2)* = DTD:,

(8)

Writing D E D ( G ) as a polynomial in the 2 ( X E ~ ) we , deduce from (5)-(8)

(9)

ED*

=

(ED)”.

Clearly if D is a bi-invariant operator, i.e., D E Z(G), then tD is invariant under all inner automorphisms g + x g x - ; moreover, t * t D= E~ * r for each t E W ( G ) in this case. Let K c G be a compact subgroup with Haar measure dk normalized by dk = 1. Let 11: G + G / K be the natural mapping, put = F 0 II, and let d x be the measure on G/K defined by

’

sK

(10)

JG,W d x = JGF(9)dg,

F E C,(G/W.

With f denoting the Lie algebra of K , let p : D,(G) + D ( G / K ) denote the homomorphism from Theorem 4.6 with kernel D,(G) n D(G)€. Since (~(E)F)= - E(F), formula (10) implies p ( ~ * = ) (p(~))*. Extending the operation F + F, we define for T E ~ ‘ ( G / Kthe ) distribution T o n G by (1 1)

T ( f ) = T(f), f ’ W~G ) ,

where

Then

T(F)= T ( F )

for F E B ( G / K ) ,

so T is determined by its values on the space 9 ( G / K ) - . If T, and T2 are two distributions on G / K , at least one of compact support, their “convolution” T, x T2 is the distribution on G / K defined by (12)

(TI x T2)(F) =

(T, * T2)(F).

Then (TI x T2)- =

T’ * T2,

PI

5. Differential Geometry on Symmetric Spaces

29 1

so x satisfies the associative law. If 6 is the delta distribution f’ + f(o) at the origin o in G / K , we have (13)

6 x T

T x 6 = T,

=

IKTr(’)dk.

If D E D ( G / K ) ,we have (14)

D(T, x T2)= TI x DT,

To verify this we select E E D , ( G ) such that p ( E ) = D. Then (15)

( p ( E ) T ) - = E(T),

T E9 ’ ( G / K ) ,

because both sides have the same value on each F (F E g ( G / K ) ) . By combining this with the fact that E commutes with left convolution t + s * t on G , (14) follows easily. We shall now see that in the case when G / K is symmetric there is an analog of (14) with D operating on the Jirst fhctor. Hence, in addition to our assumption that G is a separable unimodular Lie group and K compact, we assume now that (G, K ) is a symmetric pair, that is, there exists an involutive automorphism 8 of G such that (K&o c K c K , , where K , is the fixed point set of 8 and (K,)o its identity component. Let s denote the symmetry g K + 8(g)K of G / K onto itself. The automorphism 8 induces an automorphism of g and of D(G); these will also be denoted 8. We then have for each E E D ( G ) , Ee

=

8E.

In fact, the maps E + Ee and E + 8E are both automorphisms, so it suffices to check this for a left-invariant vector field X in G . But then

Also, if E E DK(G),we have

as an immediate consequence of

292

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

Lemma 5.1. Let (G, K ) be a symmetric pair, K compact, and D E D ( G / K ) . Then if' J E 9 ' ( G / K ) is K-invariant and T Eb ' ( G / K ) , we have

D(J x T ) = DJ x T D(T x J ) = ((D*)"T) x J . Proof. Select E E D , ( G ) such that p ( E ) = D and put j = 3, t = ?=. By the symmetry, 8(g) c K g - ' K (g EG), so putting

f"(g) =

ss

f ' 9(G), ~

f ( k g k ' ) dk dk',

we have (f'")q= (f)q. Since j ( f ) =.i(f"),we deduce (16)

j' =

;,

In particular, since j * r is bi-invariant under K , (j * r)' 8 is involutive and therefore leaves dy invariant, so

* t)'

=

Ee(j * t)- = ~ ' (*J") i

i * (Ej)'

=

(Ej * t)- = ( E j * t)'.

( E ( j * t))' = E'(j =

= (.j

* t)".

Now

Combining this with (15), we get the first relation of the lemma. Since j is bi-invariant under K and eE invariant under the conjugation g -,kgk- l , it is clear that j * eE is bi-invariant under K . Thus by (9) and (16) EE

But (eE*)' = t+*p,

* j = (1" * ~ p ) =" (j' * ~ p ) '= j * (EE')'.

SO

this proves eE * j = j

*

&(E*)B.

Taking left convolution with t and using (9, we get ( E t ) * j = (E*)'(r * j ) . Also, p((E*)') = (p(E*))" = (D*)", and now (15) implies the second relation of the lemma. Let 9'(G)q = { T E !Y(G): T

and define b'(G)@, C:(G), 9'(G),

bi-invariant under K }

. . . similarly.

Corollary 5.2. b'(G)q and C:(G) are commutative algebras under convolution.

This follows from (16), since (tl

* r2)'

= t!

* t;,

* t2)" = 22 * 1,. v

(tl

v

051

5. Differential Geometry on Symmetric Spaces

Corollary 5.3.

293

[ f ' DE D(G/K), then D* = D".

This follows by taking J = 6 in Lemma 5.1. Corollary 5.4. For (G, K ) a symmetric pair, K compact, the algebra D ( G / K ) is commutative.

In fact, if D , , D ,

E

D ( G / K ) , we have by Corollary 5.3,

D:D: = ( D l D 2 ) * = (DID2)' = D",D", = D:DT. Theorem 5.5. D E D ( G / K ) . Then

Let (G, K ) be a symmetric pair, K compact and

D(SxT)=DSx T=SxDT, jor all S , T E ~ ' ( G / K at ) , least one of'compact support. For this we need another lemma. Lemma 5.6. Let 9 ' ( G / K ) be the space of' K-invariant functions in and g through G the linear combinations 9 ( G / K ) . As 4 runs through 9', of the functions qY(e)are dense in 9(G/K). Proof. Suppose to the contrary that there were a distribution T # 0 on G / K such that T(@ce)) = 0 for all 4 E 9'and all g E G . This translates quickly into the condition T * $ = 0 for all $ E 9'(G). This implies = 0, whence the lemma.

We can now prove Theorem 5.5. If g E G , the mapping t ( g ) of G / K , extended to functions, commutes with D as well as with the right convolution f -,f x T ( T E b'(G/K)). Hence the first relation of Lemma 5.1 implies (17)

D(4"" x T ) = D(4r(g')x T,

T E b'(G/K).

By Lemma 5.6 and by the continuity of the map f ' + f x T on 9 ( G / K ) , (1 7) implies

(18)

D(.f x T ) =

(of)x

T,

~'E~(G/K).

However, 9 ( G / K ) is dense in 9'(G/K) for the strong topology, and D as well as the convolution S + S x T are continuous on 9 ' ( G / K ) (Schwartz [1966]. pp. 75, 170). This, by (18), D(S x T ) = DS x T,

S E ~ ' ,T E C .

For S E b',T E 9' the proof is similar, so Theorem 5.5, is proved.

294

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

Theorem 5.7.

(i) Let (G, H) be a symmetric pair, G semisimple, H connected. Then D ( G / H ) is commutative. (ii) Let L be a connected Lie group, L* the diagonal in L x L. Under the bijection (II, I$* + 1112 of' ( L x L)/L* onto L we have the identijcation D ( ( L x L)/L*) = Z(L). Proof.

(i) Let

(19)

9=Ij+q

be the decomposition of g into the eigenspaces of do for the eigenvalues + 1 and - 1 , respectively. Here I) is the Lie algebra of H. There exists a Cartan involution O of g commuting with ([DS], Chapter 111, Exercise B4). Let

(20)

g=f+p

be the corresponding Cartan decomposition. Let 9' be the complexification of 9, $' and q' the subspaces of 9' generated by b and q. Put u = f + i p . Then u is a compact real form of g', and since otl = Oo, we have

+ u n q', $' = u n $' + i(u n $'). u = u n $'

The subspace 5 = u n If + i(u n q') (21) is a real form of gc and (21) is a Cartan decomposition of 5. Let S denote the adjoint group Int(s), put @*= u n $', and let H* be the analytic subgroup of S with Lie algebra Ij*. Then H * is compact and (S, H*) is a symmetric pair. By Corollary 5.4, D(S/H*) is commutative, so by Theorem 4.6,

(22)

D,p(S)/D,p(S)n D(S)Ij*

is commutative.

Passing to the complexifications (22) means that (23)

Z,(g')/Zhc(g') n U(g')$'

is commutative.

Here U(g') denotes the universal enveloping algebra of 9' and Z,,(g') the centralizer of $' in U(g'). But just as (22) is equivalent to (23) (€f being the complexification of b*), property (23) is equivalent to

(24)

DH(G)/DH(G)n D(G)@

is commutative,

which by Theorem 4.6 proves that D ( G / H ) is commutative.

051

5. Differential Geometry on Symmetric Spaces

295

(ii) The action (II, I,)L* + (glll,q2l2)L* of L x L on ( L x L)/L* corresponds to the translation 1 + gllg;' on L. This proves (ii), since Z ( L ) consists of the bi-invariant differential operators on L .

2. The Connection with Weyl Group Invariants Let X = G / K be a symmetric space of the noncompact type, that is, G is a connected semisimple Lie group with finite center and K is a maximal compact subgroup. Let g = € p be the corresponding Cartan decomposition of g, let a c p be a maximal abelian subspace, let a+ c a be a fixed Weyl chamber, and let the set Z of roots, the set X + of positive roots and the Weyl group W be defined as in [DS] (Chapter IX,01). If E is a Euclidean space, let b ( E ) I> Y ( E ) 3 9 ( E ) denote, respectively, the space of C"-functions, the space of rapidly decreasing C"functions, and the space of compactly supported C"-functions. The subscripts K and W denote K-invariance and W-invariance, respectively. We shall now consider the subspaces of K-invariant and W-invariant functions,

+

= YK(P) = = b(P), 9w(a> = Y W ( 4 = &,(a) = &(a). 9K(P)

Theorem 5.8. The restriction to a is an isomorphism oj' QK(p) onto 9W(a).

Only the surjectivity remains to be proved. For this we first prove two regularity theorems for the Laplacian L,. Lemma 5.9. Let T E 9'(p) satisjy L, T = 0. Then T E b ( p ) .

Proof. If 4 E 9 ( p ) then 4 * T is a harmonic function. The mean value theorem for harmonic functions implies that if a E 9 ( p ) is a radial function with f a = 1, then (4*T)*a=#*T.

Evaluating at 0, we obtain T

* c1 = T, so that

T Eb(p).

Remark. The same argument shows that if T then T is smooth near 0.

E 9'(p),

L T = 0 near 0

As usual, we let Ck(p) stand for the space of k times continuously differentiable functions on p; here Co(p) is understood to mean C(p), the space of continuous functions on p. g

Lemma 5.10. Let T be a distribution on p such that L,T = Ck(p). Then T E Ck+'(p).

E

296

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

Proof. Fix X e p and select + E ~ ( P )such that 6 = 1 in a neighborhood of X . Assuming first n # 2 (n = dim p) we consider the convolution F = +g * S , where S is the distribution ((2 - n)Qn)-'rZ-n which satisfies L,S = 6

(cf., Chapter I, Lemma 2.35). Then L P F = +g, so by Remark following Lemma 5.9, F - T is C" near X . O n the other hand,

( 2 - n)R,F(X) = If D = D"

s,

(+g)(X - Y ) l Y 12-"

(I a I = k ) in the notation of

dY.

$1, we have

( 2 - n)R,(DF)(X) = [(D+g)(X - Y ) (Y Iz-n dY P

(D@g)(Y)IX- Y ( ' - " d Y .

We write this in the form H(X) =

s,

h(Y)IX - YI'-"dY,

h~C,(p),

and have then to prove that H E C ' ( ~ )Although . Di(IX - Y 1'-") (where Di = 8/8xi) is readily seen to be locally integrable, so that the function h ( Y ) Di(IX - Y 12-")

dY =

s,

h(X - Y ) Di(I Y Iz-") d Y

exists and is continuous, we still have to prove K = D i H . For this we replace the function Y + I Y 1'-" by the function

297

5. Differential Geometry on Symmetric Spaces

$51

so lim H,(X) = H(X)

uniformly.

E+O

Second,

I

K(X)- (DiHJ(X) = h(X - Y)CDi(IyI2-") - (Di4e)(Y)I dY =

(2 - n)

J

h(X - Y)"YI-"y,

-

&-"yi]dY,

iris&

which by polar coordinate introduction is seen to be bounded by a constant multiple of E. Thus lim (DiH,)(X) = K ( X ) &-0

uniformly. This proves K = D i H , so we have proved that F and therefore T is of class C k + ' . For n = 2 we can take S = (2n)-' log r, which satisfies L,S = 6 (cf., Lemma 2.35 in Chapter I) and proceed as before, replacing 4,(Y) by the C1-function $,(Y) = foglyl loge- 1 + e - ' l Y l

for IYI > E for I Y I I E .

We can now prove Theorem 5.8. Let f e g W ( a ) . Since K-conjugate elements in a are W-conjugate ([DS], Chapter VII, Proposition 2.2), f can be extended to a K-invariant function f on p. Since the distance between two K-orbits is minimized on a (Chapter I, Proposition 5.18), it follows that f is continuous. Let a' c a be the set of regular elements and let A(&) be the radial part of L, for K acting on p with transversal manifold a'. According to Proposition 3.13, we have

on a + ; but both sides are W-invariant, so (25) actually holds on a'. Since the function f is invariant under the symmetry sol, the function g = A , f satisfies 9"" = -9, so the function a - ' ( A , f ) extends over the hyperplane tl = 0 to a smooth function on a. Thus A(Lp)f extends from a' to a member of gW(a),which we denote by the same symbol. We shall now prove that (26)

L , ( f ) = W,)f)-

298

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

in the sense of distributions. Assuming this formula, we obtain by iteration, L ; ( f ) = (A(Lp)"'f)-. But since the right-hand side is continuous, successive use of Lemma 5.10 gives f E Ck+' ( p ) , k = 0, 1,. . . , proving the theorem. Since both sides of (26) are K-invariant, it suffices to prove that both sides give the same result when applied to any function 4 e g K ( p )Now . using Theorem 5.17 in Chapter I, and (49) in 53, we have, bar denoting restriction to a, L p ( f ) ( 4 )= J 3 ( X ) ( L P 4 ) ( X )d X P

= =

I+

( f w wd H

f(H)(A(Lp)$)

l+!La(6)6

dH

+ l+!(grad a($)d H .

Now we have the divergence theorem (cf. (26) in Chapter I, 52) SD(div F ) ( X ) d X

=

s,,

( F , n) d o

for a vector field F on R", D c R" a connected bounded open set whose smooth boundary aD has a well-defined outgoing unit normal n, d X and d o being the Euclidean measures. Using this on the vector field F = $ grad (, we obtain Green's first formula: dX Jy5. + (grad $, grad i))

=

s,,

$(grad (, n) do.

By putting $ = f6, ( = 6,the first integral on the right above becomes r

J

r

fi3La($) dH = -

a:

(grad(f'd), grad

6)d H

Ja;

Here a: is a region approximating a+ n BR(O), where BR(0) is a ball in a containing supp(6). Since 6 = 0 on da+ and 4 = 0 in a neighborhood of SR(0), the integral over aa: tends to 0 as F. -,0 and we get, using grad(f'6) = 6 grad f + j ' grad 6,

L p ( f ) ( 4 )= -

Ja+

(grad J; grad $)6 d H .

651

5. Differential Geometry on Symmetric Spaces

299

On the other hand,

(Wp)f)-(4)=

J (A(Lp)J')(H)4(H)Wf) dH a+

= J-+6La(1,)SdH

+ J-+m(srad(a))(J) dH.

We can now repeat the Green's formula application above with j and

7 interchanged. Then we end up again with the integral -

J-

(grad

6, grad j ) S

dH.

a+

This proves (26) and the theorem. Corollary 5.11. (i) The restriction to a induces an isomorphism o j g K ( p )onto &,(a). (ii) The restriction to a induces an isomorphism o j Y , ( p ) onto YPW(a).

The first statement is immediate from Theorem 5.8. For the second statement we first recall (Chapter I, Lemma 2.2) that F E Y ( R " ) if and only if for all integers k, 12 0

(27)

SUPl(1 + Ixl')(Lkf')(x)I<

00,

X

L being the Laplacian. Let j E Y,(a) and let f E b,(p) be its extension given by part (i). Then we have on a' =

~ , f+' f C m , a - ' A , j , aeT

but as pointed out above, the right-hand side is smooth on all of a and But then L, f is the K-invariant extenis, in fact, a function jlE Y,(a). sion of this function to p. It clearly satisfies (27) (for k = 1). Repeating this argument with .f' replaced by j , we obtain (27) for all k , so f~ ,4pK(p),as claimed. Corollary 5.12. Every W-invariant polynomial P on a can be uniquely extended to a K-invariant polynomial on p.

In fact, we can assume that P is homogeneous. Then P as given by Corollary 5.1 1 is a homogeneous Cm-function, hence a polynomial. Consider now the bijection 1:S(g) -+ D(G) from Theorem 4.3. It identifies the commutative algebras S(a) and D(A) and identifies the set I(a) of W-invariants in S(a) with the set D,(A) of W-invariant differential operators on A . o with constant coefficients.

300

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

We recall now the projection D -,D' ($3, No. 1) mapping differential operators on X into differential operators on A . o. From Theorem 3.2 we know that L; = L A . We shall now extend this result to other invariant differential operators. Theorem 5.13. The projection D to D,(A).

--$

D' is a bijection of D ( G / K ) on-

G maps A o into itself. Then for some a E A, g .o and Ad(a- ' 9 ) maps the tangent space (A o), into itself. Hence a - ' g E M ' , the normalizer of a in K. This shows that M'A is the subgroup of G which leaves A . o invariant. Thus, if D E D(G/K), then D' E D,(A). For the surjectivity, let q be the orthogonal complement of a in p , so Proof. Suppose g

E

= a . 0. Then a - ' g E K

a

g

(27) Let

=

€

+ a + q.

T', * * - Tp,H', * * * H , , x , , * * x , (28) be a basis of g compatible with the direct decomposition (27) and orthonormal with respect to - B ( X , Or). Let Z(p) denote the set of Ad&)invariants in S(p) and let P ~ l ( p be ) homogeneous of degree m. Writing 9

9

a ,

+ + n,,

N = h,. . . , nd,

IN1 = n,

M = (ml,..., m,),

[MI= m, +

9

.

.

+ m,,

we have

p = IN1 t (MI= m

=

Pa +

aN, H;' . . - HY'XT' . . . Xyq

C

aN,M Hy' . . . HY'XY' *

* * Xyq,

IMI>O,N

where P,eS(a). Letting M' act on this formula and noting that it leaves a and q invariant, we deduce that Pa is W-invariant and of degree m.

Writing (29)

2 = A(Z) for Z E 9, we have A(P) = &Pa) +

C

aN,MAll . * * R'ZT'*

IMI>O,N

*

37 + Q,

where Q has order c m and IN1 + IMI = m. By Theorem 1.4 in [DS] (Chapter VI), we have the bijective decomposition G = exp a (exp q)K. For Q E T ( p ) we put D, = p(A(Q)), where p : D,(G) + D ( G / K ) is the homomorphism from Theorem 4.6. If F E &(A o) and f E &(G) is determined by F(a

- 0) = f ( a exp X k ) ,

X

E

q,

k EK,

5. Differential Geometry on Symmetric Spaces

$51

30 1

we have

and f y 1 . .

-

. Xyq

=

n(xY1... X t q ) + T ,

where T E D(G) has order < 1 M I . But by the definition of A [(4) in $41 we have

if I M I > 0. (A(Xyl - .. X ~ q ) f ’ ) ( a ( h )= ) 0 Thus we conclude from (29) and (30) that for a certain R E D ( G ) of order < m

(DbF)(a * 0) = ( P , F ) ( a * 0) + (Rf)(a), for all U E A and all F € & ( A . 0). If the differential operator R is expressed in terms of the coordinate system

+ +

+ + x,X,)exp(t,T, + ... + rpTp)

exp(h,H, . . . h,H,)exp(x,X, ... (hl9. * . h,, X I , . . . ,x,, t,, . . . , f p ) , +

9

it becomes obvious, since f in these coordinates is independent of (xi) and ( t j ) , that the mapping F

+

(Rf)lA

is a differential operator on A . o of order less than or equal to that of R. Hence order(& - P,) < m. (31) Suppose now Q ~ l ( a ) the , set of W-invariants in S(a). We wish to find D E D ( G / K ) such that D‘ = Q = A(Q). For this we may assume that Q is homogeneous. Let m = deg(Q). By Corollary 5.12 we can find P E I ( ~ ) such that P , = Q. But then by (31) order(Dl, - Q ) < m. But Dlp - Q E I(a), so the desired statement follows by induction on m. Finally, suppose a D # 0 in D ( G / K ) of order rn such that D’ = 0. Let P E I(p) be the homogeneous polynomial of degree m such that order@ - D p ) < m.

302

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

Then 0;.= ( D p - D)' has order < m , whereas Pa has degree m. This contradicts (31), so the theorem is proved. While the projection D + D' is a linear bijection between the commutative algebras D(G/K) and &(A), it does not preserve multiplication. This will now be remedied by replacing the decomposition (27) by the Iwasawa decomposition g=f

+ a + n,

which has the advantage over (27) that n is an algebra, whereas q is not. Since the decomposition G/K = exp n exp a . 0, which leads to the N radial part A(D) for D E D ( G / K ) , is rather similar to the above decomposition G/K = exp a exp q ao, which led to the projection D + D', one might expect an analog of Theorem 5.13 for the mapping D + A(D). With a small but important modification this is indeed true (cf. Corollary 5.19). The proof of Theorem 5.13, however, does not generalize because while the decomposition G/K = a e A a exp q . o is preserved by the group M'A, the decomposition G/K = U a s ANa . o is only preserved by the smaller group M A .

u

Lemma 5.14. For each D E D(G) there exists a unique element D,E D(A) such that

(32)

D - D, c nD(G) + D(G)f.

Moreover,

(33)

(w-= 4 6

whenever 4 E &(G) satisjes 4(ngk) = +(g) (n E N , g E G , k denoting restriction to A.

E

K ) , rhe bar

Proof. For (32) we choose a basis X I , . . . , X, of g compatible with the decomposition g = n a f. Because of Corollary 1.10, Chapter 11, in [DS], each D E D(G) can be expressed uniquely: (34) D = a,, w ; 1 . . .f:e,".

+ +

Let D, be the sum of terms which only involve basis elements of a. The uniqueness amounts to D ( A ) n (nD(G)

+ D(G)f) = {0},

and this follows from the uniqueness of the representation (34) combined with the fact that n and f are subalgebras. For (33) one has to prove

(DT4)b) = (XD4)(4 = 0

051

5. Differential Geometry on Symmetric Spaces

for D E D(G), T E t, X E n, a E A. But T 4 e 0, so that D T 4

303

= 0; second,

1

(XD$)(a) = lim - [(D4)(aexp t X ) - (D4)(a)]. 1-0 t

This expression vanishes, since the element n, = a exp t X a - ' to N , whence (D4)(a exp t X ) = (D")L(nl'')(a) = (D$)(a).

belongs

Remark. D, is just the radial part of D for the action g + ngk of the product group N x K on G with transversal manifold A c G.

For g

E

G let A ( g ) E a denote the unique element such that g E N exp A ( g ) K .

Since D(A) is a commutative polynomial ring, each linear mapping v: a -+ C extends uniquely to a homomorphism of D ( A ) into C, denoted D + D(v). Let a* denoted the dual of a and a,* the set of all linear functions of a into C. Lemma 5.1 5.

For each linear function v : a -+ C the function 4(g) =

J ev(A(kg))dk K

is an eigenfunction of each D E DK(G). More precisely,

D4

=

D,(v)4,

D E DK(G)*

Proof. Let F(g) = ev(A(g)),D E DK(G). Then F(ngk) = F(g), so by

(33) (DF)(a) = (D,F)(a) = D,(v)F(a). (35) The functions DF and D,(v)F are both left invariant under N , right invariant under K , and by (35) they coincide on A . Hence DF = D,(v)F, and Dg( J / ( k g ) d k ) = Sx(DF)(kg) dk = D,(v)

which proves the lemma. Theorem 5.16.

With p = 4,,(g) =

m,a, ler

J e'iv+p)(A(kg)) dk,

4sv= 4v

W g ) dk,

xaer+

v E a,*.

K

Then

JK

for each S E W.

Here sv(H) = v(s-'H), H E a, as usual.

304

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

Proof. It suffices to prove

(36)

ds =

J?v(9)f(9)

J4v(s)f(s) G dg

for each ~ E C J G ) ;since 4,” and c$v are both bi-invariant under K , it suffices to prove (36) for j ’ bi-invariant under K . Under the decomposition g = nak we have by Chapter I, Corollaries 5.2 and 5.3, dg = e-2p(loga)dn da dk, so that for f ’ C:(G) ~ n

r

=

Jlda

r

f’(na)d n ,

JNe(iv-p)(l0aa)

so in the notation of Theorem 5.7. Chapter I, (37)

j G 4 v ( g ) f ( gd) g =

J eiv(log4)Ff(a)da,

f~ Ca(G).

A

The result now follows from the functional equations F f ( a s ) = F f ( a ) in the theorem cited. Theorem 5.17.

The mapping y : D + e-PD,0 ep

is a homomorphism

ofD,(G) onto &(A) with kernel DK(G) n D(G)f.

Proof. We first remark that under the canonical identification of D(A) and S(a) the automorphism D + e - P D 0 eP of D ( A ) corresponds to the automorphism p + ’ p of S(a) given by ’H = H + p(H). In fact,

e-PH(ePf’) = ( H

+ p(H))f.

To see that y is a homomorphism let D’, D“ E DK(G). Then

+ (D’ - D6)D”, nD(G) + D(G)f. Since

D’D“ - DbD: = Db(D” - D:)

where D’ - Db and D” - D: belong to D”T for T E f, we have (D’

- Db)D” E nD(G) + D(G)f,

TD” =

$51

5. Differential Geometry on Symmetric Spaces

305

and since [n, a] c n, D:(D'' - Df) E nD(G) + D(G)f. Hence (D'D"), = DbDf, and it follows that y is a homomorphism. Next we prove that for each nonconstant p E I(p) (38)

degree(y(4p))

-

Is) < degree(p).

Here q + ij denotes the mapping of S(p) onto S(a) given by the decomposition S(p) = S(a) + S(p)q. [If S(p) and S(p*) are identified by means of the Killing form, ij means the restriction qla.] Let d denote the degree of p. In proving (38) we may assume that p is homogeneous. Writing any element X E q in the form

x = z - ez = 2 2 - (z + ez), where Z E n, it is clear that p - p E nSd- '(9) + Sd-'(g)f, where Se(g) denotes the set of homogeneous elements in S(g) of degree e. Now l(L(r= s)L(rs and 4 q , q 2 ) - &Il)%)

E

c

W(9N

esdl+dz

if q 1 E Sdl(g),q2 E Sdz(g). It follows that 4 p ) - L(rs E TWG)

+ D(G)f + 1Xse(g)), e
which implies (38). To see that y has kernel equal to DK(G) n D(G)f, first let D E DK(G) n D(G)f. Since the function 4 in Lemma 5.15 is K-invariant, we have D 4 = 0, whence D,(v) = 0 for all v , so y(D) = 0. On the other hand, let D E DK(G)be such that y(D) = 0. By Corollary 4.8 we have

D

=

qp)

+ D',

where D' E DK(G) n D(G)f and p E I(p). Thus y ( l ( p ) ) = 0. By decomposing p into homogeneous components if is clear that jj has the same degree as p . Thus by (38) p = 0, so D E DK(G) n D(G)f. Finally, we prove y(DK(G)) = &(A). Let D E DK(G) and consider the function 4" from Theorem 5.16. We have D4v = D,(iv + P M V = Y ( ~ ) ( ~ V ) 4 " 9 (38') and now Theorem 5.16 implies y(D) E I(a). For the surjectivity, let q be an arbitrary homogeneous polynomial in I(a) and let d denote its degree. We prove by induction on d that there exists a D E DK(G) such that y(D) = q. This is obvious if d = 0, so suppose d > 0. Let p E I(p) satisfy = q (Corollary 5.12). By (38) the polynomial y ( l ( p ) ) - jj has degree

306

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

< d and, as we just proved, it belongs to I(a). By the inductive assumption there exists a D , E D,(G) such that y(Dl) = y(A(p)) - p. But then the operator D = A(p) - D, belongs to D,(G) and satisfies y ( D ) = q. Combining the maps p from Theorem 4.6 and the map y above, we obtain the following result. Theorem 5.18.

The mapping

r: D ( G / K )

--*

&(A)

given by r ( p ( D ) ) = y(D) for D E D,(G) is a surjective isomorphism. Corollary 5.19. Under the action of’ N on G / K with transversal manijold A .o, let AN(D) denote the radial part of’ D E D ( G / K ) . Then r is given by

r(D)

=

e-PAN(D)0 eP

and is an isomorphism of D(G/K) onto D,(A). In fact, let E E D,(G) be such that p ( E ) = D. Then ET = (Dj’)-for f’ E 6 ( G / K ) . By taking f’ N-invariant, this implies E,f = AN(D)f where f = 0. Thus T ( D ) = T(C((E))= y(E) = e-p E,eP = ebPAN(D) 0 ep. j’ Corollary 5.20.

For the Laplacian L,on X = G / K we have

In fact, this is immediate if we compare Proposition 3.8 and Corollary 5.19.

Lemma 5.21.

For each D E D ( G / K ) we have

r p ) = er(D), where s denotes the symmetry gK

+

8(g)K.

Proof. For j ’ E @ ( G ) consider the function

F*(g) = e - P ( A ( 8 ) )

s,

f’(ng)dn

and let E E D , ( G ) satisfy p ( E ) = D. Then

E,(eP(”‘g”F*(g))=

I (Ef’)(ng)dn, JN

the subscript in E, denoting differentiation with respect to g. The funcis left invariant under N, right invariant under K. tion g + eP(A(B))F*(g) Note also that by Corollary 5.2, in Chapter I, F*(a) = F J a ) for each a E

951

5. Differential Geometry on Symmetric Spaces

A . Using now Lemma 5.14 on the function tain the basic formula

(39)

4 ( g ) = ePcAce))F*(g), we

307 ob-

y(E)F, = F E , .

From (37) we have for the adjoint E* of E, and the adjoint y(E)* of y ( E ) on A :

so

It follows that y(E*) = y(E)*. But y(E)* = W(D) and y(E*)= T(p(E*)) = r(D*) = r ( D s ) (Cor. 5.3), so the lemma is proved. Corollary 5.22. Let D E D ( G / K ) . Then the projection D' on A o and the N-radial part AN(D) have the same highest order terms, i.e.,

order(D'

-

AN(D)) < order(D).

We may assume D = D,,,,in the sense of Theorem 4.9, where P E I ( p ) is homogeneous. By (31) and (38),

(40)

order (r(D) - 0')

-= orderD,

so our statement follows from Corollary 5.19. Corollary 5.22 is, of course, very plausible because, if a E A , then the manifolds N a o and a exp q . o have the same tangent space at a . 0. If a E A + , this is also true for the orbit K a . 0 , so that one would expect an analog to Corollary 5.22 for the K-radial part AK(D). We shall now prove this in a more precise form. Let al,. . . ,af be the system of simple roots in Z + ([DS], Chapter VII, $2) and A the set of all linear combinations n l a l + - . . n,a, ( n i E Z + ) .

+

308

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

Let 9 denote the ring of functions on A + - 0 which can be expanded in an absolutely convergent series on A + - o with zero constant term: .f'(exp H . 0) =

(41)

c ape-"H',

a,

E

C, a,

=

0.

PEA

Note that 1' is a power series in the variables xi = e-ai(H).The ring 93 is invariant under the action of each D E D ( A ) . Proposition 5.23. Let D E D ( G / K ) be arbitrary and let q be its order. Then there exist functions g l , . . . ,gk E 9 and operators E l , . . . , E , E D(A) of order I q - 1 such that the K-radial part AK(D) satisfies

AK(D) = e - T ( D )

0

ep +

k

2 gi E i . 1

Proof. By analogy with Lemma 5.14 we have the direct decomposi-

tion D(G) = D ( A ) Q (B(n)D(G)

(42)

+ D(G)f).

If E E Dq(G)and if ( X i ) is a basis of n compatible with the decomposition n = Coler+ 9, of n into root subspaces g., we can select D,E Dq- '(G) such that E - E,

(43)

-

x(BXi)Di~D(G)f, I

where E, is the D(A)-component of E according to (42). Let X i E g., Xi= Zi ( x ~ p Z, i ~ f )and , h = e x p H in A'. Then

x+

+ BAd(h) X i ) = +(e-a(H)(x+ Zi) + ea(H)(- + Z J ) ,

Ad(h-') Zi = f(Ad(h-') X i so yi

=

coth a ( H ) Zi - (sinh a(H))-' Ad(h-') Z , .

Hence OXi -= g,(h)Zi

mod(Ad(h- l) f),

where g,(h) = 1 - coth a(H), so g , E 9 . We substitute this into (43) and replace ZiDiby D i Z i + [Z,, D,]. Here DiZi E D(G)f and [ Z , , D,]E Dq- '(G). Repeating this process with E replaced by [Z,, Oil, etc., we obEi E Dq- '(A) such that tain by induction elements g, E 9,

(44)

E - E, -

1gi(h)Ei E (Ad(h- ') t)D(G)+ D(G)f. i

$51

5. Differential Geometry on Symmetric Spaces

309

In particular, we can take for E an operator in D,(G) for which p(8E) = D (cf. Theorem 4.6). Applying 8 to (43) and comparing with (32) and Theorem 5.17, we see that T(D) = ?(BE) = e-p OE, 0 ep. Taking adjoint on A, we see that T(D)* = eP E , o e - p , i.e., E,

(45)

= eCP B(r(D))o

ep.

On the other hand, if C E (Ad@- ')f)D(G)

+ D(G)f,

and if j ' E @ ( G / K ) , then (CJ)(h) = 0.

(46)

This is obvious if C E D(G)f. Also, if T E f, E ,

D(G), we have

E

((Ad(h- ') T)E, J ) ( h ) = (TR(h)E, f)(h) = (T(E, f)"("'))(e) = 0. Now p ( E ) = p(BBE) = D', so, by (44) and (46)

(A,P))f)(h

.0>=

This proves that AK(DS)= E ,

( W X h * 0) = ( E J ) ( h )

+ c giEi i

= e-p

e(r(D))

0

eP

+ CgiEi, i

so by replacing D by Ds and using Lemma 5.21, Proposition 5.23 is proved. 3. The Polar Coordinate Form of the Laplacian

We shall now give an expression of the Laplacian L, on X which corresponds to the polar decomposition X' = K/M x A + - 0 (cf. [DS], Chapter IX, Cor. 1.2). Let I denote the orthogonal complement of m in f (with respect to the Killing form B of 9). Diagonalizing the operators (ad H ) 2 ( H Ea), we have where f,

=

{ T E:(ad ~ H)'T = c((H)*T

for

HEa}.

310

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

Since X

X

-+

+ OX maps the root subspace g,

onto f,, we see that

dim f, = mu, the multiplicity of a. Let T p , .. . , T i = be a basis of f,, orthonormal with respect to -B, and put w, = xyz1 TTTq, where the T ; are viewed as differential operators on G. Let 1 = f K for f E b ( X ) . 0

Theorem 5.24. The Laplace-Beltrami operator L , has the following polar coordinate decomposition:

+

sinh-2(a(log a))(Ad(a- ') w, f ) ( k a ) l€Z+

for k

E

K , a E A", f

Proof.

E b(X),L A

being the Laplacian on A

. 0.

We know from Theorem 3.5 and (12) 53 that

(47)

(L,f)(a.o)

=

+

(LKa.of*)(a'o) (A(L,>f)(a.oX

where LKa.*is the Laplace-Beltrami operator on the orbit K a . 0,f'* the restriction off' to this orbit, and f the restriction off' to A . 0. Let 0 = a . o and let 4: k + k . 0 denote the natural mapping of K onto the orbit K S O(which can be identified with K I M ) . Lemma 5.25.

'

Let Xq = (sinh a (log a))- T;. Then the vectors

d&XP),

11 i Ima, a E Z + ,

form an orthonormal basis of'the tangent space ( K . O)o. Proof.

The differential d 4 maps I onto the tangent space, and since 4(exp t T $ )= a exp Ad(a- ') tT$ - 0,

we see that

d4( Tq) = dz(a)(Ad(a- ') Tq)p, where T(U) is the translation gK + agK of G / K , p is identified with (GIK),, and the subscript p refers to the component for the decomposition g = t + p. But (Ad(a- ') Tq), = +[Ad(a- ') TY

-

Ad(a) TT],

so

(48)

B(d4(T;),d$(Tf)) = aB(T7 - Ad(a2) TY, Tf - Ad(az) Tf).

$51

5. Differential Geometry on Symmetric Spaces

31 1

Expanding this and using Ad(a2) T ; = cosh 2u (log a ) Ti^

(mod p),

the result follows from (48). To conclude the proof of Theorem 5.24, consider the differential operator D on the orbit K . O defined by

where di,* = d/dxi,a . This operator is well-defined; in fact, if k . O = k , .O, then k , = km ( m E M ) and Ad(m) gives an orthogonal transformation of each fa. Moreover, D is invariant under the action of K . For the local coordinates ~ X P ( xi. C 2X;) 0

+

xi, 2

near 0 on the orbit K . O the tangent vectors This follows from Lemma 5.25 since

(ai,,), are

orthonormal.

It follows that the Laplacian L' = LK.0 satisfies (L'J)(O)=

C ( @ J ) ( O ) + lower-order terms i. I

so D - L' has order < 2 at 0 hence everywhere, by the K invariance. Assume now Ad,,(M) leaves no vector T # 0 in 1 fixed. Then by Theorem 4.9, KIM has no K-invariant differential operator of order 1 so by the above D = L K . o .Substituting the definition of X: into the definition of D and putting 0 = a . 0,we obtain

so using Proposition 3.9 for the radial part A(L,) the theorem follows for k = e. It then follows for general k by replacingj'by the function x + f ( k . x).

312

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

We must still deal with the possibility that Ad,(M) has a fixed vector in I (and this occurs for some nontrivial examples of X). Here the proof above gives Theorem 5.24 only up to a K-invariant vector field. We therefore apply the formula for the Casimir operator R in Exercise A8. Since the vector fields 2, annihilate the functions on G that are right invariant under K and since @) = Lx (Exercise A4), we obtain

. C

-2

sinh-2(/3(log a))[Ad(a- ' ) Z ,Z - f ] ( k a ) .

PEP+

Now it is easy to see that

2

1 z,z-,

= -02,

#=a

where denotes the restriction of of Theorem 5.24.

/3 to a. This then completes the proof

4. The Laplace-Beltrami Operator for a Symmetric Space of Rank One

Let X be a Euclidean space or a Riemannian globally symmetric space of rank one. We shall derive a useful geometric expression for the Laplace-Beltrami operator L, on X. For p E X let, as usual, S,(p) denote the sphere in X with center p and radius r. If X is noncompact, S,(p) is the image of a sphere in the tangent space X, under the diffeomorphism Exp,. If X is compact, let Ldenote its diameter, which is, the maximum distance between any pair of points in X. If r < L, S,(p) is the image of a sphere in X, under a diffeomorphism ([DS], Theorem 10.3, Chapter VII). This, of course, fails to hold for S,(p) (see, e.g., X = S'). However, S,(p) is a submanifold of X, being an orbit of a compact subgroup of the Lie group I ( X ) ([DS], Chapter IV, §3). In any case, S,(p) is for each r a submanifold of X and has a Riemannian structure induced by that of X. Let A(r) denote the area of S,(p), that is, the total measure of S,(p) according to the Riemannian measure on S,(p). [If dim X = 1, so X = R' or X = S', A(r) is understood to be the number of points in S,(p).] By the homogeneity of X, A(r) is independent of p. Let (Ol,.. . ,On-l ) be Cartesian coordinates on an open subset of the unit sphere S , ( O ) in X,. The inverse of the mapping (50)

(el,. . . ,On-

r)

-+

Exp,(re,,

. . . , 4-

651

5. Differential Geometry on Symmetric Spaces

313

is called a system of geodesic polar coordinates at EX. Here r runs through the intervals 0 c r c co in the noncompact case, 0 c r c L in the compact case. If we fix r, the system {O1,. . . ,On-1 } can be regarded as a system of coordinates on an open subset of S,(p). Proposition 5.26. In geodesic polar coordinates at p as described above, the Laplace-Beltrami operator Lx on X has the form

where Ls is the Laplace-Beltrami operator on S,(p). Here 0 < r < co i f X is noncompact, 0 < r < L if X is compact. Remark. Compare with Theorems 3.5 and 3.7. Proof.

Put 0, = r and let n

ds2 =

(51)

1

gij(O,, . . . , 0,) doi dej

i,j = 1

denote the Riemannian structure of X expressed in geodesic polar coordinates at p. As before, let 3 = det ( g i j ) and ( g ' j ) = (gij)-' [the inverse of the matrix (gij)]. As shown in [DS], Chapter I, the geodesics emanating from p intersect S,(p) orthogonally. Hence (51) reduces to ds2 = dr2 +

(52)

c gij(O1,. . . , 0,-

n- 1

r)dOidBj.

i. j = 1

If I denotes the identity mapping of S,(p) into X , we have I*(dr2) = 0, so that the Riemannian structure of S,(p) is given by n- 1

C

(53)

i. j = 1

gij(el,. . . , 8,-

1,

r ) doi d o j .

Using (52) we obtain

If A denotes the matrix (gij)l s i ,j s n and B denotes the matrix (Sij ) 1 _< i. j 5 n - 1 3

we have A=

(" O) 0

1 '

A-'=

("i' y),

detA=detB.

Consequently, the last term in (54) is the Laplace-Beltrami operator L, on SAP).

314

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

The isotropy subgroup of I ( X ) at p acts transitively on the set of geodesics emanating from p . It follows that the function

log and

& = a(r) + m,, . . ., &= e ea(r)

On-1)

P(UI.....B~-~)

for suitable functions a and p. Note that a’(r) is independent of the choice of {el,.. . , O n - , } . So is a(r) if we, for example, fix a(L/2) by a(L/2) = 1. NOW A(r) =

do, SAP)

where d o is the Riemannian measure on Sr(p) which is locally given by dO,-,. Remembering that a(r) is inldet BI’”d0, ... dO,-, = A d d , dependent of the choice of {el, . . . , O n - ,} we have A(r) = Cea(‘)

(C = const),

so

a&

1

&

ar

1 dA A ( r ) dr

This concludes the proof. The volume V ( r ) of B,(p) is given by V(r)= s,,p)d4, where d4 is the Riemannian measure on X, which is locally given by dr d o , . do,- It follows that

&

a .

,.

V(r) =

i

A ( t ) dt.

Let us now determine A(r) in group-theoretic terms. We can restrict ourselves to the case when the isometry group Z(X) is semisimple. Let G denote its identity component Z,(X) and K c G the isotropy subgroup at a fixed point o E X. First, we assume that X is noncompact and let g = f + p be a Cartan decomposition of the Lie algebra g of G. We assume that the Riemannian structure on X is defined by means of the Killing form of g. Let

5. Differential Geometry on Symmetric Spaces

951

315

a c p be a maximal abelian subspace (which here is one-dimensional). As shown in Chapter I, $4, we have (as a consequence of the formula for the differential of Exp: p + G / K ) (55)

where S,(O) is the sphere 1 Y J = r in p and T, denotes the restriction (ad Y)21p. Let + a and possibly 1 2 a denote the restricted roots, and for the multiplicities write p = ma, q = mza. Select H o E a such that a ( H o ) = 1. Then W

o

9

Ho)= 2CPa(Ho)2 + 4(2a(Ho)>zl = 2(P + 4q)

and the vector A , E a defined by B(H, A,) = a ( H ) satisfies B(Aa, A,) = 2(Pa(AaI2 + q(2a(Aa))'),

so a(A,) = (2p

+ 8q)- ',

A, = (2p

+ 8q)-'Ho.

In order to compute the integrand in (55), we can take Y = H,,where

H, = rIA,[-' A ,

=

r(2p

+ 8q)1'2A,.

The eigenvalues of (ad H,)'Ip being 0, a(H,J2 (with multiplicity p), and (2a(H,))2(with multiplicity q), we have

so if c (56)

=

( 2 p + 8q)- 'Iz, then

A(r) = R,+,+ 2-4

sinhP(cr) sinhq(2cr).

For the compact case we obtain in the same way (57)

= R p + , + l 2-"

c-P-4

sinP(cr)sin4(2cr),

as already done in Chapter I [Lemma 4.10 and (40) in $4;note that there the roots were labeled differently].

5. The Poisson Equation Generalized

In the Chapter I, Lemma 2.35, we derived the classical solution formula for the Poisson equation Lu = f i n R". We shall now extend this to symmetric spaces of rank one making use of Proposition 5.26.

316

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

Theorem 5.27. Let X be a Euclidean space or a Riemannian globally symmetric space o j rank one. Let j E 9(X).A solution to Poisson's equation

L,u = j is given as follows:

(i) I f X is noncompact

where (59)

(ii) If' X is compact a solution exists if' and only i J ' J ' j ( q )dq = 0. If' this condition is satisjed, the unique solution to L x u = j satisfying J' u(q) dq = 0 is given by

where

Proof of ( i ) .

We have for u given by (58)

where

and [M'f'] ( p ) is the mean value off on Sr(p). Using now Lemma 2.14, Chapter I, Proposition 4.12, and Proposition 5.26, we obtain rm

"61

where d z + 1 dA d L =' drz A(r) dr d r '

951

5. Differential Geometry on Symmetric Spaces

317

Keeping p fixed and putting F(r) = [ M ‘ f ] ( p ) , we have CL,Ul(P) = Jorn {$(r)A(r)F”(r) + $ ( w ’ ( r ) F ’ ( ~ )dr} =

lim[$(r)A(r)F’(r)],“ 1: +

Now $(E)A(E)-, 0 as Hence

E

0

+0

sorn

F’(r) dr.

and F(r) vanishes for all sufficiently large r.

CL,Ul(P)

=

F(O) = f ( P ) .

The choice of H(p, q) = $(d(p, q)) was of course motivated by the fact that the equation (60)

(L,),(H(P, 4 ) ) = a,

where 6, is the delta distribution f + f ( p ) , leads to the equation dr2

1 dAd$ +-- = 0.

A ( r ) dr dr

Proof of (ii). In the compact case we have J’L,u(q)dq = 0 for each u E b ( X ) (Prop.

2.3). Taking this fact into account, we replace (60) by the

equation

which by G(p, q ) = 4(d(p, q)) leads to the equation

whose coefficients are in general unbounded near r = 0 and r = L. The function

is a solution of (61), unbounded near r = 0 (if n > 1) but bounded near r = L since V ‘ ( t ) = A(t). With u ( p ) = G(p, q ) f ( q )dq, we have

318

INVARIANT DIFFERENTIAL OPERATORS

= -

JoL(I

-%)

F’(r)dr 1

Thus L,u = J: Moreover,

s,

[Ch. I1

U(P> dP =

s,

f ( 4 ) dq

s,

G(P7

rL

4 ) dP = 0

because J’ G(p, q ) d p is independent of q and J’ f ( q ) dq = 0. For the uniqueness, suppose u , and uq are two solutions to L,u = f’ and J’ u(q)dq = 0. Then the difference u is orthogonal to the constants on X and to the subspace L,(b(X)), so by the existence just proved, u is orthogonal to b ( X ) ; hence u = 0.

6. Asgeirsson’s Mean Value Theorem Generalized

In his paper [1937], Asgeirsson proved the following result. Let U E C 2 ( R Z nsatisfy ) the ultrahyperbolic equation (62)

azu ax:

-+

a%

a214

azu

aY

aYn

... + = = T + ’ ’ ’ + T .

Then for each r 2 0 and each point ( x o , yo) = (x?, * .

*

9

x ,0, y,,0

*

a ,

Y,”) E R2”

the following identity holds for the (n - 1)-dimensional spherical integrals:

PI

5. Differential Geometry on Symmetric Spaces

319

The theorem was proved in the paper quoted for if of class C2 in a suitable region in R2". It is obvious (taking constant in the y-variables) that (63) implies Gauss's mean value theorem

for the solutions of Laplace's equation L,.u = 0 on R". Also, (63) can be used to derive a solution formula for the wave equation LR" If =

alll -

at2 on R" (cf. Exercise Fl). We shall now state and prove a generalization of Asgeirsson's theorem to a symmetric space X of rank one. Let L = Lx be the Laplace-Beltrami operator on X and as in $4, No. 3 we consider the spherical mean value operator

(64)

( M ' J ' ) ( ~=)

11 A(r)

j ( s >dw(s)

(r 2

01.

S,(X)

Theorem 5.28. Let X be a symmetric space ojrank one (or R") and let the function if E C z ( X x X ) satisfy the differential equation

(65)

L , ( 4 x , Y))

=

L,(u(x, Y ) )

on X x X . Then for each r 2 0 and each ( x o , yo) E X x X we have

(66)

J

4x9 Yo) d 4 x ) =

SSXO)

J

4x0,

Y) d4Y).

SAUO)

Proof. If n = 1, the solutions to (65) have the form 4x9 Y ) = d4x

+ Y ) + Icl(x - Y)

and (66) reduces to the obvious relation 4 x 0 - r9Y O )

+ u(xo + r, yo> = 4 x o , yo - r ) + u(xo, yo + r).

Hence we may assume that n 2 2. First, let u E C 2 ( X x X ) be arbitrary. We form the function (67)

U(r, 4 = ( M ' , W u ) ( x ,Y ) .

The subscripts 1 and 2 indicate that we consider the first and second variables, respectively; for exampe,

320

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

Let O E X be the origin; write X = G / K , where G = Z,(X) and K is the isotropy subgroup of G at o. If g , h E G, r = d (0, 0) (d = distance) we have from & No. I, 3, h

e

where dk is the normalized Haar measure on K . Thus we have by interchanging the order of integration

w,s)

(69)

=

(M);MS2u)(x,Y ) = (MSM;u)(x, Y).

Keeping y and s fixed, we have by Propositions 5.26 and 4.12

aZu 1 au 7 +A @ ) - = L,(M;M; u(x, y ) ar

A(r)

ar

= (M‘,L,M”,)(x, y ) =

(M‘,M”,L,u)(x, Y ) ,

the last identity resulting from the fact that L , and M: act on different arguments. Similarly, by using (69),

a2u 1 au -+ -A’($ - = (M;M;L,u)(x, y). asz A@) as Assuming now u is a solution to (65), we obtain from the above

azu + -A‘(r)1 au = a2u 1 au + -A‘(s)-.

-

drz

A(r)

ar

ds2

A(S)

as

Now putting F(r, s) = U(r, s) - U(s, r), we obtain the relations (70)

a2F arz

-

1 + -A’(r)A(r)

aF aZF 1 aF - - - --A‘@)= 0, ar asz ~ ( s ) as F(r, s) = - F(s, r).

After multiplication of (70) by 2A(r) aF/as and some manipulation we obtain

Consider the line MN with equation r + s = const. in the (r, s)-plane and integrate the last expression over the triangle OMN (see Fig. 11). Using the divergence theorem [Chapter I, 52, (26)] we then obtain, if n

$51

5. Differential Geometry on Symmetric Spaces

32 1

I FIG.11

denotes the outgoing unit normal, dl the element of arc length and dot. the inner product aF aF

dr ds.

On O M N we have s 2 r so that for small I , s

so that the last integral does indeed exist since we have assumed n 2 2. We now use the following data:

aF aF On OM: n = (2-'12, - 2-'12), F(r, r ) = 0, so that - + - = 0. ar as On M N : n = (2-'12, 2-'"), on O N : A(r) = 0. The formula above then reduces to

Now we consider separately the compact case and the noncompact case.

I. X noncompact. Here we claim A'(r) 2 0 for all r 2 0. If X = R", this is obvious, so we may assume that G is semisimple. But then the

322

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

statement is clear from (56). Consequently, both terms in (71) vanish, so we can conclude that F = 0. In particular, V ( r ,0) = U(0, r), and this is the desired formula (66). 11. X compact. I n this case (57) shows that A'@) 2 0 for s in a certain interval 0 I s I r o . As before, we can conclude that V(r,0) = U(0, r ) for r in this interval 0 Ir I yo. In order to extend this to all r we approximate the solution u to (65) by analytic solutions. Let 4, $ be analytic functions on the compact Lie group G with Haar measure dg and consider the convolution (72)

u+,&

Y ) = JJ49;

*

x, 9;

. Y)4(91)$(92) d91d92.

GC

Then

for 0 I r Iro. For f~ C(X x Y ) define the function f(gl, g 2 ) = f(gl o, g 2 . 0); then (73) can be written

f

on G x G by

(74) for all h E G such that d(o, h . o) Ir o . Changing variables in the integral (72), we see that u , , ~is analytic on X x X , so a,,@ is analytic on G x G. Since (74) holds for h varying in an open subset of G, it therefore holds for all h E G. Substituting the definition (72) into (74), we see that for each fixed h E G, the function (75)

(z, w )

JK[G(z-'g1kh, w - l g , ) - ii(z-lgl, w-'g2kh)] dk

on G x G is orthogonal to all functions of the form +(z)$(w) 4 and $ being analytic. By the Peter-Weyl theorem (cf. Theorem 4.3 in Chapter V) this remains true for 4, $ E I ( G ) .But by the Stone-Weierstrass theorem these functions ~ ( z ) $ ( w ) span a uniformly dense subspace of

$51

5. Differential Geometry on Symmetric Spaces

323

C(G x G). Consequently, the function (75) is identically 0, so (66) is proved for all r. 7. Restriction of the Central Operators in D ( G )

With the notation as in $5, No. 2 let n: G -, G/K = X denote the natural mapping, and as in $4 let p : DK(G)-,D ( G / K ) be the surjective homomorphism given by ( p ( D ) f ' )0 n = D ( f 0 n), f~ &(G/K). We shall now consider the question whether Z(G) is mapped onto D ( G / K ) by the homomorphism p. At the same time we consider another mapping v of D(G) into the algebra E ( X ) of all differential operators on X. Given Y E g, let p be the corresponding left-invariant vector field on G and define v( P)E E ( X ) by

According to [DS], Chapter 11, Theorem 3.4, v is a homomorphism of g into the Lie algebra 9 ' ( X ) of all vector fields on X ; thus by [DS], Chapter 11, Proposition 1.1, it extends to a homomorphism of D(G) into E ( X ) , also denoted v. If g E G, we write also v(g)f = fr(g) (f E &(X)). Lemma 5.29. Let D

E

D(G), g E G. Then

v(D)v(g) = V(g)V(DR'").

Proof. It suffices to verify this for D = DR@)and D + v(D) are homomorphisms. But

because the maps D

-,

324

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

so

[Ad(s- '1 Ad(exp(t Yo))(0 V J f {d = (-lY{Ad(9-1) (CVo, 0 . - ~ l I > (OS41(9) =

( - 1)'

+ ( - 1Y{Ad(g- 1 ( - Po) Adb- l ) (P, .

*

.

0

mf

41(exp( - r

O

YO)^)}

1=0

41(S)l

where the unique derivation of D(G) extending the endomorphism F[Fob,?] of 6 has been denoted [To, D]. However, it is clear that [ &, D ] = yoD - DFo, so the two last expressions add up to (-1Y+'{Ad(g-')(r*..

m 0 ) c f

This proves (78). Since G is unimodular, y* = lemma is proved. Corollary 5.31. p(D*) = v ( D ) j b r each D

O

n))(s).

-P

[cf. (7) in 551, so the

E Z(G).

In fact, Z ( G ) is invariant under the mapping D

+ D*,so

(D*)R(e)= D*.

The image of Z ( G ) under the mapping p (or v) will be denoted Z ( G / K ) . As before, extend a to a Cartan subalgebra lJ of g. The complexification bC is then a Cartan subalgebra of gc. Let @ denote the corresponding Weyl group. Denoting the dual of a vector space by a star, we have the following symmetric algebras, with corresponding group action and invariants : S(g), S(g*),

Ad(G),

W, I(g*)

S(P), S(P*)?

Ad,(K),

m), 4P*)

S(a), S(a*),

W,

I(a), ](a*>

S(hC),S(lJC*>,

R

I(hC)9 W*).

551

5 . Differential Geometry on Symmetric Spaces

325

Let @ be the subgroup of leaving a invariant; then the restriction onto W ([DS], Proposition from bc to a induces a homomorphism of 8.10, Chapter VII) and therefore maps I ( I f * ) into I(a*).

mo

Proposition 5.32. The following properties of' the symmetric space GIK are equivalent:

(i) Z ( G / K ) = D ( G / K ) . (ii) The restrictionfrom g to p maps I ( g * ) onto I(p*). (iii) The restriction from bC to a maps I(bc*) onto I(a*). Proof. Consider the mapping I : S(g) + D(G) from (4) in 54. Let P E l ( g ) and let P E S(p) be determined by P - P E S(g)f. Then P E I ( p ) . Let X , , . . . , X , be a basis of g such that X,,. . . , X , is a basis of p. Since I ( P ) E D,(G), there exists by Theorem 4.9 a unique P' E Z(p) such that

(79)

(P(W))f')(&N =

[P'(a 1, . . . , a r ) ( f ' 0 n)(g exp(x, X 1

+ . . . + X, x,))](O),

and in the notation of that theorem, p(A(P)) = DAtP,),that is, ,u(I(P)) = p(I(P')). It follows that

+

I ( P - P') I ( P - P ) = I ( P - P') E D(G)f, (80) since p has kernel D,(G) n D(G)f. Now suppose P is homogeneous of degree rl > 0. Then the same holds for P - P (0 has any degree) and P' has degree d. Since P - P E S(g)f, there exists an operator D E D(G) of degree < d such that I ( P - P ) - D E D(G)€.Thus by (80)

I ( P - P') + D E D(G)f. (81) Decomposing D according to Lemma 4.7, we may take D = A(Q), where Q E S ( ~ )of degree < d . But then (81) and the directness in Lemma 4.7 imply that P - P' + Q = 0; in other words, (82)

degree(P

-

P') < degree(P).

The mapping p maps Z(G) onto D ( G / K ) if and only if the mapping P + P' maps I ( g ) onto I(p), which by (82) happens if and only if P + P maps I ( g ) onto I(p). Under the bijection between g and g* induced by the Killing form, I ( g ) and I(p) correspond to I(g*) and I(p*), respectively, and the mapping P + P corresponds to the restriction of a p E I ( g * ) to p. This proves the equivalence of (i) and (ii). For (iii) let u be the compact real form t + i p of gc, so gc = u + iu is a Cartan decomposition. Let be the subspace of bc, where all roots (of gc with respect to bc) take real values. By Corollary 5.12, I((iu)*) is isomorphic to I(f& and I(p*) is isomorphic to I(a*) under restriction.

326

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

But we have also the identifications I ( g * ) = I ( @ ) * ) = I((iu)*) (see Lemma 1.4 in the next chapter). Under restrictions from g to p, and from bC to a, we have the commutative diagram

-

4~*)

4a*)

from which the equivalence of (ii) and (iii) follows. Remark. Using Proposition 5.32, one can prove (cf. Exercise D3) that Z ( G ) = D ( G / K ) whenever G is a classical group (real or complex) but that this property fails to hold for several of the exceptional symmetric spaces.

8. Invariant Differential Operators for Complex Semisimple Lie Algebras

Let G be a connected complex semisimple Lie group acting on its Lie algebra g by means of the adjoint representation. Let t, be a Cartan subalgebra and let us adopt the notation from Example (vii) in 93. Let Z(g*) denote the set of G-invariant polynomial functions on g. As p runs through I(g*) the differential operators d ( p ) run through the set of Ginvariant differential operators on g with constant coefficients. This set is, of course, the algebra D(G .g/G) if G - g denotes the group of transformations of g generated by G and the translations. We can now generalize Proposition 3.14. Theorem 5.33. For the adjoint action of' G on g with transversal manijold $' the radial part of' the operator d(p), p E I(g*), is given by

A(d(p)) = I C - ' ~ @0)IC. Proof. The proof is based on the purely algebraic trick of taking commutators with the Laplacian. As usual, let E(g),E(t)') denote the algebras of all differential operators on g and If, respectively, and let J(g) [resp. .!($')I be the subalgebra of the G-invariant ones (resp. the Weyl group-invariant ones). Writing {Dl, D2}= D ,0 D 2 - D 20 D , if D , and D 2are differential operators, we consider the derivation

p: D

+

m4,D)

of E ( g ) and the derivation ,iid:-,i{A(d(w)), d }

551

5. Differential Geometry on Symmetric Spaces

327

of E(b'). Since the operation A is a homomorphism of J(g) into J ( l f ) , we have (83)

A(p(D>)= F(A(D))*

D E 49).

Now we need a simple lemma. Lemma 5.34. Let p be a homogeneous polynomial on g ojdeyree m. Considering p as the differential operarorj' --t we have

p"'(p) = 111 ! d ( p ) . Proof. We use induction on m. For m = 1, the result is immediate by computation. Let p = x1 . . . x,. Since p 2 ( x i )= p(d(xi)) = 0, Leibniz's rule for derivations implies

(84)

prn(xl . . . x,-

. . . x",- 1) x, + mp"'-'(xl . . . x,-

,x,) = p"(x,

0

1)

0

p(x,).

The induction hypothesis gives pm-l(xl

. . . x " - ~ )= ( m - I)! d(xl

x,-J.

Hence the first term on the right in (84) vanishes and the lemma follows. Combining (83) and the lemma, we have m ! A(d(P)) = A(Prn(P))= F"(A(P)) = F " m .

Now if A is any associative algebra and a E A, put d,(b) = *(ab - ba). Then by induction on k ,

If c E A commutes with b and has an inverse c - l and if a' = c - lac, then (d,,)k(b)= C- 'd,k(b)c.

Here we take A = E(b'), a = a(8), b = p, L' = n. Then a' = 7c- 'd(i3) n = A(d(o)). By the lemma we have d,k(4) = k ! d(q) if 4 is a homogeneous polynomial on €) of degree k. Hence if p E l ( g ) is homogeneous of degree m 0

m !A(d(p)) = ,~?"(p) = (dOr)"(F)= n-l dam(j)n = m!n-'d(F)n,

proving the theorem. The theorem can be used to prove an interesting integral formula on a compact semisimple Lie group U . Let LI denote the Lie algebra of U ,

328

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

t c u the Lie algebra of a maximal torus T c U . Let g = uc, b = tC denote the respective complexifications. We retain the notation from Example (vii) in $3 and write uX for Ad(u) X ( u E U , X E 9). Theorem 5.35. Let

dii

be a normalized Haar measure on U . Then,

if H , H ’ E b,

n(H)n(H’)

1”

e < u H , Hdu ‘ ) = w-’d(n)(n)

(det s)e<SH*H’), SE

w

where w is the order of’ the Weyl group W. Proof. Let f’ E b(u) and put F ( X ) = ju f ( u - X) du. Then if p E I(u*) we have for H E If (d(p)F)(H) = (n- d(p)(nF))(H),so the function r

+ f ( H ) = n(H) J f ’ ( ~H. ) du

( H E t)

(4arp,f)(H)= n ( H ) ( m P ) ( H )= (

WWf>(W.

U

satisfies Thus (85)

4a,p,, = W

4

f

9

P E I(u*).

Now select H‘ E b such that n(H’) # 0 and put f ( x ) = e < X * H ’Then ). d(q)f’ = q(H’)f’ for each q E S(u*), so by (85) d(p)4, = p ( H ’ ) 4 / for each p E I(u*). Thus by Corollary 5.12, = q(H’)4/ for all q E I(b*). Since H’ is regular, the simplest instance of Theorem 3.13 in Chapter I11 shows that

where C , E C. On the other hand, since n ( s H ) = (det s)n(H), it is clear ), by (86) that $ f ( s H ) = (det S ) ~ ~ ( Hwhence (87)

4/(H) = c

(det s)e<SH*H’), SE

w

where c E C.The constant c can be determined by applying the differential operator d(n) on both sides and evaluating at H = 0. The definition of 4f shows that (d(44,)(0) = d(n)(4f’(O) = d(n)n,

whereas the right-hand side of (87) gives the contribution cw.n(H‘). This proves the theorem for H E t, n ( H ’ ) # 0, and hence for all H , H’ E b by holomorphic continuation.

$51

5. Differential Geometry on Symmetric Spaces

329

We can now use Theorem 5.35 to derive a formula for the order w of the Weyl group quite different from that of [DS], Chapter X, Exercise B9. Corollary 5.36.

The order w of’rhe Weyl group is given by w

=

m(4/wp)?

where 2p is the sum of’the positive roots in A(g, b).

We put H‘ = H, in Theorem 5.35. Then by Chapter I, Proposition 5.15, the right-hand side becomes la(n)(n) (e(1/2)dH) - e-(l/2)a(H)).

,,,-

n

z>o

Dividing by n(H) and letting H + 0, we obtain the result. Remark. The formula in Theorem 5.35 gives an interesting interpretation of the characters of irreducible representations of the compact group U . This will be discussed in Chapter V, 91, where we shall also prove an analog of Theorem 5.33, generalizing Proposition 3.12.

9. Invariant Differential Operators for X = G/K, G Complex

As in Example (iii), 53, No. 4, let 6 denote the density function for K acting on the symmetric space X = G / K . In the case when G is complex this density function is up to a factor given by

S(Exp H) =

n (sinh LX(H))~,H

E a.

ztz+

Let r: D ( G / K ) + &(A) be the isomorphism from Theorem 5.18. Then we have the following description of the radial parts generalizing Proposition 3.10. Theorem 5.37. Suppose G is complex. The radial partsjor K acting on X = G / K are given by

(89)

A(D) = 6 - ” ’ r ( D )

0

6’”,

D E D(G/K).

Since the proof uses some facts from the theory of spherical functions, we postpone it until Chapter JV, $5, No. 2. There is an analog of Theorem 5.37 concerning the action of K on p. The corresponding density function is then up to a factor given by (90)

d0(H) =

n

aez

+

(a(H>Y,

H Ea,

330

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

and we have the following formula generalizing Proposition 3.13 (G complex) : A(d(P)) = n- ‘d(F)

(91)

O

71,

PE

m*),

where p denotes the restriction of p to a. The proof of this formula is identical to that of Theorem 5.33. As in Example (viii) in $3, No. 4,let J denote the ratio of the volume elements in X and in p (via Exp). Combining the restriction isomorphism I(p*) + Z(a*) with the isomorphism r, we obtain a surjective isomorphism o: D ( G / K ) + DK(p), where DK(p) = D(K . p/K) is the set of K-invariant differential operators on p with constant coefficients. By Corollary 5.20 we have o(L,) = L, - ( p , p ) . Theorem 5.38. Suppose G is complex. Then if D E D ( G / K ) , the image DExp-’of’ D under Exp-’ satisjes the relation ~ E x -p

F = J - 1’2~(D)(J’’zF)

f o r all K-invariant C*-firnctions F on p. In fact, it suffices to prove this formula on a, and there it is a direct consequence of (89) and (91) because J = 6/n2.

EXERCISES AND FURTHER RESULTS A. The Laplace-Beltrami Operator

1. Let M be a Riemannian manifold, p E M a point, {x,, . . . , x,} normal coordinates around p such that (d/dxi), (1 < i < m) is an orthonorma1 basis of the tangent space M , . Then the Laplace-Beltrami operator L is given at p by

2. Let M be an oriented manifold with a Riemannian structure g and define the * operator on the Grassmann algebra % ( M ) following [DS], Chapter 11, $7. Let = (-

and extending

(5

1

,),P+,+

* d * w,

(u

E

9Ip(M),

by linearity put A = -(d6

+ 6d)

on 9l(M).

Exercises and Further Results Let X

E9’(M)

(i) 6(0,)

and let w x be the l-form Y

=

-div X, wgrad, = df, f

E

+

33 1

g(X,Y). Show that

b(M),

and deduce that (ii) Af = LMf, where L , is the Laplace-Beltrami operator.

3. Let G/H be a pseudo-Riemannian manifold of constant curvature represented in accordance with Chapter I, Theorem 6.1, in the form O(P,4

O(P + 1,4)/O(P9

+ 1)/O(P,41,

Rp+4O(P,4)/O(P,4).

4)1

Then the algebra D ( G / H ) of invariant differential operators consists of the polynomials in the Laplace-Beltrami operator L (cf. Helgason [19591, Faraut [1979]; the latter paper contains a detailed study of the Hinvariant eigendistributions of L). 4. Let G be a connected semisimple Lie group with Lie algebra 9. Let B denote the Killing form of 9. Let X I ,... , X , be a basis of 9, put gij= B ( X , , X j ) (1 Ii, j I n), and let (gij)denote the inverse of the matrix (gij). Show that .. (i) The differential operator R = g”XiXj is independent of the choice of the basis ( X i ) and belongs to the center Z ( G ) of D(G). The operator R is called the Casimir operator. It coincides with the LaplaceBeltrami operator for the bi-invariant pseudo-Riemannian structure on G given by B. (ii) Assuming G noncompact, let G / K be the symmetric space associated with G. The operator p(R) given by Theorem 4.6 coincides with the Laplace-Beltrami operator on G / K .

--

xi,

5. (Conformal Invariance) Let M be a manifold with a pseudoRiemannian structure g. Let n = dim M. A vector field X on M is said to be conjormal if the Lie derivative O(X)has the property

O(X)g = p(X)g,

where p(X)E b ( M ) .

(i) Show that if X is conformal, then

2 p ( X ) = - div X. n (ii) The action (J:

z+-

az cz

+h + d’

z

E

c,

332

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

of the group SL(2, C) on R2 is conformal, i.e., CT*. g = T 2 g if g is the flat Riemannian structure and T E S(R2). Deduce from [DS], Chapter I, Exercise B3, that for each X E sI(2, C ) the induced vector field X + on R2 is conformal. Note also (Introduction, Exercise A4) that L = LR2satisfies

L" = h(o)L, where h(g)(z) = Icz - aI4. Deduce that X E sI(2, C),

[L, X '] = p(X)L,

(1)

where [ , 3 denotes commutator and

(iii)* Let r denote the Ricci curvature tensor of M , i.e., rji = x (negative of [DS], Chapter VIII) and K the scalar Curvature

k Rkijk

K =1 gijrij. i,j

Let M , and M,,respectively, have pseudo-Riemannian structures g , and and K, , and Laplace-Beltrami operators L, and L,. Let g 2 , scalar curvatures K

,

T :M ,

M,

be a conjormal diffeomorphism, i.e., T*g, = T2g,

Then i f f

E d(M,),

where

7E

b(M,).

c, = (n - 2)/4(n - l), n = dim

M , = dim M,,

( L , - ~ , K ~ ) ( f " ' ~ - ' (0f T ) ) = TniZ+1((L2 - c,,K2f) 0 T, so that the null spaces of the operators L , spond under T.

- c,K,

and L , - cnK2corre-

6*. Let M be a compact Riemannian manifold, g the Riemannian structure, d the distance, L the Laplace-Beltrami operator, m = dim M, (m 2 2), and

(f,?f 2 >

=

J f ; ( x ) f , ( x ) - dx,

JLf2 E L 2 ( M ) ,

M

where d x is the volume element. (i)

Let Q,(M)=

{

tl E

d(M):

U(X)

JM

dx = 0

333

Exercises and Further Results

Then L : 6 , ( M ) --t b , ( M ) is a bijection and the inverse is given by an integral operator (Gf')(x)=

s,

Y(X,

Y)f'(Y)4 4

where y(x, y ) = y ( y , x ) satisfies (a) y is C" for x # y . (b) d(x, y)"-'y(x, y) is bounded. [If m = 2, this property should be replaced by "(log d(x, y ) ) - ' y ( x , y ) is bounded".] A proof can be found in de Rham [1955], Chapter V. Theorem 5.27 gives a more explicit version for the rank-one case. (ii) For 1E C let E , be the eigenspace

E,

= {U E b ( M ) : LU = Ilr}

and A the spectrum A = {AEC:E,#O}.

Then (a) A is discrete and 1I 0 for each 1E A. (b) dim E , < cc for each 1. # 2 , . . . be an orthonor(c) In accordance with (a) and (b) let #o, ma1 system in L 2 ( M ) such that each E , is spanned by some of the 4i. Then if j ' E L 2 ( M ) , m

f'

=

C (1;&)4,,

[convergence in L 2 ( M ) ] .

0

(d) I f / ' E b ( M ) , this expansion converges absolutely and uniformly. [A proof of the results in (ii) can be found, for example, in F. Warner [1970], Chapter 6.1

7 . Let M be a manifold and p a measure on M equivalent to the Lebesgue measure. Let X be a vector field on M that is symmetric, i.e., Jyf)(x)g(x)

d p ( x ) = J M f ( X ) ( X d ( X )444,

f,9 E .ww*

Prove that X = 0. 8 . Let g be a semisimple Lie algebra and g = f + p a Cartan decomposition. Let a c p be a maximal abelian subspace, m the centralizer of a in f, and b a Cartan subalgebra of g containing a. Let superscript c denote complexification and let P , denote the set of roots /3 > 0 of gc with

334

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

respect to If that do not vanish identically on a. Let TI,. . . , T, be a basis of m orthonormal with respect to - B , H I , .. . , H,a basis of a orthonorma1 with respect to B. For each jE P , choose X P ~ ( g c ) PX, - P ~ ( g C ) - P such that B(X,, X-,) = 1 . Because of the decomposition g' = m'

+ a' + C

+ 1 (gc)P

(g'),

-PEP+

PEP+

the Casimir operator R (Exercise A4) can be written

+ n, +

c (X,X-,

+ X-pXp), = xi Tz, R, = cjHf. (Here we write X, for xP,etc., for simn = n,

PEP+

where SZ, plicity.) Let a' be the set of regular elements in a and fix a = exp H ( H E a'). If P E P , u ( - P,), we write X, = Y, + ZP(Y, E p', Z, E Y) and determine A , E a such that B ( H , A P ) = P ( H ) for H E a. If g E G , X E gc, we put Xg = Ad(g)X. From the expression above show that

+ Q, + C

0 = R,

coth(P(H))AP

PEP+

sinh-2(P(H))[(ZpZ-P)II-1+ Z , Z - ,

-2 ,EP+

-

ePIHIZa-'Z - e-/JIH)Za-'Z -P

P

P

-Dl

(Harish-Chandra [1960]; cf. Warner [1972, Vol. 111, Proposition 9.1.2.11). B. The Radial Part

l*.Let H be a Lie transformation group of a manifold V. Then a submanifold W c V satisfying the transversality condition [(29) in 931 V,. = ( H . w),,. 0 W,.

(direct sum),

MIE

W,

always exists.

2. If j ' is a radial function on a symmetric space of rank one, then

by Proposition 5.26. Derive this formula from Theorem 3.7 as well.

3. Theorem 3.7 remains true if V is a manifold with a pseudoRiemannian structure g provided the following additional conditions are satisfied : (i) For each u' E W the orbit H . M' is closed and g is nondegenerate on it. (ii) For each w the isotropy group H"' is compact.

Exercises and Further Results

335

Example. Let (G, H) be a symmetric pair, relative to an involutive automorphism 0 of G, where G is assumed connected, semisimple with finite center. Let K c G be a maximal compact subgroup invariant under CT ([DS], Chapter 111, Exercise B4). Then the K-radial part of LG," can be determined by the above variation of Theorem 3.7 (cf. Chang [1979a], Flensted-Jensen [ 19783, $4, Hoogenboom [19831).

4. The K-radial part of Lx on the symmetric space X (Proposition 3.9) can also be written

=

G/K

Here 6 is the density function n,,,(sinh a),", H,,. . . , H , is any basis of ( g i j ) is the inverse of the matrix g i j = ( H i , Hj) (1 I i , j I I ) (Harish-Chandra [1958al). a, and

5*. (Extension of Radial Part to Distributions) (i) Let M and N be two manifolds and let n: M --t N be a surjective C"-mapping such that dn,: M, --t Nn(,) is surjective for each m E M. Let dm and dn be measures on M and N , respectively, equivalent to Lebesgue measure. Then for each a E 9 ( M ) there exists a unique function fi E 9 ( N ) such that

S,(F

0

n)a dm

=

IN

~ 1 dn, ;

F

E

9(~).

By partition of unity the proof can be reduced to the case when M and N are Euclidean rectangles and n is the projection ( x l , .. ., x), + ( x ~ .,. . , x,) (cf. [DS], Chapter I, Theorem 15.5). Then if dm L n*(dn) is the "ratio" of the forms dm and n*(dn), one can put

f;(n) =

1

a(dm L n* d n )

n - '(n)

(cf. Harish-Chandra [1964a] ; also Stoll [1952] and Schwartz [1966]). (ii) Let H be a Lie transformation group of the manifold V and W c I/ a submanifold satisfying the transversality condition (29): V,

=

( H * w)w 0

w,, w E w.

Fix a left-invariant measure dh on H and measures du on V, dw on W, equivalent to Lebesgue measure. Let V* c V denote the open set H W. By (i) we can to each a E 9(H x W ) associate an f, E 9 ( V ) such that

for all F

s,,, E

9(V*).

F(h . w)a(h, w ) dh dw =

336

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

(a) Assume the measure do invariant under H and let T E 9 ' ( W * ) be H-invariant. Show that there exists a unique distribution T on W such that for all c( E 9(H x W ) ,

T(.f,) = T(A,)

where

s, s,

A,(w) =

a(h, w ) dh.

(b) If T is an H-invariant continuous function f'on V * , show that if;(A,)=

f'(w)A,(w)dw,

so T = f d w , wheref =.f'IW. (c) Suppose H is unimodular and D a differential operator on I/ invariant under H . Then

(DT)-= A(D)T, where A(D) E E ( W ) is the radial part of D. (cf., Helgason [1972a], Chapter I, $2). 6*. The K-radial parts of the operators in D(SU(p, q)/S(U, x U,))= D ( G / K ) (0 < p I q ) can be described as follows. In accordance with [DS] (Chapter VII, Exercise 5), the space a c p can be chosen as a=

where H i = E i p + i

{H,= ?,HI +

- + t,H,

: ti E R } ,

+ E p + i iand the restricted roots are given by

(1 I i I p), kji, +2fi where ,fi.(H,)= t i . Put

o(H,) =

+fi k j ;

(1 I i < j Ip),

n (ch 24 - ch 2tj), i<j

D i= w - ' S , ( L , , .. . ,L,)

0

0,

where Si is the ith elementary symmetric function. Then the algebra A ( D ( G / K ) )of K-radial parts is generated by D,, . . . , D,. (For details, as well as formulas for the spherical function 4A in the same spirit, see Berezin and KarpeleviE [1958], Takahashi [1977a], and Hoogenboom [19821.)

Exercises and Further Results

337

C. Invariant Differential Operators

1. Determine explicitly the invariant differential operators on the symmetric space GL(n, R)/O(n).

2*. Determine explicitly the algebra

w m ) / M ( P ) x O(n - P I ) of invariant differential operators on the manifold G(P9

4

= M(n)/M(p) x

O(n - P )

of p-planes in R" (Chapter I, $2). To describe the operators, let Gp,ndenote the space of p-planes through the origin and let n: G(P,n ) -,GP," denote the mapping obtained by parallel translating a plane to one , fiber F = n-'(a) is naturally identified with through 0. If Q E G p S nthe the orthogonal complement a* and has therefore a canonical Riemannian structure. Let L , denote the corresponding Laplacian. If f' E 6(G(p,n)) let j I F denote its restriction to F. Consider now the linear mapping 0 of d(G(p, n)) into itself given by

(Of') IF = JhUIF). (i) The operator 0 is an M(n)-invariant differential operator on G(P, n). (ii) The algebra D ( M ( n ) / M ( p ) x O ( n - p ) ) of M(n)-invariant differential operators on C(p, n ) is generated by 0. (See Helgason [1965a], 457-8.) For p = n - 1 the operator 0 is the operator

considered in Chapter I, $2.

3. Determine explicitly the algebra WGIM) of invariant differential operators on the manifold G / M corresponding to the hyperbolic space G / K = Oo(l,n)/SO(n). [As usual, M denotes the centralizer of A in K ; here it can be identified with SO(n - l).] In view of the fibration G/M + G / K , G / M can be viewed as the unit sphere bundle of G / K . With the orthogonal decompositions f = M + I, p = a + q we write g=m (a I q),

+ + +

338

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

so by Theorem 4.9, the problem is to determine the algebra Z(a + I + q) of M-invariant polynomials on a + I + q. We have dim a = 1, dim I = dim q = n - 1, [I, a] = q, and X' consists of a single root a. Fix H E a such that a(H) = 1 ; let T,, . , . , T,- be an orthogonal basis of I such that -B(&, 7;) = B ( H , H) (1 5 i In - 1). Put Xi = [ T , HI. Then B(Xi, Xi) = B(H, H) (1 I i I n - 1). With the basis H, T,, . . , , q-,, X,, . . . , Xn-, of a + I + q put

,

17-12 =

1 T:,

T.X=

1T . X i ,

1 x 1 2

i

=

Ex:. i

Prove that (i) The algebra Z(a + 1 + q) is generated by H, IT 12, T . X, 1x1'. The corresponding operators D,, D I T I 2D, T . X , Dlx12 therefore generate D(GIM)* (ii) The generators above satisfy the following commutation relations : [ D H , D1X121 = -2DT.X, [DHI

DlT121 = -2DT.X,

[ D H DT.x] ~ = -D1,12

- Dlxp.

(For these results and interesting geometric interpretations of the operators see Reimann [1982]; see also Ahlfors [1975].) 4. The homomorphism y: D,(G) + & ( A ) from Theorem 5.17 is defined by means of a choice of positive restricted roots. Show, however, that y is independent of this choice.

5 . (i) Let N denote the three-dimensional Heisenberg group ($4, No. 1). The algebra Z ( N ) of biinvariant differential operators is generated by alax,. RX, with (ii)* Consider the four-dimensional Lie algebra n4 = the brackets [X,, x21 = x,, cx,,X,l = x4, all other brackets being 0. Then n4 is nilpotent and the center Z(n4) of its universal enveloping algebra is generated by X, and X: - 2X2X4 (cf. Dixmier [19583). 6*. Let G be a simply connected nilpotent Lie group with Lie algebra 9. Under the adjoint action of G on 9, g induces an algebra '9 of vector fields on g ([DS], Chapter 11, Theorem 3.4). Let D E Z ( G ) and let cD be the corresponding distribution (05, No. 1). Viewing eD as a distribution on g (by exp), we obtain a differential operator Do on g with constant coefficients by (convolution on 9). D o f = f * eD

Exercises and Further Results

339

If Dexp-’is the image of D under exp-’, then

Do - D e x P - ’ E E(g)g+ (cf. Kashiwara and Vergne [1978a], which has also more general results). Thus the action of D on distributions invariant under all inner automorphisms is completely determined by the constant coefficient operator Do. 7*. The symmetrization 1:S(g) + D(G) maps I ( g ) bijectively onto Z(G). Although I ( g ) and Z(G) are both commutative algebras, 1is not in general multiplicative on them. If g is semisimple, Harish-Chandra has defined purely algebraically [1951a] a ring isomorphism between Z(g) and Z(G) (cf. also Theorem 1.9 in Chapter V). For g arbitrary Duflo [1977] has established a surjective ring isomorphism y: Z(g) + Z(G): Let j(X) be the Jacobian determinant of the exponential map exp: g --+ G,i.e.,

([DS], Chapter 11, Theorem 1.7). Then j’” is analytic in a neighborhood I/ of 0. View an element P of S(g) as constant coefficient differential operators d(P) on g. Given P E S(g), let P“ E S(g) be determined by (Wb)4)(0)= (a(P)0”’*4))(0> for all 4 E 9 ( V ) . Then y(P) = 1 ( P ‘ ) for P E Z(g). For g nilpotent an isomorphism was given by Dixmier [1959]. 8*. Consider the space gI(n, R) of all n x n real matrices X = (xij). Consider the Capelli differential operator D on gl(n, R) given by

[ ((&)

(Df)(X) = (det X) det

1 5 i, j

5Sf](x)*

Then D is invariant under left and right multiplications by members of GL(n, R). Thus it gives a biinvariant differential operator on the group G = GL+(n,R) of n x n matrices of positive determinant. For Iz E C define D, by (D,f)(X) = (det X)-’D((det X)’f)(X). Then D, = I”

+ E,,-lI”-’ +

+ E I A + D,

where Ei E Z(G). Moreover, Z(G) is the polynomial ring

Z ( G ) = C I E , , - l , . . . ,E l , D].

340

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

For n = 2, Z(G) is generated by

(cf. Rais [1977] (unpublished), Carter and Lusztig [1974], and Dixmier [19753). D. Restriction Theorems

1, This exercise outlines Harish-Chandra's proof of Chevalley's restriction theorem: The restriction P + Pla is a bijection of I(p) onto !(a) (Corollary 5.12). (a) Let pc and ac denote the complexifications and !(pC) [resp., !(aC)] the sets of K-invariants (resp. W-invariants) in S(pc) and S(aC). Prove that

I(pc)

is integrally closed.

For this use the fact that S(pc) is integrally closed (Appendix, Lemma 3.4) and that if x E S(pc) lies in the quotient field C(!(pC)) of !(pc), then x E !(pC>.

(b) Let J = !(pC)(aC(restriction to aC). Then S(aC)

is integral over J .

For this, for X E pc, let T, = (ad X)21pc and consider the characteristic polynomial det(A2!

-

T,) =''1

+ ~ , ( x ) A ~ ' +- ~- . + prPl(x)A2',

where p i E !(pC). In particular, det(L2!

- TH)= A2'

+ pl(H)L2r-2+ - - + pr-l(H)L2',

so that each root a E Z is integral over J. (c) If H, E a, H, E ac satisfy p(H,) = p ( H , ) for all p E J, then H I = sH, for some s E W. Since 5 det(12 - T,,), det(A2 - THO) it follows that each a E I; is real on H,, so H I E a. Since each p E I(p) has the same value on Ho and H,,it follows from Weierstrass's approximation theorem on the real space p that H, and H , are Ad&)-conjugate. Now use [DS], Chapter VII, Proposition 2.2.

Exercises and Further Results

34 1

(d) The field extension C(S(aC))/C(J)is normal. In fact, the quotient field C(S(aC)) is obtained by adjoining all the roots of the polynomial

F(1) = 1 2 ' namely, all a E Z, to C(J).

+

p$-2

+ . + Pr-IA', * *

(e) Let Q be an automorphism of C(S(aC)) leaving C(J)pointwise fixed. Fix H o E a. Then there exists an H , E ac such that p " ( ~ =~ p) ( ~ , )

for all p

E s(aC).

In fact, Q permutes the roots of F(1) and hence leaves S(aC) invariant; thus p + p"(Ho) is a homomorphism of S(aC) into C. (f) By (c), H , = s H , for some s E W, so if q E I(aC), then q"(H0) = d H , ) = q(H0);

thus since H , E a was arbitrary, q" = q. Hence by Galois theory, Z(aC) c C(J). But I(&) is integral over J and J is [by (a)] integrally closed. Thus Z(ac) = J, as claimed. (For more details see Helgason [1962a], Chapter

x.1 2*. Let U / K be a symmetric space of the compact type, U simply connected, K connected. Let the notation be as in [DS], Chapter VII, @. Let & , ( U / K ) denote the set of K-invariant functions in b ( U / K ) and let grX (a,) denote the set of r,-invariant functions in &(a,). Given F E b,( U/K),define f E &(a,) by f ( H ) = F((exp H ) K ) ,

H E a,.

By [DS], loc. cit., the map F + f maps &,(U/K) into &rz(a*). Prove that the mapping F + f is surjective (Dadok [1982]).

3*.Let the notation be as in Proposition 5.32, so that I(IJC*) is the set of polynomials on IJc invariant under the Weyl group @ = W(gc,bC) and I(a*) is the set of polynomials on a invariant under the Weyl group w = w(g, el. (i) If G / K is a classical symmetric space (i.e., neither G nor K exceptional), then the restriction mapping (*)

pE

+ jj E

I(a*)

is surjecriue. (ii) For the symmetric spaces E6/F4, E , / E 6 . T, E , / E , x SU(2) the restriction mapping (*) is not surjective. (For the proof see Helgason [1964a]; for a related positive result see Theorem 3.16 in Chapter 111.)

342

[Ch. I1

INVARIANT DIFFERENTIAL OPERATORS

4. Let the notation be as in $5, No. 2. Call a function on a Euclidean space Rn Lipschitzean if for each R > 0 there exists a constant C , such that

If(x) - f(Y)I ICRIX - Y I

1x1 < R ,

I YI < R. Prove that each Weyl group-invariant Lipschitzean function 4 on a exfor X, Y E R”,

tends to a K-invariant Lipschitzean function 0 on p. [Hint: Use Prop. 5.18 in Chapter I.]

E.

Distributions

1. In the notation of $5, No. 1 show that if ~ E ~ ( G / K ) , . T E ~ ’ ( G / K ) , then

(f x T ) ( g .0)

=

s,

(T x f ) ( g . o ) =

f ( g h - ’ . o ) dT(h), f(h-’g.o)dT(h).

G

2*. If T E ~ ’ ( G such ) that T * 9 c 9,then T E b’(G) (cf., Ehrenpreis [1956]).

3. In the notation of $5, No. 1, let s, t E b’(G),D E D(G).Then s

Using this for s = 3, t 5.5 (Koornwinder).

=

* (D*j t

=

DS * t .

T, D E A(I(p)), derive another proof

of Theorem

F. The Wave Equation

1. Write down the solution of the wave equation

by using Asgeirsson’s mean value theorem on the function U(X1, ..., x,,Y~, . . . , Y n ) = u ( x l r . . . r x n , t ) r

which satisfies the equation

a2v

-+

ax:

(Asgeirsson [19371).

a2v a2v a2v . . . +--1+ * . * +ay,z ax,” - a y l

Y , = t,

343

Notes

2*. As in Chapter I, &consider I the hyperbolic space H" with Riemannian structure ds2 of constant sectional curvature - 1. Show that the Lorentzian manifold H" x R has constant scalar curvature

K

= -n(n -

1)

(cf. Exercise A5). As shown there, the natural wave equation on H" to consider is

Using Theorem 5.28, find an explicit solution formula for this equation. Show that for n odd, Huygens's principle holds.

gl-2. The characterization in Theorem 1.4 is from Peetre [1959, 19603 (with a proof by L. Carleson). We have followed the presentation in Narasimhan [1968] of this as well as of the results 1.2 and 1.3. Distributions on manifolds are treated in detail by Schwartz [1966]. The Laplace-Beltrami operator (cf. Beltrami [1864]) has played a prominent role in Riemannian geometry in each of the forms (12), (13), (26) in 52. Its relation to the operator dd + Sd on forms is explained in de Rham [1955], $26; for another proof see Exercise A2 in this chapter. Proposition 2.4 and Theorem 2.7 are from Helgason [1959], Chapter I, and [1970a], Chapter IV. 53. Nos. 1-3. This material is based on Helgason [1972a], Chapter I, [1965b], and [1972b]. Of the results in 43, No. 4, Examples (ii) and (iii), the noncompact versions of (v) and (vii) are from Harish-Chandra's papers, [1958a, 1956b], and [1957a], respectively, with entirely different proofs. See also Berezin [1957] and KarpeleviE [1962]. Theorem 3.15 is from Helgason [1972a], and so is the proof in the text for Example (vii). $4. Theorem 4.3 is a modification of results of Harish-Chandra and Schwartz (see Harish-Chandra [1953], p. 192, and [1956b], p. 111). For Corollary 4.5 see, in addition, Gelfand [1950b]. The description of D ( G / H ) (Theorems 4.9-4.1 1) is from Helgason [1959]. The nonreductive case has been studied by Hole [1974], Koornwinder [1981] and Jacobsen [1982]. Proposition 4.12 occurs in Giinther [1957] for harmonic spaces and in Berezin and Gelfand [ 19561 and Helgason [1959] for symmetric spaces. 45, No. 1. The convolution calculus on distributions on G / K (in particular, Theorem 5.5) is from Helgason [1973a]. The commutativity of @ ( G ) for G / K symmetric (Corollary 5.2) was first found by Gelfand [1950a]. Corollary 5.4 is stated in a different form in Harish-Chandra [1954c], Lemma 1, and is proved in Selberg [1956], where Corollary 5.3 is also stated. The result is indicated in Gelfand [1950a], but the proof described in Berezin et al. [I9601 is insufficient because it is based on the statement that Z(G) gets mapped onto D ( G I K ) by applying the operators in Z(G) to functions on G which are right invariant under K. In general, this surjectivity fails (example: E,/F,; see Proposition 5.32 and Exercise D3), although it happens to hold for all classical G (real or complex). The present proof of part (i) of Theorem 5.7 was written in 1960, following

344

INVARIANT DIFFERENTIAL OPERATORS

[Ch. I1

a discussion with Harish-Chandra. In [19631, Lichnkrowicz proved the same result under the weaker assumption that G / H has invariant volume element. A further (algebraic) generalization was given by Duflo [1979a]. 55, Nos. 2-3. Corollary 5.12 is an unpublished result of Chevalley which is utilized in Harish-Chandra [1958a], whose proof of the result is outlined in Exercise D1 in this chapter. The proof in the text, which also gives the more general Theorem 5.8, is due to Dadok [1982]. Theorem 5.13 is from Helgason [1977a], and Theorems 5.16 and 5.17 and Prop. 5.23 from Harish-Chandra [1958a]. Formula (39) is from Helgason [1964a], where it is used to prove local solvability of each D E D ( G / K ) . Theorem 5.24 is a special case of Harish-Chandra's expression of the Casimir operator on G in terms of the Cartan decomposition G = KAK (cf. Exercise A8). 85, Nos. 4-8, Proposition 5.26 and the solution to Poisson's equation (Theorem 5.27) are from Helgason [1959] and [1962a], Chapter X. Part (ii) of Theorem 5.27 is also in Allamigeon [19613. Theorem 5.28 and its generalization to Riemannian homogeneous spaces are from Helgason [1957b], [1959]. The proof of Theorem 5.28 is a generalization of Asgeirsson's original proof [1937] for the case R".Theorems 5.33, 5.35, and 5.37 are from Harish-Chandra [1957a] and [1958a], $14, the material in 55, No. 7 and Theorem 5.38 from the author's paper [1964a].

CHAPTER Ill

INVARIANTS AND HARMONIC PO LY N0 MIALS A linear action of a group on a vector space extends to an action on the symmetric algebra, giving rise to invariant polynomials and corresponding harmonic polynomials. In 51 we prove (under general assumptions) that each polynomial is a polynomial in the invariants with harmonics as coefficients. The harmonics are also described in terms of the variety defined by the invariants. For the linear isotropy action in case of a symmetric space the results hold in a strengthened form, discussed in $5. In 52 we prove one analogous result for the exterior algebra. In $3 we consider the special case of the standard action of the Weyl group. This case is particularly interesting, because the harmonic polynomials, the invariants, and the corresponding eigenfunctions have profound connections with the root structure of the corresponding Lie algebra. In §4 we study the orbits for the complexified isotropy action for a symmetric space. Here the orbits have important relationships to the invariants and the harmonic polynomials.

61. Decomposition of the Symmetric Algebra. Harmonic Polynomials

Let E be a finite-dimensional vector space over R and G a group of linear transformations of E. The action of G on E induces an action of G on the ring of polynomials on E. The fixed points, the G-invariants, form a subring. The G-harmonic polynomials h are the common solutions of the differential equations Dh = 0 where D is a G-invariant differential operator of constant coefficients annihilating the constants. Under some general assumptions about G we shall prove results of the following kind. 1. Each polynomial on E is a sum x k ikhk where ik is invariant, h, harmonic. 2. The G-harmonic polynomials are of two types:

(a) Those which vanish identically on the algebraic variety N , determined by the G-invariants. (b) The powers of the linear forms given by the points in N , . 345

346

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

Each G-harmonic polynomial h is a sum h = ha + h, where ha is of type (a) and h, a sum of harmonic polynomials of type (b) above. We proceed now with a more detailed treatment. Let E* denote the dual of E, S(E*) and S(E) the corresponding real symmetric algebras. Then S(E*) consists of the polynomial functions on E. If X E E let d ( X ) denote the differential operator given by

The mapping X -+ d ( X ) extends to an isomorphism of the symmetric algebra S ( E ) (resp. the complex symmetric algebra S'(E) = C 8 S(E)) onto the algebra of all differential operators on E with constant real (resp. complex) coefficients. (In Chapter I1 we dropped the superscript c for simplicity of notation.) Each g in the general linear group GL(E) acts on E and on E* by ( g . e * ) ( e )= e*(g-' . e). These actions extend to automorphisms of S(E) and S(E*). Let G c GL(E) be any subgroup and let I(E) denote the set of G-invariants in S ( E ) and I + @ ) c I ( E ) the set of G-invariants without constant term. Similarly we define I + ( E * ) c I(E*) c S(E*). An element h E S'(E*) is said to be G-harmonic if d(J)h = 0 for all J E I + @ ) . Let H'(E*) denote the set of G-harmonic polynomial functions and put H(E*) = S(E*) n H'(E*). Let I'(E) and I"@*), respectively, denote the subspaces of S'(E) and S'(E*) generated by I ( E ) and I(E*). Each p € S ' ( E * ) extends uniquely to a polynomial function on the complexification E', also denoted p. Let N , denote the variety in E' defined by N , = {XE E' :j ( X ) = 0 for all j E I + @ * ) } . Remark. The terminology G-harmonic is of course suggested by the case when G is the orthogonal group. In this case the invariants are the polynomials in the square of the distance and the G-harmonic polynomials are the polynomial solutions of Laplace's equation. Let B be a nondegenerate symmetric bilinear form on E x E ; let B also denote its bilinear extension to E' x E'. If X E E' let X * denote the linear form Y -,B(X, Y) on E. The mapping X -+ X * (XE E ) extends uniquely to an isomorphism p : P + p of S'(E) onto S'(E*). Under this isomorphism B gives rise to a bilinear form ( , ) on E* x E* which in turn extends to a bilinear form ( , ) on S'(E*) x S'(E*); the formula for ( , ) is (1)

(P9

4) = (J(P)d(O).

911

1. The Symmetric Algebra. Harmonic Polynomials

347

Since a ( X ) ( B ( Y ,.)) = B ( X , Y ) , ( , ) does indeed extend B. On the other hand, let Sk(E*) denote the space of elements in S(E*) which are homogeneous of degree k. We define a bilinear form ( , )’ on Sk(E*) x Sk(E*) by

(2)

1

(X1”*Xk,yl**‘yk)’=

aeOk

n k

(Xi,Yn(i)>

i= 1

for x i , y j € E*, 6,being the symmetric group on k letters. This tilinear form is indeed well defined because the right-hand side is linear in each xi (and each y j ) and is independent of their order. We then extend ( , )’ to a bilinear form on Sc(E*) x Sc(E*). It is clear [say by (4) below] that ( x k , y”)’ = ( x k , y k ) , so since the powers xk (xEE*) span Sk(E*) (Chapter I, Exercise Dl), ( , ) and ( , )’ are identical. In particular, ( , ) is symmetric and now (1) shows that it is nondegenerate. Moreover, if p , q, r E SC(E*) then which shows that multiplication by q is the adjoint to the operator a(Q). Now suppose G leaves B invariant; then ( , ) is also G-invariant and (3)

p(Z’(E)) = Z’(E*).

Let P be a homogeneous element in Sc(E) of degree k. If n E Z , n 2 k, then the relation d(P)((X*)n)= n(n - 1 ) . . - (n - k + l)p(X)(X*)n-k can be verified by a simple computation. In particular, if X E N , then (X*)”is a harmonic polynomial. Let H1(E*) denote the vector space over C spanned by the functions (X*)”, (n = 0, 1,2,. . . ;X E NG)and let H,(E*) denote the set of harmonic polynomial functions which vanish identically on N , . If A c Sc(E*) is a subspace and k E Z+, Ak denotes the set of homogeneous elements in A of degree k ; A is called homogeneous if A = Ak. The spaces Z(E*), H(E*), H1(E*) and the ideal Z+(E*)S(E*) are clearly homogeneous. If C and D are subspaces of an associative algebra A then CD will denote the subspace of A spanned by the products cd ( c E C , d E D).

(4)

1

Theorem 1.1. Let G be a compact group of linear transformations of’a vector space E over R. Then

S(E*) = Z(E*)H(E*), that is, each polynomial p on E has the form p = x invariant and h, harmonic.

(5)

k

ikhk where ik is G -

348

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

Example. If G is the orthogonal group O ( n ) acting on R" the invariants are the polynomials in x: + . . + x i so ( 5 ) gives the classical decomposition of any polynomial p ,

where the polynomials h, are harmonic, i.e., LRn(hk)= 0 (cf. Introduction, $3, No. 1). Proof of Theorem 1.1. Let B be a positive definite quadratic form invariant under G. Using a basis of E orthonormal with respect to B it is easy to see from (2) that ( , ) is positive definite on S(E*) x S(E*). For a given k E Z+ consider the direct decomposition

Sk(E*) = (I+(E*)S(E*))k @ Vk, (7) V k being the orthogonal complement with respect to ( , ). Let 1 Ik and let j E I+@*)'. Since multiplication by j and d(J) are adjoint operators, (7) implies

(sk-I(E*), d ( J ) V k ) = 0, whence V k c Hk(E*). The converse inclusion follows in the same way, so we have the orthogonal decomposition Sk(E*) = (Z+(E*)S(E*))k (8) Now ( 5 ) follows by iteration.

+ Hk(E*).

Theorem 1.2. Let E be a Jinite-dimensional vector space over R and let G be a connected semisimple Lie subgroup of GL(E) leaving invariant a nondegenerate symmetric bilinear form B on E. Then

S(E*) = l(E*)H(E*). Proof. The form B has a unique extension (also denoted B) to a bilinear form on E' x E'. Let g be the Lie algebra of G, gc its complexification, and let w be an arbitrary compact real form of gc. Since g is a subalgebra of gI(E), the Lie algebra of GL(E), gc is a subalgebra of gI(E'), the Lie algebra of all linear transformations of E', which again is the Lie algebra of GL(EC). Let U and GC be the connected Lie subgroups of GL(E') (considered as a real Lie group) corresponding to u and gc, respectively. The elements of G extend uniquely to linear transformations of E' whereby G becomes a Lie subgroup of GC leaving B invariant. This implies that (9)

E(T * Z , , Z 2 ) + B(Z1, T * Z , ) = 0,

T E 9, Z,,2 2 E E'.

911

1. The Symmetric Algebra. Harmonic Polynomials

349

However, since (Tl + iT2).Z = T, .Z + iT2 Z for T,, T2 E g, Z E E', and since B is C-bilinear, it follows that (9) holds for all T € g c ; hence, by the connectedness of GC, B is invariant under GC. Lemma 1.3. There exists a real form F of E' on which B is strictly positive definite and which is leji invariant by U . Proof. By the usual reduction of quadratic forms the space E is an orthogonal direct sum E = E - + E + of subspaces E - and E + on which - B and B, respectively, are strictly positive definite. Let J be the linear transformation of E' determined by

JZ=iZ

(ZEE-),

JZ=Z

(ZEE').

Then the bilinear form B(Zl, Z , ) = B(JZl, JZ,),

Z , , Z , E Ec

is strictly positive definite on E. Let O(B), O(B') c GL(E') denote the orthogonal groups of B and B , respectively, and let @BE)denote the subgroup of O(B') which leaves E invariant, i.e., O(BE)= O(B')n GL(E). Now we have U c G C c O(B) = JO(B')J-'.

Since SO(n) is a maximal compact subgroup of SO(n, C), the identity component of the group JO(Sz)J-' is a maximal compact subgroup of the identity component of JO(B')J-'. By an elementary special case of Cartan's conjugacy theorem ([DS], Theorem 2.1, Chap. VI) this last group contains an element g such that g-'Ug c JO(Bk)J-'. Then the real form F = gJE of E' has the properties stated in the lemma. In fact, UF = U g J E c g J O ( B E ) J - ' ( J E ) c F ,

and if X E F then since J - l g - ' J

E

O(B') we have

B ( X , X) = B'(J-'X, J-'X) = B ' ( J - ' g - ' X , J-'g-'X) 2 0, B' being positive definite on E. This proves the lemma. Since a polynomial function is determined by its restriction to a real form it is not difficult to reduce Theorem 1.2 to ( 5 ) by means of Lemma 1.3. For the sake of clarity we carry this out by explicit maps. Since B is nondegenerate on E x E , F x F and on Ec x E' there are induced surjective isomorphisms pl : S'(E)

-P

S'(E*),

p,: S'(F)

--f

S'(F*),

p: S(E') + S((E')*).

350

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

By restriction of a complex-valued function on E' to E and F, respectively, we get the surjective isomorphisms

1,: S((E')*)

+ S'(E*),

1, : S((E')*)

--+

S'(F*).

Since S(E') = S(((Ec)*)*), we obtain by restricting functions on (E')* to E* and F*, respectively, the surjective isomorphisms AI : S(Ec) + S'(E),

-

A2: S(E')

--+

S'(F).

We then have the commutative diagram

S'(E)

4

S(E')

S'(F)

Corresponding to the actions

U on F,

G on E,

GC on E'

we consider the spaces of invariants I'(E),

I"E*),

IC((F), I'(F*),

Lemma 1.4. Let 1= &A;',

I(E'), I((E"*).

A = h , A ; ' . Then

A(I"(E*)) = I'(F*),

A(I'(E)) = Ic(F).

Proof. Since G c GC it is clear that Al(I((Ec)*)) c I'(E*). On the other hand, let p E Ic(E*). If 2 E g let d, denote the unique derivation of Sc(E*) which satisfies (d, e * ) ( X ) = e * ( Z . X ) for e* E E*, X E E. Then (10)

d , * p = 0.

Let (Xi)be a basis of E, ( x i ) the dual basis of E*, (zi) the basis of (E')* dual to ( X i ) considered as a basis of E'. Then (10) is an identity in ( x i ) which remains valid after the substitution x i + zi (all i). This means that

6, * (1; 'W = 0, where d2 is the derivation of S((Ec)*) which satisfies (1 1)

(6,. e * ) ( X ) = e*(Z X ) 1

for e* E (E')*, X E Ec. However, 6, can be defined for all Z E gc by this last condition and (11) remains valid for all 2 E gc. Since GC is connected, this implies 1; '(p) E I((E')*). Thus we have proved Al(Z((Ec)*)) = If@*); similarly l,(I((Ec)*)) = Zc(F*), so the first statement of the lemma

911

1. The Symmetric Algebra. Harmonic Polynomials

35 1

follows. The second statement follows from the first if we take into account (3) and the diagram above. Lemma 1.5. Let P E S'(E), q E S'(E*). Then

W P )(& =IW )( P ) q ) . Proof. First, suppose P = X E E, q = p , ( Y ) ( Y E E ) . Here both sides of the equation reduce to B ( X , Y). Therefore, the derivations q -, a(AX)Lq and q -,I(a(X)q) of S'(E*) coincide on E*, and hence on all of Sc(E*). Since the mappings P -,d(AP), P -+ a(P) are isomorphisms, the lemma follows.

Combining the two last lemmas we obtain the following result. Corollary 1.6. A(H'(E*)) = Hc(F*).

With U acting on F we deduce from Theorem 1.1 S'(F*) = Z"F*)H'(F*).

Applying the isomorphism Theorem 1.2.

A-l,

Lemma 1.4, and Corollary 1.6, we obtain

Theorem 1.7. Let E be a jinite-dimensional vector space over R, and G c GL(E) a subgroup leaving invariant a nondegenerate symmetric bilinear form B on E. Then

(a) Zf G is compact and B positive definite, (12) H'(E*) = H , ( E * ) + H , ( E * ) (direct sum).

(b) Zf G is a connected semisimple Lie subgroup of GL(E), (12) is also valid. We begin with a simple lemma. Lemma 1.8. The space H , ( E * ) is homogeneous. Proof. We recall that the elements of S'(E*) are automatically extended to polynomial functions on E" without change in notation. Let J = I',(E*)S'(E*). Then N G is the variety of common zeros of elements of the ideal J. By Hilbert's Nullstellensatz (see, e.g., Zariski and Samuel [1960], p. 164) the polynomials in Sc(E*) which vanish identically on N G consitute the radical f l of J, that is, the set of elements in S'(E*) of which some power lies in J. Now let p E f l and let p = pk be the decomposition of p into homogeneous components. Writing this as p = q + p , we see that, by the homogeneity of J, the relation p" E J implies P : E J ; in other words, p , ~ f l By . linearity of f i ,q E f l and now

ro

3 52

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

pk E f l follows inductively. Thus f l is homogeneous, and since H,(E*) = H'(E*) n f l the lemma is proved. For the theorem consider first Case (a). For k E Z + let M denote the orthogonal complement of Hl(E*)k in H'(E*)k, the superscript k indicating the space of homogeneous components of degree k. Let q E H'(E*)k, Q = p- '(4). Then q E M o (d(Q)h)(O)= 0 o c?(Q)(X*)~ =0

for all h E for all X E N , .

In view of (4) this amounts to q vanishing identically on N , ; consequently, M = H,(E*)k. This proves formula (12) since all terms in it are homogeneous. Next we consider Case (b). We use the notions introduced during the proof of Theorem 1.2. Because of Lemma 1.4 the varieties N u and N , coincide. It follows that A(Hi(E*)) = Hi(F*) (i = 1,2) and now Case (b) reduces to the previous case. Example. Again we consider the example G = O(n) acting on R".

In this case

H,(R")

(13)

= 0,

so by (12), (14) Hc(Rn) = spanned by

This was already proved in the Introduction, $3, No. 1. Corollary 1.9. With the notation of Theorem 1.7 suppose the ideal Z"+E*)S'(E*) equals its own radical. Then

H"E*) = H,(E*).

This is clear since (15)

H,(E*) = H'(E*) n J

=

0.

Theorem 1.lo. Let E be a finite-dimensional vector space over R and let G c GL(E) be a Lie subgroup which is either (a) compact (possibly .finite) or (b) connected and semisimple. Then the algebra I(E*) is finitely generated. Proof. Let us first consider the compact case. Again let

J = I:(E*)sc(E*),

913

1. The Symmetric Algebra. Harmonic Polynomials

353

By Hilbert's basis theorem (cf., e.g., Zariski and Samuel [1958], p. 200) the ideal J has a finite basis. It follows that there exist finitely many homogeneous invariants j,, . . . ,j , E I',(E*) such that each homogeneous invariant j E I:@*) can be written (16)

j = pljl

+ . . . + pJ,,

where all P k E S'(E*) are homogeneous and degree@,) = degree(j) degree(j,). Applying the linear transformation g E G and integrating over the compact group G we obtain j = iljl

+ ... + i l , ,

where i, = jGg pk dg. Applying (16) to the homogeneous invariant ik we obtain by induction, j E C u l , . . . ,j,] as desired. Passing to Case (b), let 9, g", u, G, Gc, U be as in the proof of Theorem 1.2. Consider also the isomorphism 1, of S((E')*) onto S'(E*), which we proved satisfies

n,(r((E')*)) = l'(E*).

(17)

A polynomial function on E" is U-invariant if and only if it is GCinvariant. This is readily seen by expressing the invariance condition by means of the Lie algebras u and gc (cf. proof of Lemma 1.4). Now Case (a) implies that Z((E')*), and therefore I"(,!?*), by (17), is finitely generated. Writing now S = S((E')*), I = I(@')*), let Sk denote the space of homogeneous elements in S of degree k and put Ik = I n Sk. We shall now find the generating function dim(lk)tk of the sequence dim(Zk)in the case in which G has finite order N = I GI.

1;

Theorem 1.11

.

The generating function is gioen by

m

1 1 dim(Ik)tk= 1 (det(l - is))-', 0

geG

where I is the identity operator on E'. Proof. Let F' c E' be the subspace fixed by each BEG. Then the operator P = N - ~ e . go satisfies g o P = P for each go E G. Thus PE' = F' and P2 = P. In particular, 1

dim(F') = -

IGI

1Tr(g),

geG

354

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. 111

Tr denoting trace. On the other hand, if g(k)denotes the endomorphism of Sk induced by g we have, by diagonalizing g,

Now the theorem follows by applying (19) to the endomorphism g'k).

$2. Decomposition of the Exterior Algebra. Primitive Forms

Let E be a finite-dimensional vector space over R as in $ 1 and let A(E) and A(E*), respectively, denote the Grassmann algebras over E and its dual. Each X E E induces a linear mapping S ( X ) of A@*) given by k= 1

where 2 k denotes omission of xk. Then S ( X ) is an antiderivation of A(E*) in the sense that A 4 + (- 1)'P A S(XI4 S ( X ) @ A 4) = (cf. [DS], Chapter I, Exercises BS-6) if p has degree r. The mapping X + 6 ( X ) extends uniquely to a homomorphism of the tensor algebra T ( E ) over E into the algebra of all endomorphisms of A(E*). Since

S(X 0 X ) = S(X)2 = 0

there is induced a homomorphism P -, 8(P) of A(E) into the algebra of endomorphisms of A(E*). As will be noted below, this homomorphism is actually an isomorphism. Now suppose B i s any nondegenerate symmetric bilinear form on E x E . The mapping X + X * ( X * ( Y ) = B ( X , Y ) ) extends to an isomorphism p of A(E) onto A(E*). We obtain a bilinear form ( , ) on A(E*) by the formula If xl,

. . .,x k , y,, . . . ,y1E E* (XI A

then by direct computation,

... A xk,y1

A

. * *

A

y,) =O

or (2)

(- l)(1'2)k(k') det[B(p- 'xi, p- ' y j ) ] ,

921

2. The Exterior Algebra. Primitive Forms

355

depending on whether k # 1 or k = 1. It follows that ( , ) is a symmetric nondegenerate bilinear form. Also if Q E A(E), q = p(Q) then the operator p + p A q on A(E*) is the adjoint of the operator S(Q). It is also easy to see from (1) and ( 2 ) that the mapping P + 6 ( P ) [ P E A(E)] above is an isomorphism. Now let G be a group of linear transformations of E. Then G acts on E* as before and acts as a group of automorphisms of A(E) and A(E*). Let J ( E ) and J ( E * ) denote the set of G-invariants in A(E) and A(E*), respectively; let J + ( E ) and J + ( E * ) denote the subspaces consisting of all invariants without constant term. An element p E A(E*) is called G-primitive if 6 ( Q ) p = 0 for all Q E J + ( E ) . Let P ( E * ) denote the set of all G-primitive elements. Theorem 2.1. Let B be a nondegenerate, symmetric bilinear form on E x E and let G be a Lie subgroup oJCL(E) leaving B invariant. Suppose that either (i) G is compact and B positive dejinite or (ii) G is connected and semisimple. Then

(3)

A(E*) = J(E*)P(E*).

The proof is quite analogous to that of Theorems 1.1 and 1.2. For case (i) one first establishes the orthogonal decomposition

(4)

A(E*) = A ( E * ) J + ( E * ) + P(E*)

in the same manner as (8) in $ 1 . Then ( 3 ) follows by iteration of (4). The noncompact case (ii) can be reduced to the case (i) by using Lemma 1.3. We omit the details since they are essentially a duplication of the proof of Theorem 1.2. Example. Let I/ be an n-dimensional Hilbert space over C. Considering the set V as a 2n-dimensional vector space E over R the unitary group U(n) becomes a subgroup G of the orthogonal group O(2n). Let = xk + i & (1 Ik I n ) be an orthonormal basis of V, z ~ , . .. , z , the dual basis Of V*, and put xk = +(zk -k Zk), yk = -$(Zk - &) (1 k 5 n). Let F denote the vector space over C consisting of all R-linear mappings of V into C. The exterior algebra A(F) is the direct sum

1

A(F) =

Fa,b,

0ba.b

where F,,b is the subspace of A(F) spanned by all multilinear forms of the type (ZalA

* . .A

Zaa) A

(zol A

. * .A

job),

356

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

< P2 c ... < b b I n. The where 1 I a1 < a2 c ... c a, s; n, 1 I space J(E*) of G-invariants is given by the space J of invariants of V ( n ) acting on F. It is clear that J = z , , b .la,*where Ja,b = J n Fa,b. Now if pi E R (1 I i I n) the mapping

(z,,. . . ,2,)

+ (eiPIZl,

. . .,e'PnZ,)

is unitary. As a consequence one finds that J a , b = 0 if a # b and that if J E J,, ,then

Now, there always exists a unitary transformation of V mapping Zai-,Zi so 1 is a constant multiple A y,) it is clear that J ( E * ) is the algebra generated by u = x , A y,. From Theorem 2.1 we can therefore derive the following consequence.

(1 I i 5 a). It follows that A, ,...,,= A,, ,..,, of z , A 2,)". Since z, A 5, = -2i(x,

z=l

Corollary 2.2. Each q E A@*) can be expressed

where u =

xu A y , and each

Pk

is primitive; i.e., 8(U)(pk) = 0.

$3. Invariants for the Weyl Group We shall now amplify and extend the results of R l , 2 in the case in which W is a finite group of linear transformations of E generated by reflections, in short, a finite rejection group. By a rejection we mean (as in [DS], Chapter X) a linear transformation of order 2 whose fixed points form a hyperplane. Let w or 1 Wl denote the order of W. The Weyl groups associated with semisimple Lie groups and symmetric spaces are finite reflection groups. 1. Symmetric Invariants

Theorem 1.10 can be put in a more precise form for a finite reflection group. Theorem 3.1. Let W be a finite rejection group acting on an n-dimensional real vector space E. Then the algebra Z((Ec)*) of invariants is generated by n homogeneous elements which are algebraically independent.

931

357

3. Invariants for the Weyl Group

Example. Let W be the symmetric group acting linearly on C" by bei = en(,,,e, being the ith coordinate vector. Here the algebra of invari-

ants is generated by the elementary symmetric functions

1z i , c zizj, . . ., i

2, ' * ' z,,

i#j

and W is generated by reflections. As before let S = S((Ec)*), Z = Z((Ec)*), I , c Z be the space of invariants without constant term, J the ideal Z,S, and H the space H((E')*) of harmonic polynomials. Let Sk, Zk, and Hk denote the respective subspaces consisting of homogeneous elements of degree k. Let deg stand for degree. Lemma 3.2. Let j , , . . . ,j , E I such that j , $1 j,Z. ; Zf q,, . . .,q, are homogeneous elements such that ~ ~ j s=q0,sthen q1 E J .

ES

Proof. The proof goes by induction on deg(q,). If deg(q,) = 0 then j i q i = 0 gives, by averaging over W, j , j,Z, contrary to assumption. Let deg(q,) 2 1, let c r W ~

1:

q1 = 0 because otherwise the relation

€17

be a reflection, and let the hyperplane z = 0 constitute the fixed points of cr. From the assumptions we deduce 1 y ji(qi - a.qi) = 0. But qi - aqi vanishes on the hyperplane z = 0, so qi - aq, = zri, where r i € S . Hence j i r i = 0, and so by the induction hypothesis r , E J; Le., a - q , = q1 mod J. Since these reflections cr generate W this congruence holds for all cr E W. Averaging over a we deduce q1 E J. We now prove Theorem 3.1. Let j , , . . . ,j , E I , be homogeneous elements forming a minimal basis of the ideal J. During the proof of Theorem 1.10 it was shown that j , , .. .,j , generate I. We shall now show that they are algebraically independent. Suppose this were not so. Let P(y,, . . , ,y,) be a polynomial of minimal degree >O, such that

1:

P ( j , ( Z ) ,. .. ,j , ( Z ) ) = 0 for all Z E E. Let di = deg(ji). Writing P as a sum of monomials y:' . . y:' we may assume P only contains such monomials for which the degree a l d , + . - + a, d, of the polynomial Z jl(Z)"l . . j,(Z)". is a certain fixed number h. Let +

a

ap Pi@) = -(jl(Z), . . . ,j , ( Z ) )

(1 I i I s).

aYi

The Pi are invariant homogeneous polynomials. Since deg P > 0, aP/ayi # 0 for some i, so by the minimality of deg P the corresponding Pi is #O. We relabel the indices such that P,, . . ., P, are precisely the Pi which are #O and such that among P,, . . . , P, none is in the ideal of Z

358

INVARIANTS AND HARMONIC POLYNOMIALS

generated by the others whereas Pr+ 1, . . ., P , are in the ideal Thus r

P r + j= C u i j P i ,

st

1 Ij

[Ch. 111

C; ZPi.

- r,

i= 1

where each nonzero uij is a homogeneous polynomial of degree

h - d , + j - ( h - di) = di - d r + j . With Cartesian coordinates zl, . . .,z, on E' we differentiate the relation PO.,, . . . ,j , ) = 0 and obtain

However, the polynomial inside the parentheses is homogeneous of degree di - 1. By Lemma 3.2 we conclude for each i (with P , , . . . , P , playing the role of j l , . . . ,j,),

where each B, is homogeneous. Considering the degrees we see that B, = 0. We multiply this relation by zk, sum over k, and use Euler's formula for homogeneous polynomials. This gives 1-r

diji

S

+ IC Oil dr+rjr+i= C A m j m , = 1 1

m=

where Ai = 0. This shows that j i is in the ideal C m f i j m Scontradicting the minimality of the basis jl, .. . ,Is. Thus the j,, . . . ,j s are algebraically independent. It remains to prove that s = n. Let K = C(jl,. . . , J s ) be the field generated by the j , , that is, the quotient field of 1 = COl,. . . , j s ] . Also let Q = C(zl, . . . ,z,) be the quotient field of S = C[z,, . .., z.]. If p e S then the polynomial (1)

R(X) =

n (X - a-p)

= X"

+ q1XW-' + ... + 4"

O€W

has coefficients in I . Since R(p) = 0 it follows that p, and therefore each element in Q, is algebraic over K. Denoting by tr deg the transcendence

031

3. Invariants for the Weyl Group

359

degree of a field extension, we have by general theory Zariski and Samuel [1958], p. 100). (2)

tr deg(Q/C) = tr deg(Q/K) + tr deg(K/C).

Since the extension Q/K is algebraic this implies n = tr deg(K/C). Since any two transcendence bases of a field extension have the same cardinality, we conclude that s = n, so Theorem 3.1 is proved. Theorem 3.3. Let W be a jinite rejection group acting on the real vector space E of dimension n. Let j l , . . . ,j , be homogeneous generators f o r the algebra I((EC)*)of W-invariants. Let d l , . . . , d, be their respective degrees. Then n

n

ndi= w, i= 1

x ( d i - 1) = r, i= 1

where w is the order of W and r the number of reflections in W. Moreover, the degrees di are unique. Proof. First we observe from (2) that the j i are algebraically independent. Let j E I k , the space of homogeneous invariants of degree k. Then j is a linear combination of monomials j;' . - j p for which

a, d ,

(3)

+ + a, d, = k.

By the algebraic independence of the ji it follows that dim(Zk) equals the number of nonnegative integral solutions ( a l , . . . ,a,) to (3). Hence m

(4)

1dim(Ik)tk= ( 1 - tdl)-l . . . ( 1 - tdn)-l. 0

Combining this with Theorem 1.11 we conclude

if the c,, are the eigenvalues of 0 counted with multiplicity. Letting t + 1, only the term 0 = 1 on the right gives a contribution, so we obtain the first formula of the theorem. Next we differentiate ( 5 ) with respect to t and then let t + 1. A term corresponding to 0 on the right gives a nonzero contribution exactly if 0 has eigenvalue 1 with multiplicity n - 1. But this happens exactly if 0 is a reflection; the contribution in question is

360

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

We conclude that

and now the first formula of the theorem leads to the second one. For the uniqueness let J,, . . . ,J, be another set of homogeneous polynomials of respective degrees D,, . . . , D, generating I. By reordering we may assume d , I d , I - . I d, and D, i; D, I - I D,. Because of (4) we have (1 - t d r )= (1 - tDi). Considering the lowest power of t we get d , = D, and by cancellation, d , = D,, etc. This proves the uniqueness.

Hi

Hi

2. Harmonic Polynomials

We can now strengthen Theorem 1.1 a bit in the case of a finite reflection group. Theorem 3.4. Let W be a jnite reflection group and let the notation be as above. Then dim H = w and the mapping 4: j @ h +j h extends to a linear bijection of I @ H onto S. Moreover,

(6)

c (dim Hk)tk

=

(1

+ t + . . + tdi-l1.

1s i s n

kLO

Proof. We know from Theorem 1.1 that C#I is surjective. To prove that it is injective we must show that if ar,,i,h, = 0, where a,,, E C and {i,} and {h,} are homogeneous bases of the vector spaces I and H , respectively, then ar,, = 0. We write the relation in the form

XI,,

“(7ar,sir)

= 0.

Put I, = x,a,,,ir. We have to prove that each I, = 0 and for this it suffices to consider the case in which each I, is homogeneous and deg h, deg I , the same for all s. Suppose there were an I, f 0. We write it in the form

+

1 s = C am,, ...,m.,sJ1‘ml . . .j:n

with nonzero coefficients aml,...,m n , S . Then

and at least one of the monomials j y l ...jrn is not in the ideal in I generated by the others. The corresponding term

c am,....,m,,shs S

931

3. Invariants for the Weyl Group

361

then belongs to J according to Lemma 3.2. But by Eq. (8) of $1 this term will have to vanish and then the linear independence of the h, gives the contradiction a, m.,s = 0. The identification 163 H = S implies the identity

1 (dim Ik)tk 1(dim H')t' = 1 (dim S")t" k2O

120

m2O

for the generating functions. Since the right-hand side equals (1 - t)-" the formula for (dim H')t' follows from (4). Putting t = 1, we obtain the formula dim H = w from Theorem 3.3. Corollary 3.5.

The variety N , reduces to (0) so

H=H2,

Hl = O

in the notation of Theorem 1.7.

In fact if Z # 0 is in N , then by Eq. (4) of $1 all the powers (Z*)k would be W-harmonic and this would contradict the finite-dimensionality of H. We can now determine the harmonic polynomials more explicitly. Theorem 3.6. Let W be a Jinite rejection group acting on the Jinitedimensional real vector space E. Let nl, . . ., 6, be the rejections in W ; a1 = 0,. . . , a, = 0 the corresponding rejecting hyperplanes; and let II = ai. Then

n;

(i) H = d(S)n; that is, the harmonic polynomials constitute the linear span of the partial derivatives of n. (ii) If f E C"(E) satisjies d(P)f = 0 for each P E I + , then f E H . In other words, each W-harmonic function is a polynomial. Proof. First we prove (ii). Considering the polynomial (1) for p = xi we see that x r = - q l x ; - ' - q w , so x y e I + ( E * ) S ( E * ) (1 < i < n). It follows that d(P)f = 0 whenever P is a homogeneous polynomial of degree 2 n w (because then each term will contain a power Xq, a 2 w). Hence f is a polynomial of degree e n w . For (i) we first prove some general facts about n. As before let il,. . . ,j n be homogeneous algebraically independent generators of I. Lemma 3.7.

The

Jacobian

identity:

where c is a constant # 0.

determinant

satisfies

the following

362

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

Proof. Let q denote the left-hand side. Then by the chain rule for the Jacobian, q is skew; that is, it satisfies a.q = (det a)q (a E W). In particular, q vanishes in each hyperplane ai = 0, and so is divisible by the polynomial ai. Hence A divides q and now the relation follows from the sum formula in Theorem 3.3. It remains to prove c # 0. For each i, 1 I i I n, the polynomials x i , j l , . , . ,j , are algebraically dependent. We choose a polynomial Q i ( x i , z l , ..., z,) of minimal degree ei > 0 in x i such that Qi(xi,j l , , . . ,j , ) = 0. Applying d/dxk we obtain

We write this as a matrix identity A B = C,

where aQi

A . = -( x i , j l , . . . , j r ) , Br

B

az,

ajr axk

=rk

and C is the diagonal matrix given by Cii = aQi xi,j l , . . . ,j,).

axi

By the minimality of ei, det C # 0 so det B # 0 as desired. Corollary 3.8.

A

is skew and each skew polynomial is divisible by

II.

Now it is clear that A E H because if J E I , , a(J)n is skew, hence divisible by A. Hence we also have a(S)n c H . In order to prove the converse inclusion we prove the following result of independent interest. Lemma 3.9. If a(Q)n = 0 then Q E I , S . Proof. Let B be a positive definite bilinear form on E x E invariant under W. Let Q + q be the corresponding bijection of S(Ec) onto S((Ec)*). For the lemma we may assume Q homogeneous. We have seen that S"" c I, S, so clearly the lemma holds if deg q 2 nw. Assuming it holds for deg q = m + 1 we shall show that it holds for deg q = m and thus by induction in general. Let a be any reflection in W and LX = 0 the corresponding hyperplane. Then a(q)n = 0 implies d(aq)x = 0, so since deg(aq) = m 1, we have

+

n

(7)

Akjk,

Crq = k=l

A , ES.

531

3. Invariants for the Weyl Group

363

Applying o and subtracting, we obtain (since a divides A , - aA,) q = -oq (mod J ) . Since the reflections o generate W , this implies q = (det s)sq (mod J ) for each s E W. Averaging over W we obtain q 5 q* (mod J ) , where q* is skew. By Corollary 3.8, q* = ni, where i is a homogeneous invariant. If deg i > 0 we have q E J as desired. If deg i = 0 so i = c (a constant), then 0 = d(q)n = cd(n)(n),so c = 0. This proves the lemma. For Theorem 3.6 it remains to prove H c d(S)n or, equivalently, (d(S)n)1c H I ,

the orthogonal complement taken with respect to ( , ). But if (d(S)n, p )

=

0 then (d(P)n, S )

= 0,

so d(P)n = 0. By Lemma 3.9, p E J = HI,so the theorem is proved.

3. The Exterior invariants

Let W again be a finite reflection group acting on E and let denote homogeneous algebraically independent generators for the algebra of invariants. We shall now describe the W-invariant forms on E with polynomial coefficients. As usual d denotes exterior differentiation. Proposition 3.10. Each W-invariant p-form on E with polynomial coeficients may be expressed uniquely as

(8)

1

i l <... < i

ail,...,i, dji, A

*

*

A

ail,.... i, €1.

dji,,

) dj for j E l , O E W , it is clear that each form (8) Proof. Since ~ ( d j = is invariant. Next we prove that the (); forms dji,A . . A djipare linearly independent over the quotient field Q = R(xl,. , . ,x,,) so that they form a basis over Q of the space of p-forms with polynomial coefficients. Indeed, suppose

(9)

C

i l < ...< i,

ki,,,..,i,dji, A . * *

A

dji, = 0,

< ep) and let e p + l , .. . , e, be where ki,,...,i,EQ. Fix a p-tuple ( e l < the complementary indices. Multiplying (9) by djeP+,A A dj,, we obtain from Lemma 3.7.

k,, ,,..,epdjlA ... A dj, = kke,,..., , c n d x ,

A

A

dxn = 0,

364

[Ch. 111

INVARIANTS AND HARMONIC POLYNOMIALS

whence kel:.,.,e, = 0, proving the linear independence. It follows that each p-form w with coefficients in Q can be expressed (10)

0

=

il <

C... i, ail ,..., <

i,

dji, A . . A

4,’

ail ...., i, E Q.

Now assume w is invariant with polynomial coefficients. Again multiplying by die,+, A - . . A dje, we deduce w

A

d j e p + ,A ... A dje, =

+ C U ~ , , . . . , ~ ~ TAC ~ ~ A ,

dx,,

so ael,...,epnis a polynomial. O n the other hand, applying CTEW to (10) and averaging over W, we see that each aer,...,e,is W-invariant. Since ~ ~ , , , , , is, ~a , polynomial n and skew we conclude from Corollary 3.8 that ael,,..,e, E I and the proposition is proved. 4. Eigenfunctions of Weyl Group Invariant Operators

Let g be a semisimple Lie algebra over R, B its Killing form, 8 a Cartan involution of 9, and g = t + p the corresponding Cartan decomposition. Let a c p be a maximal abelian subspace and W = W(g,13) the corresponding Weyl group acting on a (cf. [DS], Chapter IX, $1). Then W is generated by the reflections s, as a runs through the set Z of restricted roots. We shall now use the previous results of this section, a playing the role of E. We shall determine the functions f ’ on a which are eigenfunctions of each d(P), P E I(a). In the language of Chapter 11, $4.1, we are looking for the joint eigenjinctions of the operators in D(W.a/W), W . a denoting the group of transformations of a generated by W and the translations.

-

Lemma 3.11. The homomorphisms of I(a) into C are precisely

P P(PX where p is an element in (aC)*. Two such homomorphisms ~ cide f u n d only f p W.V. xp:

x,, and xy coin-

Proof. Equation (1) shows that the ring S(aC) is integral ouer I(&), in the sense that each element in S(aC) satisfies an algebraic equation with coefficients in I(ac) and leading coefficient one. By Theorem 3.5 of the Appendix each homomorphism of Z(a) into C extends to a homomorphism of S(a) into C and is therefore given by evaluation at some point p€(aC)*. Furthermore, suppose for two such p, v€(ac)* we have P ( p ) = P(v) for all P € I ( a ) . If the orbits Wsp and W.v were disjoint we can, since these orbits are finite, find a polynomial Q such that Q > 0 on W . p and Q 0 on W .v. Then the sum Q* = COG w ~Q .belongs to I(a), yet takes different values on p and v. This proves the lemma.

-=

031

3. Invariants for the Weyl Group

365

Corollary 3.12. If j,, . . . , j , ( I = dim a) are algebraically independent generators oj’ I , then the mapping e E ac -, ( j l ( e ) ,. . . ,j,(e))E C’

(11)

induces a bijection oJ’aCfW onto C‘.

In fact, if (t1,.. . ,5’) E C’,then the mapping ji + ti (1 I i I I ) gives, by the algebraic independence of the j i , a homomorphism of l((ac)*) into C which by the lemma is given by the evaluation at a point e E a‘. Given p E (aC)*, let 8, = &,(a) denote the joint eigenspace &,(a) = { f &(a) ~ : d(J)f’ = J ( p ) j for all J E l(a)}. (12) According to Lemma 3.11 this is the most general eigenspace of the operators J ( J ) , J E l(a). Let W, denote the subgroup of W leaving p fixed. According to [DS, Chapter VII, Theorem 2.151, W, is a finite reflection group acting on a. Let P(a) c S(a) denote the set of W,invariants and H” the corresponding set of harmonic polynomial functions.

Theorem 3.13. For each p E (aC)*,

dim &,(a) = I WI. Moreover, if’s runs through a complete set of representatives of the cosets of’W, in W and h runs through a basis of’ HSP,the junctions

(13)

e

+

h(e) expCsp(e)I

form a basis of’&,(a). Remark. In particular, if p is regular (cf. Exercise 2) the functions e l form a basis of 8”. We start by proving the weaker result

dim &,(a) I I W I .

(14)

For this let h l , . . ., h, be a basis of H (where w = I W l ) and let H I , .. . , H,, be the corresponding members of S(aC) [under the identification of ac and (aC)* by means of B ] . Let j ’ ~ & , ( a ) and put ci(f)

(1 Ii I w).

= (a(Hi)f’)(p)

Since S = I H it is clear that if cl(f’) = . the mapping

1’

+

-

. = c,(f)

=0

then f’ = 0. Thus

(c1(1’), * . c,(f)) 9

is a one-to-one linear mapping of &‘,(a) into C” so (14) follows.

366

lNVARIANTS AND HARMONIC POLYNOMIALS

[Ch. 111

Lemma 3.14. For each P E P(a) and each h E H@ the function e E a, f ( e ) = h ( 4 expCp(41,

satisfies

mf’= P(P)f*

(15)

I n particular, f’ E &,(a). Proof. Consider the autornorphism P -+ P” of S(a) given by Pp(A) = P ( p A). Then d ( P ) f = d(P)(ech) = ePee-”d(P)(ej’h) = eV(Pp)(h)= e’(P”(0)h) epd(Pp- P”(O))h,

+

+

But if P E ’I then PI E I”, so 8(PP - P’(0))h = 0. Thus d ( P ) f = P ( p ) f as desired. This lemma shows that the functions (13) belongs to &,(a). Since by Theorem 3.4, dim HSfl = I W,,I = I W,l, and since we have inequality (14), Theorem 3.13 will follow if we prove that the functions (13) are linearly independent. As s varies the linear forms sp are different; hence they are different on a line in a through 0 on which the basis elements h are not identically zero. Thus it just remains to prove the following elementary result.

..

Lemma 3.15. Let c l , . ,c, be distinct complex numbers and p l , . . . ,pn polynomials on R with complex coeflcients. Assume n

1pi(t)eci‘= 0

(16)

for all

t E

R.

i= 1

Then p 1 = = pn = 0. This lemma is a consequence of Exercise D5 of Chapter I.

5. Restriction Properties With the notation of 83, No. 4, let b denote a Cartan subalgebra of g containing a and If its complexification. Then in addition to the Weyl group W above which acts on a‘ we have the Weyl group @ of the pair (g‘, If). Corresponding to these groups W and we have the algebras ](a*) and )’‘@I of invariant polynomial functions on a‘ and b‘, respectdenote the subgroup of @ leaving a invariant. Then we ively. Let know from [DS] (Proposition 8.10, Chapter VII) that the restriction s + s I a is a homomorphism of onto W. If p E S((Ij‘)*) its restriction to a‘ will be denoted by jj. By the above

mo

(17)

p

E

* p E r(a*).

031

3. Invariants for the Weyl Group

367

While this restriction mapping is not always surjective (cf. Chapter 11, Exercise D3) the following positive result holds for the rational invariants. Theorem 3.16. Let J c I(a*) be the image of I(If*) under the restriction mapping p -+ p. Then the quotient Jields C(J) and C(I(a*)) coincide. Proof. From Eq. (1) we know that S(Ijc*) is integral over I ( € f * ) . The restriction p + p gives a homomorphism of S(tjc') onto S(a*) mapping I(tjc*) onto J. Hence S(a*) is integral over J. Now suppose C(J) were a proper subset of C(I(a*)). Then there exists a a E I(a*) such that a 4 C(J). But as remarked above a is algebraic over C(J). Since I(tjC*) is finitely generated, the homomorphic image J also has finitely many generators, say tl,.. . ,<,. Let

p,(x) = x"

+ f;(<)x"-' + '.. +

( n 2 2)

In(<)

be the polynomial with coefficients in the field C ( J ) = C(tl,. . . ,{,) of lowest degree having a as a zero and leading coefficient 1. Let f(<)denote the product of the dominators of the A(<) and let q(5) denote the product of f(t) with the discriminant of the polynomial f ( t ) p = ( x ) .Then q ( t ) is a polynomial function on a', not identically 0 (because of the minimality). Consider now a compact connected Lie group U with Lie algebra t + ip. Put a, = ia and let A , c U be the corresponding analytic subgroup. Select H, E a, such that (18)

q(<)(H,) # 0,

exp RH, dense in A,.

Consider the homomorphism A: p -,p ( H , ) of J into C. The image of p,(x) under 1 is a polynomial with discriminant 20 which therefore has n distinct roots, say al, . . . ,a, E C. For each fixed i (1 5 i I n) we wish to extend 1 to a homomorphism ,Ii:C[tl,. . . , t,, a] -, C by putting Ai(a) = ai. To see that this is possible, suppose p(5, x) is a polynomial with coefficients in J = C[tl,. . . ,5,] which has a as a zero. Then p,(x) divides p ( t , x); i.e., p(5, x) = p,(x)q(t, x); the coefficients of q ( t , x) are found by long division, so they appear as rational expressions in tl,. . . , 4, with denominators dividing f(t). Since rl.(f(t))# 0 [by (18)] the image of p(5, x) under rl. has x = ai as a zero. Thus IZi is well defined. Since S(a*) is integral over J and a fortiori over C[tl, . . . ,t,, a], there exists a homomorphism A i :S(a*) -,C extending Ai (Appendix 03). Then there exists HiE ac such that Ai(p) = p ( H i ) for all p E S(a*). In particular,

P W * ) = PWl) = P W , )

for P

E

I(bC*).

368

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

which (cf. Lemma 3.11) implies that H,, H,, H, are all conjugate under Since H, E a, and since the subset b n f + a, c bC, where all roots a E A(@, bc) are purely imaginary, is invariant under we obtain

m.

H,, H,

E (4 n f

+ a,)

n ac = a,.

Now select u,, u2 E U such that Ad(ui)H, = H i ( i = 1, 2). Then ui exp t H , u;

= exp tHi

so by (18) ill and u2 belongs to the normalizer of a, in U . Using [DS] Proposition 8.8, Chapter VII describing this normalizer we can therefore conclude that H , and H, are conjugate under W. But then, since a E I(a*), a1 = A1(a) =

a ( H l ) = a(H,)

= A2(a) = a,,

which is a contradiction. Remark. According to Exercise D 3 in Chapter I1 J # I(a*) for some of the exceptional symmetric spaces, so by Proposition 5.30 of Chapter 11, Z ( G / K ) # D ( G / K ) in general. The theorem above suggests that this should be remedied by going over to the quotient fields of Z ( G ) and of D(G/K).

$4. The Orbit Structure of p 1. Generalities

In this section we make considerable use of the results proved in Appendix &1,2. Let g be a semisimple Lie algebra over C, go any real form of g. Let 8 be any Cartan involution of go; we extend it to an involutive automorphism of g, also denoted 8. We now vary go too. Lemma 4.1.

All involutive automorphisms q of g arise in this fashion.

Proof. Let gR denote g considered as a Lie algebra over R and qR the automorphism of gR corresponding to q. According to [DS] (Exercise B4 for Chapter 11), qR commutes with a Cartan involution of g R ; in other words, q leaves a compact real form o of g invariant. Let g = f + p, go = f, + po be the decompositions into eigenspaces of 8, and put w = f, + ip,. Then D = 4u for a suitable automorphism 4 of g. Since 4 - ' ~ 4 leaves u invariant, u = u n g1 + u n g-l, where ge is the eigenspace of 4- 'q+ with eigenvalue E (= f 1). But then u n g1 + i(u n g- ,) is a Car-

4. The Orbit Structure of p

§41

369

tan decomposition of a real form gb of 9. If 8' is the corresponding Cartan involution, r] is the extension of @'$I-' to an involution of 9. Let G denote the adjoint group of 9, i.e., G = Int(gR) with the associated complex structure ([DS], Chapter VIII, 52). Let K O , U c G denote the analytic subgroups corresponding to f, and u, respectively. Let the automorphism of G induced by 8 also be denoted by 8, let K , denote the fixed point group of 8 and K the identity component (K,),. We are now interested in studying the orbits for the action of K (and K,) on p. Let a, c p o be a maximal abelian subspace, a c p its complexification, and A ( = exp ad,(a)) the analytic subgroup of G with Lie algebra a. Let F = {a~A:a~=e}.Itisclearthata~F*8a=a-'=a,soFcK,. Lemma 4.2. K, = F K .

(u

+

Proof. Let g E K,. Using the Cartan decomposition g = u iu = f, ip,) and the corresponding decomposition G = U exp(iu) of

+

G we have g = up where U E U , p = exp X ( X ~ i u ) By . uniqueness Ou = u, 8 p = p . But exp is one-to-one on iu so OX = X. Thus

X

E

iu n f = if,

so P E K . Also by [DS] (Chapter V, Theorem 6.7), we have u = k,ak, [ k , , k, E K O , a E exp(ia,)], so Bu = k , a - ' k , . But then 8u = u implies a E F, so g E F K . Definition. An element X E p is called semisimple (resp nilporent) if ad,(X) is a semisimple (resp. nilpotent) endomorphism; X E p is called quasi-regular if its orbit under K has maximal dimension. The set of semisimple (resp. nilpotent, quasi-regular) elements in p will be denoted '(resp. Jy; and 9). If X E p then (ad X)' maps p into itself. Thus we by 9 generalize the definition in [DS] (Chapter 111, $3) and put for X E p, AEC (1)

p(X, A) = { Y E p: ((ad X)' - A)"'(Y) = 0 for some m}.

The element X E p is said to be regular if dim p ( X , 0) = min dim p(Y, 0). YEP

Let r denote the set of regular elements in p. It will be proved later that the regular elements are the quasi-regular elements which are also semisimple that is,

(2)

r=WnY

The elements in 9 n A'" are called principal nilpotent elements.

370

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. 111

2. Nilpotent Elements We begin by relating nilpotent elements to K-invariant polynomials on p. Then we shall investigate nilpotent elements by imbedding them in subalgebras of g isomorphic to d ( 2 , C). As in 41 we consider the symmetric algebras S(p), S(p*), the sets I(p), I(p*) of the corresponding K-invariants and the sets I+(p), I + ( p * ) of K-invariants without constant term. Theorem 4.3. Let X

E

p. Then

X E N o f(X)

= 0 for

a l l f E Z+(p*)S(p*).

Proof. Suppose f ( X ) = 0 for all f~ I + S. Then in particular, S ( X ) = 0 for all G-invariant polynomial functions on g. The characteristic polynomial det@ - ad X) therefore reduces to AJimg so ad X is nilpotent ( [ D S ] ,Chapter 111, $1). Conversely, suppose X E X To prove f(X) = 0 for all Y E I + S we may assume X # 0. By [ D S ] , Chapter IX, Theorem 7.4, we can select H, YE^ such that

(3)

[H, X ]

=2x,

[H, Y ]

=

-2Y,

[X, Y ] = H .

Writing H = H,+ H, (H,E f, H, E p), we have 2 x = CH,XI = CH,,XI

+ [ H p , XI9

so since X E p, [ H , , X ] = 2 X . It follows that exp(C ad H,) X = (C - {O})X,which has 0 in its closure. By continuity f ( X ) = f(0) = 0. a

Definition. A sI,-triple in g is a triple (H, X, Y ) of nonzero elements in g satisfying relations (3) above. The 51,-triple is called normal if H E f, X , Y Ep. Lemma 4.4. If ( H , X , Y ) and ( H , ,'A Y') are two 51,-triples in g then Y = Y'.

Proof. We have [ X , Y - Y'] = H - H = 0,

so Y - Y ' E kernel(ad(X)). Also, (ad H + 2Z)(Y - Y') = -2Y

+ 2Y' + 2(Y - Y ' ) = 0.

But as remarked in the proof of Lemma 7.6, Chapter IX in [DS], ad H 21 is nonsingular on the kernel of ad X . Hence Y = Y'.

+

Lemma 4.5. Let n c g be a nilpotent subalgebra. Suppose H E g satisJes ad H(n) = n. Then eadcn)H= H n.

+

§41

4. The Orbit Structure of p

371

Proof. The inclusion ead(")(H)c H + n is obvious. Conversely, we shall prove that if Z E n then H + Z E eadcn)(H). The central descending series %?On= n, . . . ,V + ' n = [n, VPn] being defined as in [DS] (Chapter 111, §2), it suffices to prove H + Z E ead'")(H)+ %?pn for all p 2 0. (4) In fact, V n = 0 for p sufficiently large. We prove (4) by induction, the case p = 0 being obvious. Suppose we have Y, E n, Z, E %?,-In such that H + Z = eadYp(H)+ Z,. (5)

Since ad H gives a bijection of n onto n leaving %?,-'n invariant, there exists Zo E WP-'n such that 2, = [ Z , , HI. But then ead(Yp + 20)( H ) = H + [ Y , + Z o , H ]

eadYp(H)= H

+1[Y,+Z,,[YP+Z,,H]J

+ [Y,, H] + icyp, [Y,, HI] +

+

.

a

*

,

. . a .

Both expansions are finite. Their difference belongs to P o , HI

+ wpn,

so using ( 5 ) we obtain

ead(Yp+ZO)H EH

+ Z + qPn.

This proves (4) and the lemma. The group K acts on the set of 51,-triples (H, X, Y), normal or not, by k.(H,X, Y) = ( k . H , k * X , k *Y). Proposition 4.6. Any X E JV - (0) is contained in a normal 51,triple. Any two such triples are K-conjugate. This induces a bijection between the set of all K-orbits in JV - {0} and the set of all K-conjugacy classes of normal 51,-triples. Proof. By [DS] (Chapter IX, Theorem 7.4) used on gR, there exists an

51,-triple (H*, X, Y*) containing X. We now modify it to make it normal. Let H* = H + H, (HE f, H, E p). Then 2 x = [H*, XI = [H,XI

+ [H,,XI,

so [H, X] = 2X, [H,,X] = 0. Writing similarly Y* = Y : + Y,: we conclude from [X, Y*] = H* that [X, YX] = H. By Lemma 7.6 in [DS] (Chapter IX), the relations [H,X] = 2X, H E [X,91 imply that there exists an element Y E g such that relations (3) hold. Decomposing again Y = Y + Y, we see from (3), since H E € , X E ~ that , (H, X, Y,) is also f an 51,-triple. Thus by Lemma 4.4, Y = 5 , so (H,X, Y) is a normal 51,triple.

3 72

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

Suppose (H', X,Y ' ) is another normal 51,-triple containing X . Let - H so [ X , H , ] = 0 and H, E f. The centralizer gx of X in g is invariant under ad H . From 01 of the Appendix used on CH + CX CY we know that ad H and therefore the restriction (ad H)IgX is semisimple and also that the eigenvalues of ad H on gx are in 2'. Let g: c gx be the subspace spanned by the eigenvectors for ad H with strictly positive eigenvalues. By Corollary 1.5 of the Appendix,

H, = H'

(6)

+

g: = gx n [ X , 91.

The centralizer fX of X in f is invariant under ad H. Putting :€ fX n g : , we have from (6) f: = fX n [X, g], which clearly contains

=

H, = [X, Y - Y]. Now by the Jacobi identity g: is a subalgebra which is readily seen to be nilpotent. Hence €5 is nilpotent and we obtain from Lemma 4.5 exp(ad €T)H= H + €5. (6') In particular, there exists a k E K such that k X = X and kH = H H, = H . Then Lemma 4.4 implies Y' = kY. We have now proved the two first statements of the proposition. The last one follows immediately, the mapping K X --+ K ( H , X , Y ) being the desired bijection.

+

Lemma 4.7. Let ( H , X , Y ) and (H', X , Y') be two normal sl,-triples. Then they are K-conjugate if and only i j H and H are K-conjugate. Proof. We may assume H = H', and have to show that if ( H , X , Y) and ( H , X', Y') are normal sI,-triples they are K-conjugate. Let go, fo, and po denote the centralizers of H in g, f, and p, respectively. Since 0H = H we have go = fo po. Let KO c G denote the analytic subgroup corresponding to ad,(€'), and let g2 c g, f2 c €, p2 c p be the eigenspaces of ad H for eigenvalue 2. Again OH = H implies g2 = f2 + p'. Also [go, g'] c g2 so [f', p2] c p2. For Z E p2 consider the map @': KO -,p' given by @'(k) = Ad(k)Z. Its differential is d e ( T ) = [T,21. Let V c p2 be the set of 2 for which d@f is surjective; i.e., [fo, Z] = p'. We claim that X, X' E V. From Corollary 1.5 of the Appendix applied to CH + CX + CY acting on g we deduce [X, go] = g2. From the direct sum decompositions of go and g2 we then deduce [ X , €O] = p' so that X E V. Similarly, X' E V. Consider now the linear mapping -d@f, that is, ad Z : fo + p2 in terms of fixed bases of f0 and p'. Then surjectivity amounts to at least one of the minors of order dim p2 being fO. Thus V is the union of open, intersecting, connected sets ( [ D S ] ,

+

$41

373

4. The Orbit Structure of p

Chapter VII, Cor. 12.5); hence V is connected and open in p2. However, V is invariant under KO and for each Z E V , KO . Z is open in V. Thus V consists of just one KO-orbit. Hence X' = k . X for some k E KO c K , so that k ( H , X , Y ) = ( H , X', k Y ) . But then k Y = Y' by Lemma 4.4. Theorem 4.8. There are only jinitely many K-orbits in .K Proof. By Proposition 4.6 it suffices to show that there are only a finite number of K-conjugacy classes of normal 51,-triples. If X denotes the set of H E f for which there exist X , Y Ep such that ( H , X , Y ) is a normal 51,-triple it suffices by Lemma 4.7 to show that X (which is clearly stable under K ) consists of finitely many K-orbits. Now by [DS] (Chapter 11, Proposition 6.6), we have the direct decompositions

(7) where jo c f,, j c t are the respective centers. Let Ijlo be a maximal abelian subalgebra and let br be the subalgebra of t generated by Then $! is a Curtan sirbulgebra off (i.e., the direct sum of j with a Cartan subalgebra of [f, €3). Let ljo c go be a maximal abelian subalgebra containing $!, and let $ = b0 + ib,. Then H , E $, implies [OH,+ H,, $J = O so OH, + H , E bl0, whence 6 H , E bo. Thus $, = br, @ (I), n pol, so bo consists of semisimple elements in go. Thus is a Cartan subalgebra of g. By [DS], Chapter V, Theorem 6.5, all Cartan subalgebras of [f, f], and hence all Cartan subalgebras of f, are K-conjugate. By Corollary 1.5 of the Appendix each H E X is semisimple and so is each component of H in the decomposition (7) (since each element in 3 is semisimple and commutes with H). By [DS] (Chapter IX, Proposition 4.6) for [f, €IRwe conclude that H is conjugate to an element in 5,. Also, by $1 of the Appendix, the eigenvalues of H are integers with norm less than dim g. In particular, 0 I Ict(H)I I dim g for each root ct of g with respect to 6. But each ct(H) is an integer and an element in lj is determined by the values the roots take on it. Thus the intersection K . %' n br is finite and the theorem is proved. 3. Regular Elements We next turn to the set r of regular elements in p as defined in $4. No. 1. For X E p, (ad X), maps p into itself and we consider the characteristic polynomial of the restriction (ad X ) * Ip, (8)

det(U - (ad X ) , I p ) = A"

where a, $ 0.

+ a,-

I(X)A"-l +

+ al(X)A1,

374

[Ch. I11

INVARIANTS AND HARMONIC POLYNOMIALS

Proposition 4.9. The element X E P is regular a,(X) # 0. I n particular, t is connected.

if and only

if

Proof. Using [DS] (Proposition 1.1, Chapter 111) on (ad X)'(p, we

have det(A1 - (ad X)' I p)

=

+ . + b(X)),

Adl(X)(A"n-dl(X)

where d,(X) = dim p(X, 0), b(X) # 0. Thus d,(X) 2 I with equality holding if and only if a,(X) # 0. But d,(X) = 1 if and only if X is regular, so the proposition is proved (cf. [DS], Chapter VII, Corollary 12.5 for the connectedness). For X E p, (ad X)' maps € into itself, so, in analogy with p(X, A), we define for A E C €(X, A) = { T E f ; ((ad X)' - A)m(T)= 0 for some m}. (9) Proposition 4.10. An element X

E

p is regular

if and only

if

dim €(X, 0) = min dim €(Y, 0). YEP

Proof. Consider the eigenspace decompositions of (ad X)' according to [DS] (Chapter 111, 41):

(10) (1 1)

P = P(X, 0) + p(X, A,)

+

+ p(X, 1,) f = t(X, 0) + f(X, P I ) + . * . + t(X, pm) 1 . .

(1,Z 01, (pj # 0).

Since ad X maps f into p and p into f, it follows immediately from the definitions that ad X maps p(X, A,) into f(X, A,) and maps f(X, p j ) into p(X, pj). Hence (Al,.. . , A k ) = ( p l , . ..,/A,) ( m = k). Also, if u # 0 in p(X, A,) (resp. f(X, A,)), then ad X(u) # 0, so the mapping f(X, A,) is a bijection. Thus (10) and (11) show that dim p(X, 0) and dim f(X, 0) reach their minimum value for the same X, so the proposition is proved. In analogy with the definition of a Cartan subalgebra in [DS] (Chapter 111, §3), we make the following definition. ad X: p(X, Ai)

Definition. A Cartan subspace of p is a subspace a c p satisfying the conditions (i) a is a maximal abelian subspace of p. (ii) For each X E a, ad, X is semisimple.

If a, is a maximal abelian subspace of p, its complexification + ia, is a Cartan subspace. Following the method of [DS] (Chapter 111, §3), we shall now give a more general construction. a,

§41

4. The Orbit Structure of p

375

Theorem 4.11. Let H E p be regular. Then H is semisimple, p(H, 0) equals the centralizer pH of H in p and i s a Cartan subspace of p. Proof. The proof is similar to that of [DS] (Chapter 111, Theorem 3.1). Let Al,. . . , be the nonzero eigenvalues of (ad H)’;put

and for any subspace 5 c g and an element YE^ let 5” denote the centralizer of Y in 5. Let ad H = S + N be the decomposition of ad H of g into its semisimple part S and its nilpotent part N. Then it is known (Appendix, Prop. 2.7) that S = ad H,, N = ad H,, where H,, H,, E g. Since OH = -H the uniqueness of the decomposition implies H,, H , E ~ . Also, (ad H)’ = (ad H,)2 + N o , where N o = 2 ad H, ad H,

+ (ad H,,)’.

Since this is a nilpotent operator commuting with (ad HS)2, (ad H)’ has semisimple part (ad H,)’.Thus by [DS] (Prop. 1.1, Chapter 111) and the semisimplicity of ad H,, (13)

a = p(H, 0) = p(H,, 0) = pHa.

Similarly, since (ad H,)’- A is the semisimple part of (ad H)’ have (14)

- Iz

we

P(H, 4= p w , , 4.

But if Y E pHs then (ad Y)’ maps p(H,, A) into itself, so as in [DS] (Chapter 111, $3) we can for each Y E p(H, 0) consider d( Y) = det((ad Y)’ 1 p’). By (12) and (14) d(H,) # 0, so, since d is a polynomial function on a, we have d(Y) # 0 for all Y in a dense subset of a. For all such Y, (ad Y)2 is nonsingular on p’ and maps each p(H, Ai) into itself. Thus if Z E p(Y, 0) and we decompose Z by (10) (with X = H) we find Z E p(H, 0). Hence p(Y, 0) c p(H, 0), so by the regularity of H, p(Y, 0) = p(H, 0). Thus (ad Y)’ I a is nilpotent; by continuity this is so for all Y E a. The same reasoning with f instead of p gives (using Proposition 4.10) (ad Y)’ IfHais nilpotent for each Y E a. Combining these results and the relation gHa= fHs + pHs,we find (15)

(ad Y) I gHs is nilpotent for Y E a.

376

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

The semisimple element H, lies in a Cartan subalgebra I), of g, so the centralizer g H a is reductive and has the form (16)

gH* = g'

+ center(gHa),

g' =

[gHs, gHs],

where the ideal g' is semisimple (cf., [DS], Chapter IX, $4;Chapter 111, Exercise B.l). If Y E g' n a then by (15) ad Y is nilpotent on g', so for the Killing form B' on g', B'(Y, Y) = 0. The orthogonal decomposition g' = g' n fHa + g' n a therefore implies B'(g', g' n a) = 0. Hence g' n a = 0 and g' c f ; so, since the decomposition (16) is &stable, a c center(#"), which in turn is contained in bS. This proves the theorem. We shall now prove, in analogy to the real situation ([DS], Chapter V, Lemma 6.3) that all Cartan subspaces of the complex space p are conjugate under K. Let a c p be an arbitrary Cartan subspace. In analogy with [DS] (Chapter VI, Lemma 3.6, and Chapter VII, Lemma 11.3), we consider the joint eigenspace decompositions of the operators (ad,(H))' (H E a). For A E a*, put g, = { Z : [ H , Z] = A(H)Z for H

Then dg, = g-,,

g=

g,, and g,

+ g-,

=

f,

E

a}.

+ p,,

where

f, = {TEf : (ad H)'T = A(H)'T for H E a}, p,

=

{XE p : (ad H)'X

=

A(H)*X for H

E

a}.

Let A(a) = {A E a* : A # 0, g, # 0) and put a' =

{H E a : A(H) # 0 for A E A(a)}.

Since H E a is semisimple, ad H and (ad H)' have the same null space. Hence fo = m, the centralizer of a in f, and p o = a. Also, if A€A(a), H E a', then ad H interchanges f, and p, bijectively. Thus (17)

f=m+

Cf,, 1 E A(4)

Cp,.

p=a+

1E A(o)

Comparing now with (8), we see that the restriction alla equals the product of the members of A(a) with multiplicities. This gives the following description of the set r n a. Lemma 4.12.

Let a' = { H Ea : A(H) # 0 for all

A E A(a)}.

§41

4. The Orbit Structure of p

377

Then if H E a', we have pH = a, X E a : d i m px = mindim p" Y€0

X

E a : dim fX =

min dim tY Y€a

Corollary 4.13.

The set K . a' is open in p.

For this it suffices to verify that the mapping (k, H) + k . H of K x a' into p has a surjective differential at (e, H ) and hence at (k, H) E K x a'. For this we must show [f, H ] + a = p. However, this is obvious from (17) since ad H(f,) = pi. Theorem 4.14. Any two Cartan subspaces of p are conjugate under

K . Moreover, (18)

r

=

K a'.

Proof. Each X E r lies in a Cartan subspace so if a, ( i E I ) are the K . a;. If two summands, say Cartan subspaces of p, we have r = K . a; and K . a; intersect we can select H i E a;, H j E a;, which are conjugate under K. But then, by Lemma 4.12, p"' (=a,) is conjugate to p H J (= aj), so the two summands coincide. Using Corollary 4.13 and the connectedness of r we deduce that all summands coincide and that all a, are K-conjugate. The argument also proves that r = K a'.

u,

Corollary 4.15. The regular elements in p are the semisimple quasiregular elements; i.e.,

(19)

r=Yn9.

For this let X E r. We have seen that X E Y and that f(X, 0) = fX has dimension dim m [cf. (17), (18)]. By its definition, W is the set of points X E p for which the map k + k - X of K into p has maximal rank; expressing this by rank of Jacobians we see that W is open in p; thus by Proposition 4.9, r n 9 # 0. Also, since fZ is the Lie algebra of the isotropy subgroup of K at 2 we see that W is the set of 2 E p minimizing dim fZ. Select 2 E r n 9.Then dim fX = dim m = dim kZ = min dim f", YED

378

so X then

INVARIANTS AND HARMONIC POLYNOMALS E 9. This

[Ch. I11

proves r c Y n 9. On the other hand, if X E Y n 9 dim f(X, 0) = dim fX = min dim f Y YEP

I min dim €(Y, 0), YEP

so

x € r. 4. Semisimple Elements

Proposition 4.16. Let a c p be a Cartan subspace and let p, = Y denote the set of semisimple elements in p. Then

Y = K-a.

This statement is similar to [DS] (Chapter IX, Proposition 4.6) about Cartan subalgebras, and the proof is also quite analogous. Let H €9. Then ad H maps p into f and dim p = dim p" + dim ad H(p). Consider now the mapping T : pH

x (ad H ( p ) ) -,p

+

given by r ( X , T ) = eadT(X H ) . Its differential at (0,O)is given by d r o ,o,(X, T ) = X

+ [T, HI.

If this vanishes then (ad H)'(T) = 0. Since T = ad H(Y), ( Y E p) we have (ad H ) 3 ( Y ) = 0 so the semisimplicity of ad H implies T = 0 whence X = 0. Thus the image r(pH x (ad H ) p ) contains an open subset of p and therefore a member of r. Thus p H contains an element X such that Z = X H E K . Then pz is a Cartan subspace containing H. Because of Theorem 4.14 this proves the proposition.

+

Lemma 4.17. Let X E p and let X = X , + X , be its additive Jordan X, E JV (since OX = -X). decomposition so that [ X , , X,] = 0, X,E 9, Then X,EK T (closure). Proof. We may assume X, # 0. Then X n ~ J l r n g X aBy . [DS] (Chapter 111, Exercise Bl), gxa is reductive and its center consists of elements which are semisimple in g. By Propositions 2.12 and 2.16 of the Appendix, gxs is reductive in g, and X , is a nilpotent element in the semisimple algebra g' = [gX: s'*]. But then by [DS](Chapter IX, Theorem 7.4), there exists an element H E g' such that [H, X , ] = X , . Decomposing

§41

379

4. The Orbit Structure of p

+

H according to g x s = fXs px", we may take H E g' n f. But then the group U = exp(ad CH)satisfies U c K,

Hence X , E U

U * X= X ,

+C*.X,,

c* = c - (0).

xcK T .

From this lemma and Theorem 4.3 we have Corollary 4.18. Let j be a K-invariant polynomial on p without constant term. Then

j(X) =jM,>,

A X , ) = 0.

Theorem 4.19. Let X E p. Then the orbit K . X is closed if' and only i j ' X is semisimple. Proof. From Lemma 4.17 we know that a closed orbit necessarily consists of semisimple elements. On the other hand, suppose X E p is semisimple. Let X' = X: + X ; be in closure (K X)-.With j as in Corollary 4.18 we have j ( X : ) = j ( X ' ) = j ( X ) = j ( X , ) . Because of Proposition 4.16, Corollary 3.12, and Chapter 11, Corollary 5.12 (extended in an obvious way to complex a) we have X , = k . Xi for some k E K. Then k . Xi belongs to the centralizer gx* which is reductive in g. By Prop. 2.16 and Cor. 2.10 in Appendix, k - X i E g' = [gx*, gX1] and an element in g' is nilpotent if and only if it is nilpotent as an element in g. Thus, by Theorem 4.8, we can fix elements Z , , . . . , Z, E n gxs such that each nilpotent (in g) element in gx=n p is conjugate to some Zi under the subgroup K X s of K fixing X , . It follows that k . Xk = u . Zi for some ti E K x a and some i. This proves that X' = Xl + X i = k - ' X , k - ' u Z i = k - ' u ( X , + Zi) SO

+

r

(K * X)- =

UK

. ( X , + Zi).

i= 1

The space ( K . X ) - is locally compact, so, by the Baire category theorem ([DS], Chapter 11, @3), one of the orbits K . ( X , Zi), say 0,= K . ( X , + Zl), has an interior point in ( K X ) - . By homogeneity, 0 , is open in (K . X ) - . Then the complement (K . X ) - - 0 , contains an open orbit, etc. Thus (K X)- = 0,u . .. u 0, where for each i, Oiu ... u 0, is closed and contains the orbit Oi as an open subset. In particular 0, is closed. By the first part of the proof it consists of semisimple elements. On the other hand, the relation X , = k X : above showed that the semisimple elements in (K . X ) - form a single orbit which thus must be 0,. In particular, X E 0,, so K X = 0,, which is closed.

+

380

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. 111

5. Algebro-Geometric Results on the Orbits In this subsection we describe two basic results from Kostant and Rallis [1971] about the K , orbits in p. The proofs will be omitted since they require several tools from algebraic geometry. These results as well as those in $5 which rely on them will not be needed in the present volume. As in $4, No. 2, consider the symmetric algebras S(p), S(p*) and the subalgebras I+(p) c I ( p ) c S(p) and I + @ * ) c I(p*) c S(p*). Similarly, let I(a) and I(a*) denote the spaces of Weyl group invariants acting on S(a) and S(a*), respectively. By Corollary 5.12 of Chapter 11, the restriction p + p ( a induces an isomorphism of I(p*) onto I(a*). By Lemma 4.2 the groups KO and K induce the same transformations of a; consequently each p E I(p*) is &-invariant. In accordance with the stated isomorphism and Theorem 3.1, let j l , . . . ,j , be homogeneous algebraically independent generators of I(p*) and let j : p + C' denote the mapping j ( x ) = ( j l ( x ) ,. . . ,j , ( x ) ) (x E p). According to Corollary 3.12 and Theorem 4.3, j is surjective and j - ' ( O ) = A As before, let W denote the set of quasiregular elements in p. Theorem 4.20. The intersection W n JV is a dense open sitbset o j JV and is a single KO-orbit. More generally, if ( E C', W nj - ' ( ( ) is a single KO-orbit and is open and dense in . j - '(5). For 5 = (tl,. . . ,5') E C' let I, = I,(p*) denote the ideal in I(p*) gener-

ated by ( j k - ( k ) ( k = 1, ..., l). Then I, S(p*) is the ideal in S(p*) generated by I,. The corresponding set of common zeros in p is just j - '(5). According to Hilbert's Nirllstellensatz a polynomial j~ S(p*) vanishes identically on j - ' ( ( ) if and only if some power of it belongs to I,S(p*). Here a stronger result holds. Theorem 4.21. Let 5 E C' and j~ S(p*). Then j'Ij-'(()

=ooj~I~S(p*).

In other words, the ideal I,S(p*) is eqiral to its own radical. 45. Harmonic Polynomials on p

We recall that the elements in I @ * ) are also K O invariant. Just as in $1 we call a polynomial h on p harmonic (or K-harmonic) if d(j)h = 0 for each j E I + ( p ) . Let H(p*) denote the space of harmonic polynomials.

051

38 1

5. Harmonic Polynomials on p

From Corollary 1.9 and Theorem 4.21 (for 5 lowing result.

we now have the fol-

= 0)

Theorem 5.1. The harmonic polynomials on p are spanned by (X*)k (k = 0,1,. . .) where X runs through X E X

We next prove an analog of a statement in Theorem 3.4, sharpening Theorem 1.1. Theorem 5.2.

The linear mapping

4: m*) €3 W P * )

+

m*)

given by 4(j €3 h) = jh is a bijection.

The surjectivity of 4 being clear from Theorem 1.1, we have to prove that if h l , . . . ,h, E H(p*) are linearly independent and i,, . . . ,i, E Z(p*) are linearly independent and if for some a,, E C,

4CC as,4 €3 h,)

=

C as,ishr = 0,

then all a,, = 0. Equivalently, we must show that if j l , .. . , j p E Z(p*) satisfy h,j, = 0 then j , = 0 for each r. First we observe that the restrictions h , l N (1 I iI p ) are linearly independent. This in fact follows from Theorem 4.21 (for t = 0) and the directness of the decomposition S(p*) = H(p*) + Z+(p)S(p*) [cf. Eq. (8) of $13. Using (by Theorem 4.20 for = 0) an element X E .N for which KO - X is dense in N it follows that the functions hi on KO (1 5 i I p ) given by h:(k) = h i ( k . X ) are linearly independent. This means that as k runs through K O the vectors (h,(k-' . X ) , . . .,h,(k-' X)) span CP.Thus we can select k,, . . .,k,E KO such that the translated polynomials prs = k, h, (1 I r, s 5 p ) satisfy det(p,,(X)) # 0. By continuity, det(p,,(Y)) # 0 for all Y in some neighborhood U of X . But then the functions h,l(KO* Y)(1 5 r 5 p) are linearly independent for each Y E U. If the polynomials j,, . ..,j , were not all 0 there would exist a Y E U such that the vector (jl( Y), .. . ,jp(Y)) # 0.. Since each j , is constant on K O Y it follows that j,h, IKO * Y f 0, contradicting the relation j,h, = 0.

<

E=

Corollary 5.3. For each

< E C' the direct decomposition

Q*)= I,W*) @ W * ) holds. Proof. Let i E Z(p*) be a homogeneous element. Writing

(1)

i =

C akl . . . k l j i l (k)

* *

'j:' =

C akl . . . k l o l (k)

- t l ) k'l. 0'1 - ti)k'+ if. '

382

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

where deg(i') c deg(i) we obtain by iteration the direct decomposition I(p*) = I, 0 C. Consequently,

m*)8 H(p*) = (1, 8 H ( p * ) ) CD H(p*).

(2)

Applying the mapping

4 in Theorem 5.2, we obtain

(3)

S(P*) = I , H ( p * ) 0 H(p*),

the directness coming from the injectivity of 4. But since S(p*) = I(p*)H(p*) we have I,S(p*) c I , H ( p * ) c I,S(p*) so (3) gives the corollary. Remark. So far we have only used Theorems 4.20 and 4.21 for the

case

< = 0. In the next result we shall, however, need the general 5. Let

Theorem 5.4.

r be a &orbit h E H(p*),

(4) then h

in p of maximal dimension. 'I

hlr = 0,

= 0 on p.

Proof. Pick X E r and put 5 = j ( X ) , the map j : p + C' being as in §4. Thus r c 9 nj - ' ( { ) . But then Theorem 4.20 implies that r equals W n j - ' ( < ) and is dense in j-'((). Thus (4) implies that h vanishes identically on j - ' ( { ) , whence by Theorem 4.21 and the directness in Corollary 5.3, h = 0.

EXERCISES AND FURTHER RESULTS 1. Let Q ( X ) = x i

0

+ a2

+ x i - x:+' - ... - X P2 + 9 ,

= -+

ax:

a2 a2 a2 ... +2 - * . . - -, ax, - t axp+1 ax:+,

and C the quadric Q ( X ) = 1. Show that for each polynomial P on RP+g there exists a polynomial H on R p + qsuch that

OH

= 0;

H = P on C.

2. With the notation of 53, No. 4, let G / K be the symmetric space corresponding to the Cartan involution €J of g. Let M be the centralizer of a in K. Given I E a: let A, E a' be determined by B(H, A,) = I ( H ) ( H E a) and let K , denote the isotropy subgroup of K at A,. The element I is

Exercises and Further Results said to be regular if a(A,) # 0 for each restricted root 1 is said to be singular. Prove the following statements:

383 a#

0; otherwise

(i) l E a,* is regular o K , = M o al # Iz for each a # e in W. (ii) Let &,(p) denote the joint eigenspace

&,(PI

=

If E &(PI: W f = P ( w - for P E W I .

The K-invariant functions in &,(p) are the constant multiples of the function IC/~(X) = j:iB(k.

AA. X )

dk

(XE PI,

dk being the normalized Haar measure (cf. Prop. 4.8, Chapter IV). Show that if h is a K,-harmonic polynomial on p then the function

f(x)= j(X)eiB(""*x ,

(X E P)

belongs to &,(p). (iii) Let h E S ( p ) be an M-harmonic polynomial. Then, if 1 E a,*, h(k X)eiB'AA*k'X) dk = h(O)IC/,(X).

3. In the notation of 53, No. 4 the natural representation of W on the space H of harmonic polynomials is equivalent to the regular representation of W (Chevalley [1955a]). 4*. In the notation of Theorem 3.3 let for k (0 5 k I n) wk denote the set of a E W which fix some k-dimensional subspace of E but fix no subspace of higher dimension. Let wk = Card(Wk). Then n

n

generalizing both identities of Theorem 3.3 (cf., Solomon [1963]).

5*. Let a + I(a) be the length function on the Weyl group (cf. Chapter IV, $6). Then

(cf., Bott [1956] and Solomon [1966]). Deduce from Theorem 3.4 that dim H k = Card{a E W: I(a) = k).

384

INVARIANTS AND HARMONIC POLYNOMIALS

[Ch. I11

6*. Let R be an irreducible reduced root system, B = { a 1 , .. . , al} any mi. Then basis. For each a E R’, a = miai, put m(a) =

c:

I

(cf. Kostant [1959a]). 7 . Show that the group F from Lemma 4.2 has cardinality 2‘ if I = dim, a. 8. With the notation of $4 show that p E S(p) is K-invariant if and only if it is &-invariant (use Lemma 4.2 and Corollary 5.12 of Chapter 11). 9*. Let (If,X, Y) be a normal &-triple where X, Y E JK Then the plane Y + p x intersects each &-orbit in W exactly once. Furthermore, the restriction mapping u + ul(Y

+ pX)

is a bijection of Z(p*) onto S(Y + p x ) (cf. Kostant and Rallis [1971], Chapter 11, $3; here S( Y + p x ) = S(p*) I (Y + px)).

NOTES $1-82. Theorems 1.1, 1.2, 1.7, and 2.1 are from Helgason [1962, 19631. Theorem 1.1 was proved independently by Kostant [1963], and Corollary 1.9 was proved in a different way by Maass [1959]. An example [for G = SO(2n l)] for which the ideal J = I:(E*)S(E*) is not its own radical was given by Hesselink [1979]. If p is a finitedimensional irreducible representation of SL(2, C), combined results of Popov and Dixmier [1981] show that J = if and only if dim p 5 5. Theorem 1.11 is from Molien [1898]. Corollary 2.2 (Lepage decomposition) is proved in a different way in Weil [l958] (Chapter I); it is a “flat” analog of the Hodge decomposition of harmonic forms on Kahler manifolds. $3. Theorems 3.1 and 3.4 are due to Chevalley [1955a], and Theorem 3.3 is used in his paper [1952]. A generalization is given by Solomon [1963] (cf. Exercise 4), where Proposition 3.10 is also proved. Theorems 3.6 and 3.13 are proved in Steinberg [1964]; the latter result was also proved in Harish-Chandra’s unpublished paper [19601. Corollary 3.12 occurs in Kostant [1963] (Proposition 3.10) with a different proof. Theorem 3.16 was proved by the author [1964a] and independently by Harish-Chandra. In 53 we have used some proofs from Flatto [1978]. & is lmostly due to Kostant and Rallis [1971]. We have sometimes [say, for (6)] given more elementary proofs in order to make the exposition more self-contained. Theorem 4.19 is from Bore1 and Harish-Chandra [1962], $10 (in the group case) and Kostant and Rallis [1971], Prop. 16, in the general case. The proof of the “if” part is a generalization of the proof of Proposition 7 of $1 in Varadarajan [1977].

fi

&

+

CHAPTER IV

SPHERICAL FUNCTIONS A N D SPHERICAL TRANSFORMS In this chapter we begin a detailed study of spherical functions and their applications to analysis on a semisimple Lie group. Since these functions arise as representation coefficients [see (8) in $41 we begin in $1, No. 1 with a review of some standard notions and results in representation theory. In 41, No. 2 we review some facts about representations of compact groups K and relate these to representations of a bigger group G =) K. $32, 3 deal with elementary properties of spherical functions on arbitrary Riemannian homogeneous spaces. In $4 we specialize to the three types of symmetric spaces, prove specific integral formulas for the spherical functions, and exhibit them as representation coefficients. The case of a complex group G has some special features, exemplified by Theorem 4.7 and Proposition 4.10. In 555, 6 we give an explicit expansion of the spherical function into ordinary exponential functions. Not only does this expansion describe the behavior of the spherical function at infinity, but in addition it can be used to relate the theory of the spherical transform to ordinary Fourier analysis. This serves as the basis of the Paley-Wiener theorem in $7, where, in addition, detailed information is required about the coefficients in the indicated expansion. In 47 we also prove the inversion formula and the Plancherel formula for the spherical transform. $8 is devoted to the determination of the bounded spherical functions, or, equivalently, the maximal ideal space of the algebra of K-bi-invariant integrable functions. $9 deals with the Paley-Wiener theorem for the spherical transform on the flat space p where the spherical function expansion is not available. In $10 we prove geometric properties of the Cartan and Iwasawa decompositions which are then applied to spherical functions.

$1. Representations 1. Generalities

In the preceding chapters representations have played a rather peripheral role; they will now be involved in a more significant way. In this subsection we therefore develop some of the basic notions from the theory of infinite-dimensional representations; occasional results from topological vector space theory will be used without proof. In Chapter II,§4, we defined the concept of a representation 7c of a locally compact group G on a locally convex topological vector space I/ 385

386

SPHERICALFUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

over C ; we also defined the concept of irreducibility of R . It is sometimes convenient to have the following reduction in the assumptions. Lemma 1.l.Let G be a locally compact group and V a FrCchet space. Let K : G + Aut( V ) be a mapping satisjying the conditions

0) ~ ( x Y=) ~(x)nCv), (x, Y ) E G, d e ) = 1. (ii) For each u E V, the mapping x + n(x)v is continuous from G to V. Then the mapping (x, v ) -,R(X)U from G x V into V is continuous so is a representation of G on V.

K

Remark, For the proof, which is a variation of that of the BanachSteinhaus theorem, see, e.g., Bourbaki [Integr. Chapter VIII, $2, Proposition 11. The lemma actually holds for barreled topological vector spaces; in particular, it holds if V is the inductive limit of Frkchet spaces, and also if V is a Monte1 space. Thus Lemma 1.1 applies to the function spaces 9 ( M ) , b ( M ) and their duals 9 ’ ( M ) , b ’ ( M ) , M being a manifold with a countable base. Let F be any function space on G invariant under left translations such that the mapping (9,f ) + f L @ )is continuous from G x F into F . The representation 1 of G defined by I(g)f = f L ( # )is called the left regular representarion of G on F. Similarly, one defines right regular representations r by r(g)f = f R ( g - ’ ) . Let V ’ denote the dual space of V. If v E V and A E V’ the function g 4 A(n(g)v) is called a representation coeficient. In harmonic analysis on G one investigates these functions as well as how they can serve as building blocks for “arbitrary” function spaces on G . Two representations (K,, V,) and ( x z , V,) of G are said to be equivalent if there exists a linear homeomorphism A of V, onto V’ satisfying

(1)

A R , ( x )= R ~ ( x ) A , x E G.

A continuous linear mapping A of V, into V, satisfying (1) is called an intertwining operator. If V is a Hilbert space and R a representation of G on V with each n(x) unitary then R is called a unitary representation. Suppose now G has a left and right invariant Haar measure dx and that the locally convex space V is complete. For the representation R of G we define for each f E C,(G) the continuous linear transformation ~ ( f )V : - , V by R(f)V =

s,

f ( x ) n ( x ) v dx,

v E V.

0 I1

1. Representations

387

Suppose now G is a Lie group with Lie algebra g. A vector U E V is said to be differentiable if the mapping 7,: x + n(x)u of G into V is differentiable. Let V" be the space of differentiable vectors in V. It is clearly invariant under n(C). Moreover, if Y is a vector field on G the mapping x + (Yz,)(x) of G into V is also differentiable. If X E g, u E V m ,we put 1

n(X)u = lim - [n(exp tX)u

(3)

t-+o

-

u],

t

knowing that this limit exists (as an element of V). Lemma 1.2. For X E g we have n ( X ) V m c V" and the mapping --+ n ( X ) is a representation of g on V m .

X

x

Proof. Let be the left invariant vector field on G , determined by X . Then if V E V"

(4) n(x)n(X)u = (Bz,)(x), so n(X)v E Vm. Let YE g and replace x in (4)by x exp t Y ; then we obtain (5)

( n ( ~ ) nY().- n( Y )n(x))u = ((13P - PX)t,)(e).

But if I is any continuous linear functional on V we have [since 1 0 t, is a scalar-valued function] I 0 ([X,Y ] - ( t , ) ) = [ X , Y ] - ( I t,) = (139 = I o ( ( 1 3 9 - P8)t"). By the Hahn-Banach theorem, we conclude 0

PX)(A

0

z,)

[ X , Y]-(z,) = (238 - W)(t,), so by (4)and ( 5 ) n ( X ) n ( Y ) - n ( Y ) n ( X ) = n ( [ X , Y ] ) , proving the lemma. Lemma 1.3.

Let f~ 9(G).Then n(f')V c V".

Proof. We have for u E V

so (5')

n(x)n(f)o = n(fL'"')u.

388

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Thus the mapping z,(/)" is the composite of the differentiable mapping x +J'(x) of G into 9 ( G ) and the continuous linear mapping f - t n ( f ) u of 9 ( G ) into V ; hence z,(/)~ is differentiable as desired. Corollary 1.4.

V" is dense in V.

Proof. Let J E ~ ( G ) be positive, j G f ( x ) d x = 1, and u E V,

E

> 0.

Then F

n ( f ) ~- u

=

J

f(x)(x(x)u - U) dx,

G

and if q is one of the seminorms defining the topology of V there exists a neighborhood U of e in G such that q(n(x)u - u) < E for x E U. But then if f has support contained in U, (6) implies q ( a ( f ) u - u) I E, whence the corollary. Definition. The representation of g on V" defined by Lemma 1.2

is called the diflerential of

A.

It is denoted by

A

(and sometimes by dn).

In view of [DS) (Chapter 11, Proposition 1.1), a induces a representation of the universal enveloping algebra U(g) on V". The representation A of G on V is said to be quasisimple if A maps the center of G into scalars and if the differential d n maps the center of U ( g ) into scalars. (This definition differs slightly from that in Harish-Chandra [1953] (§lo).) Again let V be an arbitrary locally convex space. Let V' denote the dual of V and let V' be given the strong topology (that is, the topology of uniform convergence on bounded subsets of V). The mapping which sends a vector u E V into the continuous linear form fi: 1 + 1(u) on V' is an injective map of V into the second dual space (V')'. If this map is surjective, that is, if P = (V')', then V is called semirejexiue. It is known that any closed subspace W of a semireflexive space V is again semireflexive, (Kothe [1969]). If A is a continuous linear transformation of V its transpose ' A : V' -t V' is defined by ('A(l))(u) = A(Au); it is again continuous. If A is a linear homeomorphism of V onto itself then ('A)-' = ' ( A - ' ) . Let A be a representation of a locally compact group G on a locally convex space V. The mapping it: g + '(n(g-')) is a homomorphism of G into Aut(V'). If V is sernirejexiue, it is known that the mapping (9, A) --* ic(g)1 of G x V' into V' is continuous (Bruhat [1956], $2) so ii is a representation; it is called the representation contragredient to n.

611

1. Representations

389

If V is a Hilbert space with inner product ( , ) then the mapping which to each u assigns the linear form v':w + ( w , u ) is a conjugate linear bijection of V onto V'. If A: V + I/ is a continuous linear mapping with adjoint A*, we have 'A(u') = (A*u)'. We shall now establish an important connection between unitary representations and positive definite functions. Definition. A complex-valued continuous function compact group G is called positive definite if

4 on a locally

i. j = 1

for all finite sets x l , . . . ,x,, of elements in G and any complex numbers

a ] , . . . ,a,,.

4 satisfies the conditions m- = m, I4 w I 4 ( 4

A positive definite function

4(4 2 0,

which are easily derived from the definition. In particular, a positive definite function is necessarily bounded. Let n be a unitary representation of a locally compact group G on a Hilbert space fi. Let ( , ) denote the scalar product in fi. For each fixed vector e E fi, the function x + ( e , n(x)e) is a positive definite function on G. The continuity is obvious, so let x l , . . .,x , be any finite set of elements in G and a l , . . ., ct,, any complex numbers. Then n

C

a

(e, n(x; lxj)e)aiEj =

C i.

(n(xi)e,n(xj)e)aiiij

j= 1

i, j = 1

On the other hand, a positive definite function 4 -f 0 on G gives in a canonical way rise to a unitary representation of G as follows: Let V' denote the set of all complex linear combinations of left translates 4L(x) (x E G) of 4. We define a scalar product in V, by the formula (7)

cj

-

if f = X i ai4L'xi),g = flj4L'yJ). Now ( J , g ) = aig(xi)= x&fcYj) so it is clear that (7) depends only on f and g but not on their special expressions in terms of 4. Since ( f , f ) 2 0 we have the Schwarz inequality [(f,g)I2 I(f, f ) ( g , g ) . Since f ( x ) = (f, 4L(x)) it shows that (f,f)=O if and only i f f = 0. The completion of V, in the norm l l f l l = (f, f)"' is a Hilbert space xl,. Each x E G gives rise to an endomorphism

1

390

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

f -P f L ( x ) of V,; this endomorphism preserves the inner product ( , ) and so extends uniquely to a unitary operator n(x) of 8,. The mapping x + n(x) satisfies n(xy) = n(x)n(y) for all x , y E G. Moreover, if f~ V, we have = 2 C a i i i j I # J ( x ; ' x j )-

C aiEjI#J(x;'x-'xj) - C aiiiiI#J(x;lxxj), i, i

i,j

i,i

which tends to 0 for x -,e. Thus for a dense set of vectors a E $, the mapping x + n(x)II is continuous for x = e, hence for all x E G. From the inequality Il.(x)b - bll 5 IIn(x)(a

- b)ll + Iln(x)a - 411 + Ila - bll

it follows that for each b E @,, the mapping x + n(x)b is continuous. Thus A is a unitary representation of G on $,. Finally, if e denotes the vector in Js, which corresponds to the vector I#J E V' we have

44-4 = (e,

n(x)e).

We shall call A the unitary representation associated to the results above we have

4. Summarizing

Theorem 1.5. Let A be a unitary representation of a locally compact group G on a Hilbert space $. For each vector e E J s the function (e, n(x)e) is a positiue definite function on G. Conversely, to any positive definite function #IJ f 0 on G corresponds a unitary representation R of G such that I#J(x)= (e, n(x)e) for a suitable vector e.

2. Compact Groups

In this subsection we develop some basic general facts about harmonic analysis on compact groups and their homogeneous spaces. The results are based on Schur's lemma, Schur's orthogonality relations, and the Peter-Weyl theorem, all of which will be stated without proof. This analysis will be developed further in the next chapter. We recall that a compact group of complex matrices leaves invariant a positive definite Hermitian form; this means that any finite-dimensional representation of a compact group K is unitary for a suitable positive definite inner product. Schur's lemma asserts that such a representation 6 of K is irreducible if and only i f each operator A commuting with all 6(k) ( k E K ) is a scalar. In particular, if K is abelian any finite-dimensional irreducible representation of K is one-dimensional.

011

1. Representations

39 1

Let V‘ and V” be two finite-dimensional vector spaces over C. Their tensor product V’ @ V” is by definition the dual space of the space of all bilinear forms on V‘ x V”. If u’ E V’, u” E V”, the element u1 @ Ut1 E V’ @ V”

is defined by u’ @ u”(B) = B(u’, d’) if B is any bilinear form on V’ x V”. If E‘ E Hom(V‘, V),E E Hom(V”, V”) then E’ @ E” is the endomorphism of V’ @ V” defined by

E

@ E(u’@ u”) = E u ’ @ Eu“.

Let K‘ and K” be representations of the group K on V‘ and V”, respectively. Then the mapping k -+ n’(k) @ ~ “ ( kis) a representation of K on V’ @ V”, denoted K‘ @ K“ and called the tensor product of d and z”. On the other hand, we can assign to each k E K the endomorphism (K’ n”)(k) of V’ x I/” defined by

+

(K’

+ n”)(k) = ( ~ ’ ( k n”(k)); ),

then K‘ + nrris a representation of K on V’ x V” called the direct sum of K‘ and d‘. Let K be a compact group and dk the Haar measure on K, normalized by JK dk = 1. Let 6 be a unitary representation of K on a finite-dimensional Hilbert space V, with inner product ( , ). Let Xd and d(6), respectively, denote the character and dimension of 6 and put ad = d(6) conj(2,).

For u, u E V, the representation coefficients k + (6(k)u, u ) satisfy the orthogonality relations of Schur: If 6 and 6’ are irreducible then (bar denoting complex conjugation)

+

where 6 6’ means that 6 and 6’ are not equivalent. If ul,. . .,vd(& is an orthonormal basis of V, and the matrix ai,(k) is determined by S(k)uj = 6,(k)ui, then (8) means that the functions d(6)”26i, are orthonormal in L i( K ) . According to the Peter-Weyl theorem they form a complete orthonormal basis of Lz(K). This can be expressed by the orthogonal Hilbert space decomposition

xi

(9)

L Z ( K )= d€K

392

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

where fi is the set of equivalence classes of finite-dimensional unitary irreducible representations and H, is the subspace spanned by the functions (representation-coefficients)on K : k

+ (6(k)u, u ) ,

u, u E V,.

We can also write (10)

H , = {F E C ( K ) :F(k) = Tr(G(k)C), C E Horn(&, V,))

and, as the notation in (9) indicates, If, depends only on the equivalence class of 6. The mapping 7, which to the element (u, u) E K x K associates the endomorphism f + fL(u)R("-') of H , is a representation of K x K on H,. With the orthonormal basis (vi)of V, fixed, consider for each i the subspace

of H a ; H i , , is the subspace spanned by the ith row of 6. Similarly, we consider the subspace

of H , , i.e., the subspace spanned by the jth column of 6. Theorem 1.6. Let 6 E R.

(i) For each i, H i , , is right invariant under K and the right regular representation of K on it is equivalent to 6. (ii) For each j , is left invariant under K and the left regular representation of K on it is equiualent to the contragredient representation 8. (iii) The representation 7, of K x K on H , is irreducible. Also, (&jk

=

Kronecker delta).

(iv) The normalized character $ = xa/d(6) satisfies the functional equation (13)

J?(kuk-'v)

dk = +(u)+(u).

On the other hand, if $ f 0 is a continuous function satisfying (13), then $ = xa/d(6)for some 6 E k.

911 Proof.

1. Representations

393

(i) For the right regular representation r we have 46)

r(k)6ij = 6 c ( k - ' = )

C dpJ{k)6ip,

p= 1

proving (i). (ii) The contragredient s' operates on the dual space Vg = b.We define the inner product ( , ) on b by (u', w ' ) = (w, 0 ) . For a linear transformation L of & let L' denote the linear transformation u' -+ (Lo)' of b.If L has matrix expression ( I i j ) in the basis u l , . . . ,"u,,(~),then L' has in the basis u;, . . . , v ; ( ~ , . Since 6(k) = ' a @ - ' ) we matrix expression find (&(k)v>Xw)= u;{b(k-')w) = ( w , 6(k)Uj) = (6(k)Uj)'(W)

(G)

so (14)

&(k) = ~ ( k ) ' , 8 is unitary, Sij(k) =

m),

the last relation holding in terms of the bases (vi) of V,, (0:) of l$. For the left regular representation I we have with fixed j d(d) ..

(I(k)aij)(u)= 6 ( k - ' ~ = ) ~aji(u-'k)~

=

C d(k),idPJ{u),

p= 1

so, comparing with (14), we see that I on Ha, is equivalent to 8. (iii) Let f E Hd - (0) and V, the K-bi-invariant subspace generated byf. Then V, contains the function

which for some j is #O. Thus H i , &n V' # 0 for each i. Similarly, n V' # 0 for each j. Now (i) and (ii) imply that V' contains each H i V aand each Hd,j , so (iii) follows [using (8)]. For (iv) consider the integral A(u) =

s,

d(kuk-') dk E Horn(&, Vs).

By the invariance of dk, A(#) commutes with each d(1) (1 E K ) . By Schur's lemma, A(u) is a scalar multiple of the identity Z,,(s). Taking traces, we obtain

Multiplying by 6(u) on the right and taking traces again we obtain (13).

394

[Ch. IV

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

For the converse we select by the Peter-Weyl theorem a that a(k)$(k)dk # 0. Then by (13) $(u) J$(u)cJ(u) du =

ss

$(kuk-

'U)CJ(U)

CJ

E

such

du dk,

which by successive change of variables becomes

It follows that $ = xs/d(o) as stated. This finishes the proof. The characters form a natural description of R.Two equivalent representations of K have the same character. The converse is a simple consequence of (8) if we recall that each finite-dimensional representation of K can be decomposed into irreducible ones. Let M be a closed subgroup of K . We shall write out (9) more explicitly for L 2 ( K / M ) viewed as the subspace of L 2 ( K ) consisting of functions right invariant under M . If ~ E C ( K is ) right invariant under M then so is each term in the decomposition f = fa from (9). Thus we have to describe the subspace of functions in H , which are right invariant under M . Let V f = { u E V,: 6(m)u = u for m E M } .

1,

and put l(6) = dim Vf. If F E Hf then F ( k ) = Tr(G(k)C) where C E Hom(V,, V,) is unique. Since F(km) = F(k) we deduce 6(rn)C = C (mE M) so CV, c V y . The mapping F + C is a linear injection of Hf into Hom( V,, Vf). The latter space has dimension d(6)1(6) whereas Hf contains the 1(6)d(6) linearly independent functions (1 I j Il(S), 1 I i I d(6)) (6(k)uj, ui> provided the basis (ui)of V, is chosen such that the first 1(6) elements belong to vy. The mapping F + C is therefore a bijection and

Hy

(15)

=

{ F : F(k) = Tr(G(k)C), C€Hom(V,, V f ) } .

With a basis ( u i ) of V, compatible with the direct decomposition = V, the functions

V f @ (Vf)'

(1 Ii I d(6), 1 Ij Il(6)) di, form a basis of Hf. Moreover, we have the orthogonal Hilbert space decomposition

L 2 ( K / M )=

@ Hf, 8,

where

R,

is the set of elements in

Ez,

R for which

V y # 0.

911

1. Representations

395

Given a representation II of K on an arbitrary vector space V, a vector is said to be K-Jnite if the vector space {II(K)u}spanned by the orbit n(K)u is finite-dimensional. The vector u is said to be K-Jnite of type 6 if the representation of K on {n(K)u} given by K decomposes into finitely many copies of 6. Let V ( 6 ) denote the space of K-finite vectors of type 6. Suppose now V is locally convex and complete and recall definition (2). U E I/

Lemma 1.7. The mapping n(aa) is a continuous projection of’ V onto V(6). In particular, V ( 6 ) is a closed subspace of’ V. Proof, The orthogonality relations (8) imply aa * a6 = a d , so n(ad) [defined by (2)] is a continuous projection. Also, if u E V u E K we have

(17)

n(u)n(aa)u= d(6) = d(6)

J- xdu- ‘k)n(k)udk. K

But the members of H i , in particular x i , are K-finite of type 6 under the left regular representation of K . Hence by (17), n(a6)uE V(6). On the other hand, suppose u E V(6). We shall prove n(a& = u. For this we may assume that the action of K ( K )on {n(K)u} is irreducible. It is then equivalent to 6 and we can take the space V, as {n(K)u} and u as a basis element u j . Then n(k)uj = di,(k)ui, so

xi

,.

n(aa)uj= d(6)

z(k) JK

1 Gi,(k)vi dk = uj i

by (8). Remark. The same argument shows that if 6 then n(a6)u’= 0.

+ 6’ and if u’ E V(6’)

Corollary 1.8. If’ K is the lefi regular representation of’ K on V =

L 2 ( K ) ,then V ( 8 ) = Ha

(18)

and the decomposition (9) equals

More generally, if M c K is a closed subgroup and W = L 2 ( K / M ) , then for 6 E K v ,

W(8) = H Y .

396

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

In fact, we have remarked that H, c V(8). But i f f € V = L2(K),

s,

(dCra.)S)(u) = 4 6 ) Xa(k)f(k - '4dk =

~ A u k '-) f ( k ) dk,

46) K

so n(ai)f E Ha by (10). We can now prove an analog of (9) for the general representation n of K on an arbitrary locally convex complete space I/. Lemma 1.9.

The sum Z a E k V ( S ) is direct and it is dense in V.

Proof. The directness is clear from the remark following Lemma 1.7. Next suppose I E V' vanishes on each V(6). Then for each k E K , u E V, we have I(n(a,)n(k)u) = 0. With Xa(k) = z(k-') this equation reduces to

Xa * 4J = 0,

(20)

where I , is the function k + I(n(k)u). Hence by (19) I , = 0, whence I = 0, so the lemma follows via the Hahn-Banach theorem. Corollary 1.lo. An irreducible representation of a compact group on a complete locally conuex space V is jnite-dimensional.

In fact, by Lemma 1.9 V contains a K-finite vector u. The space (n(K)u} is finite-dimensional, hence closed and so equals V. Thus dim V < 00. Suppose now the group K is a compact topological transformation group of a locally compact Hausdorff space X with a countable base. If f and 4 are continuous functions on X and K , respectively, we write

(4 * f X x ) =

(21)

K

4 ( k ) f ( k - ' * x) dk.

Putting X ( X )= C,(X), we consider for each compact subset S c X the subspace Xs(X)c X ( X ) of functions with support contained in S . This space is a Banach space in the topology of uniform convergence; Si,where S , , S , , . . . is an increasing sequence of comwriting X = X s , ( X ) and acpact sets with nonempty interior, we have X ( X ) = cordingly give X ( X )the inductive limit topology. This topology does not depend on the choice of the Si. Furthermore, this topology is the strict inductive limit topology in the sense that for each i the topology of

Ui

ui

013

1. Representations

397

Xs,(X)is here the relative topology of X , i , ,(X).It is often convenient to take the Sito be K-invariant. With 6 E I? acting on V, we consider also the space (22)

X ( X , Hom(V, V,))

of compactly supported continuous functions on X with values in Hom(V,, V,). In the same manner as for X ( X ) we consider the subspace Xs(X,Hom(V,, V,)) of X ( X ,Hom(V,, V,)) and give this latter space the strict inductive limit topology given by the sequence of Banach spaces .Xs,(X, Hom(V,, V,N. We consider now the space

(23)

X6(X)= { F E X ( X ,Hom( V,, V,)) :F(k x ) = G(k)F(x)}, +

and for the natural representation of K on X ( X ) [assigning to k E K the linear transformation f ( x ) -+ f ( k - x ) ] let X,(X) denote the subspace of K-finite vectors of type 6. The spaces X6(X)and X,(X) are given the topologies induced from (22) and X(X),respectively. For any continuous function f on X we put f ' ( ~ )= d(6)

s,

f(k-

'

*

~ ) 6 ( kdk. )

Proposition 1.11. The mapping Q : F ( x ) -+ Tr(F(x)) is a homeomorphism of Xd(X) onto X , X ) . The inverse is given by f + fa. Moreover, the maps

d(6)Xs * f E .XAX), X ( X )-+ f a E P ( X )

P:fE W X ) q:fE

+

are continuous open surjections and the subspace X A X ) of X ( X ) is closed. Proof. Let F e X 6 ( X ) and let (ui) be an orthonormal basis of V,.

Then

so ~ ( k -. x )

= 6(k-')F(x) implies

398

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

which in turn shows that each F,,, and therefore Tr(F), belongs to Xi(X).Also, if f ( x ) = Tr(F(x)), we see by writing the orthogonality relations (8) in the form d(6)

[ 6,Xk)6(k) dk = E i i

JK

'

that f'(x) = d(6)

J',Tr(G(k-')F(x))G(k)dk = F(x).

In particular, Q is one-to-one. It is also surjective because if g E X i ( X ) then g' E Xd(X) and Tr(g') = d(6)Xa * g = g [using (25)]. We know from Lemma 1.7 that p is a continuous projection of X ( X ) onto the closed subspace S A X ) . This implies that p is open. In fact, let N be a neighborhood of 0 in X ( X ) and M c N another such, satisfying p ( M ) c N . Putting N o = M u p(M), we have P(N0)

= {fENo:P(f) =f

) = No n . X U ) ,

so p ( N ) 3 p(No) is a neighborhood of 0 in X j ( X ) . Since Q 0 q = p it follows that q is a continuous open surjection. We recall that an LF-space is by definition the countable strict inductive limit of Frkchet spaces. Thus X ( X )and X ( X , Hom(Vb, V,)) are LFspaces. In case the separable space X is a manifold, 9 ( X ) is also an LFspace, the strict inductive limit of the Frechet spaces g S , ( X ) . Proposition 1.12. The spaces X'(X) and @ ( X ) are LF-spaces.

This is a consequence of Proposition 1.11 and the following general lemma (which is necessary since a closed subspace of an LF-space is not necessarily an LF-space). Lemma 1.13. Let E = Ei be a strict inductive limit of Frkchet spaces El c E 2 c M c E a closed topological subspace and M i = M n Ei.Assume p : E + M is a continuous open surjection which for each i maps Ei onto M i . Then M is the strict inductive limit of (Mi)i=l,2,,.., each M i having the topology induced by E. . . a ,

Proof. By general theory, Ei carries the relative topology of E, so by completeness it is closed. The restriction p i = p ( E i gives a continuous surjection of Ei onto M i . We must prove that a convex subset V c M is a neighborhood of 0 if and only if for each i, V n M i is a neighborhood of 0 in M i ( i = 1, 2 , . . .). The "only if" being obvious, we assume V n M i is a neighborhood of 0 in M i ( i = 1,2,. . .). Then p;l(Vn M i ) = is a

@I

2. Spherical Functions : Preliminaries

w+

399

us

convex neighborhood of 0 in E i and W, c so W = is convex. Also, Ei n = (j2 0) because if ei E E i , p(ei)E V n M i + j then p(ei)E Vn M i and so ei E Consequently, Ei n W = so by the definition of an inductive limit, W is a neighborhood of 0 in E. Finally, p ( W ) = V, so, since p is an open map, V is a neighborhood of 0 in M as desired.

w.+j w.

w,

w..

$2. Spherical Functions : Preliminaries 1. Definition

Let G be a connected Lie group, K a compact subgroup. Let n denote the natural mapping of G onto X = G / K and as usual we put o = n(e) and f = f n i f f is any function on X . As in Chapter I1 let D(G) denote the set of all left invariant differential operators on G , DK(G)the subspace of those which are also right invariant under K and D(G/K) the algebra of differential operators on G / K invariant under all the translations z ( g ) : x K + g x K of G / K . 0

Definition. Let 4 be a complex-valued function on G / K of class C" which satisfies &n(e)) = 1; 4 is called a spherical function if (i) @ ( k ) = 4 for all k E K , (ii) D 4 = ,lD4 for each D E D ( G / K ) , where AD is a complex number. It is sometimes convenient to consider the function = 4 0 n on G instead of 4. We say that 6 is a spherical junction on G i f and only if q5 is a spherical function on G / K . Then a spherical function on G is characterized by being an eigenfunction of each operator in D,(G) and in addition satisfying the relations &e) = 1, &kgk') = $(g) for all gEG and all k, k' E K . The last condition will be called bi-invariance under K . We shall now see that spherical functions can also be characterized by means of an integral equation. Let dk denote the invariant measure on K , normalized b;JK dk 2 1.

6

4

Lemma 2.1. Let U be an open subset of G such that U k c U for each k E K . Let F be an analytic function on U.Then the function

x -,J / ( x k ) dk is also analytic on U .

400

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Proof. For simplicity, let us assume U = G. Let xo E G and let { x l , .. . ,x,} be a system of coordinates valid in an open neighborhood V of x o . There exists a finite set of coordinate neighborhoods U , c K whose union equals K and a neighborhood N of xo in V such that the function F(xk) is given by a power series

F(xk) = P a ( x I , .. . ,x , , k,, . . . ,kp), x E N, k E U,,

where {kl, ..., kp} is a system of local coordinates on U , . Consider a partition of unity 1 = 4, subordinate to the covering { U,} of K . Then

1.:

which on integration over K gives a power series valid on N. Proposition 2.2. Let f be a complex-valued continuous function on G, not identically 0. Then f is a spherical function i f and only r

for all x , y

E

G.

Proof. Let D E D(G). Then all the differential operators Ad(k)D,

(k E K), lie in a finite-dimensional subspace of D(G) and we can form the integral

Do =

I

Ad(k)Ddk,

which is an operator in DK(G). The mapping D + D o is a linear mapping of D(G) onto DK(G). Let F be a function in b(G), bi-invariant under K. Then [DF](k) = [DF](e), so JK

JK

and

(3) Now suppose ,f is a spherical function on G. Then f has the form f = 4 0 7c where 4 is a spherical function on G / K . The space G/K has a Riemannian structure invariant under the action of G. It is not difficult to see that this Riemannian structure is necessarily analytic. The Laplace-Beltrami operator with respect to this structure has analytic coefficients when expressed in terms of analytic local coordinates on G / K . In addition, this operator is elliptic and consequently, by a theorem

2. Spherical Functions : Preliminaries

§21

40 1

of S. Bernstein, its eigenfunctions are analytic (see F. John [1955], p. 57). In particular, the function $, and therefore also the function f, is analytic. Now, let x be a fixed element in G and consider the function F(Y) =

J f W Y ) dk

(Y E GI,

K

which is clearly bi-invariant under K. Let D and Do be as in (3). Since Do./' = A,f we have CDOFl(Y) =

1

CDofl(xky) dk = A D F ( Y )

K

and consequently CDo(f(e)F - W f ' ) I ( e > = 0. Combining this with (3), we obtain CNf(e)F - F(e)f)l(e> = 0 for all D E D(G). Since f(e)F - F(e) f is an analytic function (Lemma 2.1), we conclude from Taylor's formula ( [ D S ] , Chapter 11, $1, (6)) that f(e)F = W4.f

Since f(e) = 1, this is just (1). On the other hand, let f be a continuous function on G , not identically 0, satisfying (1). Select x o E G such that f ( x o ) # 0. Now (1) implies that f ( x k ) f ' ( x o )= f ( x o ) f ( x )= f ( x o ) f ( k x ) for all x E G, k E K . Thus f is bi-invariant under K. Putting y = e in (1) gives f(e) = 1. In order to see that f~ b(G), select p E B(G) such that p(y)f(y) d y # 0, dy denoting a left invariant measure on G. Then

iG

which shows that f e b ( G ) . For each Do E DK(G) we get from (1)

402

SPHERICAL FUNCTIONSAND SPHERICAL TRANSFORMS

[Ch. IV

Putting y = e we get [Do f ’ l ( x ) = [Do f’I(e)f(x), which shows that f is spherical. From (4) it follows that a spherical function is determined by its system of eigenvalues. More precisely, we have (4)

Corollary 2.3. Let 4 and be two spherical functions on G such that Doc$ = AD# and Do41 = ADq51 for all Do E DK(G). Then 4 = 41.

In fact, Equation (3) implies that [D4](e)= [D41](e) for each D E D(G); being analytic, 4 and 41 must coincide.

2. Joint Eigenfunctions We recall (Chapter 11, $4) that a joint eigenjunction on X = G/K is an eigenfunction of each of the operators D E D(G/K). Let p: D ( G / K ) C be a homomorphism and let E, denote the corresponding joint eigenspace; i.e., E , = { jb (~G / K ) :Df = p(D)f for each D

E D(G/K)}.

Of course each spherical function belongs to a unique E,. Proposition 2.4. The joint eigenfunctions on G/K are characterized by the following integral equation: Each joint eigenspace E, # 0 contains exactly one spherical function 4. The members f of E, are characterized by the equation (5)

jKf ( x k y K )dk

=f (xK)+(yK),

x, Y E G.

Proof. The first statement is immediate from Corollary 2.3. Next, let

f~ b ( G / K ) and x E G. Putting

we find for D E D ( G / K ) (6)

(DFx)(Y) =

jD,(f (XkYK))dk K

=

s,

( ~ f ) ( x b Kdk. )

If f~ E, this last integral equals p(D)F,(y). Thus F , is a K-invariant member of E, and is therefore proportional to 4. Evaluating at y = e, we obtain (5).

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2. Spherical Functions : Preliminaries

403

On the other hand, suppose f is a continuous function satisfying (5). The argument of Proposition 2.2 shows that f is necessarily of class C". By ( 5 ) F , = f ( x K ) 4 , so putting y = e in ( 6 ) gives f ( x K ) ( D 4 ) ( 4= ( D f ) ( x K ) , which shows that f~ E,. A function u on G / K is said to be harmonic if Du = 0 for all D E D ( G / K ) which annihilate the constants. In the case in which G / K is twopoint homogeneous D ( G / K ) is generated by the Laplace-Beltrami operator L (Chapter 11, Corollary 4.11), so the harmonic functions are the solutions to the equation Lu = 0. In general the harmonic functions are the members of the eigenspace E,, containing the spherical function 4 = 1. From Proposition 2.4 we obtain the following result. Proposition 2.5. (Godement) The harmonic functions on G / K are characterized by the mean-value property r

(7)

In the case in which G is the group M ( n ) of rigid motions of R" and K = O ( n ) this result reduces to Gauss's mean-value theorem for solutions of Laplace's equation in R".

3. Examples We shall now determine the spherical function for some simple examples including the three in the Introduction.

I. X = R2, G = R2,K = { e } . Here D(G/K) consists of all differential operators with constant coefficients. We have obviously Proposition 2.6. tial junctions

The spherical functions on G / K are the exponen( x , y ) + eax+fly,

u, /3 being any complex numbers.

11. X = R2,G = Mo(Rz) (the e-component of M(2)), K = SO(2).

Here we have as a special case of Lemma 2.4 of the Introduction

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SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Proposition 2.7. The spherical functions on R2 = G / K are the functions

(8)

4 k Y) =z;- /02neiA(x cos o + y sin o) dB

, Ibeing an arbitrary complex number.

Of course 4(x, y) can be written $((x2 + yZ)l/’) and then the functional equation (1) can be expressed in terms of $. If g, h E G denote translations in the direction of the x-axis of distance r and s, respectively, and k is a rotation around the origin of angle 0 then the point gkh . o has distance (r2 + s2 + 2rs cos 0)1/2from 0. The functions (8) are therefore characterized by 211 /02n$((r2

where &x, y) = $((x2 have $ ( I ) = Jo(Ar).

+ s2 + 2rs cos B)1’2) dB = $(r)$(s),

+ Y ~ ) ” ~As ).

shown in the Introduction, $2, we

111. X = S2, G = S0(3), K = SO(2). Here we consider K as the subgroup of rotation of S2 leaving the north pole o = (0, 0, 1) fixed. The circle

{(sin r cos 4, sin r sin 4, cos r ) : 0 I

4 I 2n)

is the K-orbit of a point at geodesic distance r from Theorem 3.1 of the Introduction the function

0. According

to

(r, 4) -+ (al sin r cos 4, a2 sin r sin 4, a3 cos r)”

on S2 is an eigenfunction of the Laplacian L with eigenvalue -n(n + 1) provided the constants ai satisfy a: a: u: = 0. Thus we have the following result.

+ +

Proposition 2.8. cisely the functions

The spherical functions on S2 = G / K are pre-

4A P ) = Pn(cos(d(o,P))) where Pn is the Legendre polynomial

P,(cos r ) =

2n

J

(cos r 0

+ i sin r cos u)” du.

The functional equation (1) can now be expressed in terms of P,(cos r). Consider the rectangular xyz-coordinate system in which S2

§21

2. Spherical Functions : Preliminaries

405

is given by x2 + y 2 + z2 = 1 and o is the point (O,O,1). In the equation

/ p h ) dk

= f(s)f(h)

let g and h denote rotations around the y-axis through the angles r and s, respectively, and let k denote a rotation of angle u around the z-axis. Then the point kh o is given in coordinates

-

kh . o = (sin s cos u, sin s sin u, cos s), and since the rotation g has coordinate expression

E}+{ ,yr : siq{;] -sin r

z

0 cos r

the coordinates of g k h o are (cos r sin s cos u + sin r cos s, sin s sin t i , -sin r sin s cos u a

+ cos r cos s).

Equation (1) therefore takes the form (9)

/02'P,,(cos r cos s - sin r sin s cos u ) du

=

P,(cos r)P,(cos s).

IV. G = SL(2, R),K = SO(2), X = G / K . The group G acts transitively on the upper half-plane Im z > 0 by means of the mappings az b a b z-bg*z=if g = d ) E sL(2, R). cz d

(

+ +

The isotropy subgroup of G at i is K, so X can be identified with the upper half-plane. Let g and f denote the Lie algebras of G and K, respectively, and as usual we have g = f + p, where p is the orthogonal complement of t in g with respect to the Killing form B of g. Since g has complexification sI(2, C ) we have by [DS] (Chapter 111, 58),

B(Z, Y) = 4 Tr(ZY),

2, Y E 9.

The restriction of 3B to p x p gives rise to a G-invariant Riemannian structure on X with respect to which X has constant curvature (G is transitive). According to [DS], Chapter V, 53, this curvature is WCZ, YI, cz,

Yl).

if 2, Y is an orthogonal basis of p. For example, we can take

z = 21- (01

),

1 0

Y = i 1( o 1

0

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SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Then

and so the curvature of X is - 1. In geodesic polar coordinates (6, r), say at the point i E X,the Riemannian structure on X is given by dr2 + (sinh r)2 do’, (cf. [DS], Chapter I, Exercise G) and the Laplace-Beltrami operator is

a2

L=-+-dr2

coshr a sinh r dr

+--(sinh1 r)’

Each spherical function 4 on G / K has the form the function $(r) satisfies the differential equation

a2 a02 ’

4 ( p ) = $(d(i, p ) ) where

coshrd$ = a$ (r > 0), dr

d2$

p-+ -sinh r

for some complex constant a. As shown in the Introduction, $4, No. 2, the general solution to this equation is given by the Legendre function r2n

4

(11) p(p

+

$(r) = P,(cosh r) = .LJ (cosh r + sinh r cos u)P drr, 27t 0 1) = a, and the following result holds.

Proposition 2.9.

The spherical firnetions on GIK are given by

4 4 ~=) P,(cosh

d(i, P)), where P , is given by (11) and p is an arbitrary complex number. Denoting by A the group of diagonal matrices a=

(:

d!!l),

d > 0,

we have G = K A K . In order to find the explicit form of the functional equation (l), let

Then a l k a z can be written k,ak2 where k,, k 2 E K and

cosh r = cosh r , cosh r2 + sinh rl sinh r2 cos 2u.

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3. Elementary Properties of Spherical Functions

407

Hence the functions (1 1) are characterized by the integral formula (12)

P(cosh r cosh s

+ sinh r sinh s cos u ) du = P(cosh r)P(cosh s).

V. Compact groups. In this example let U be an arbitrary compact connected Lie group. Let G denote the product group U x U and let K denote the diagonal in U x U . Then G / K is identified with U under the mapping ( t i 1 , u,)K -+ Under this identification, the mapping z ( i i 1 ,t i 2 ) of G / K corresponds to the mapping u -+ iiluu;l of U . Consequently, a differential operator D on G / K belongs to D ( G / K ) if and only if, when considered as a differential operator on U , it is invariant under all left and right translations. Thus D ( G / K ) is identified with the center Z ( U ) of D(U), (Chapter 11, Corollary 4.5). If we interpret functions on G / K as functions on U , the functional equation (1) becomes

which characterizes those eigenfunctions of the operators in Z( U ) which are invariant under all inner automorphisms of U and satisfy 4 ( e ) = 1. From Theorem 1.6 we know that the solutions to (13) are precisely 4 = Xa/d(6) where is the character of an irreducible representation 6 of U and d(6) denotes its dimension.

$3. Elementary Properties of Spherical Functions Let G be a Lie group and dg a left invariant measure on G. The space C,(C) of complex-valued continuous functions of compact support can be turned into an associative algebra over C, the multiplication being the convolution product

and the addition being the pointwise addition of functions. This algebra is called the group ulgebru of G . If 0 is an analytic automorphism of G which preserves the measure dg, then the mapping f -+ f o is an automorphism of the group algebra. Let f denote the function x -,f ( x - ' ) .

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SPHERICAL FUNCTIONS AND SPHERICALTRANSFORMS

[Ch. IV

Let K be a compact subgroup of G and dk the normalized Haar measure on K. As in Chapter 11, 45, No. 1, we put fb(x) = I K I K f ( k x k ' )dk d k whenever this integral exists. Then the image Ci(G) = (Cc(G))b is a subalgebra of Cc(G) and consists of the functions in Cc(G) that are bi-invariant under K. Theorem 3.1. (i) If (G, K) is a Riemannian symmetric pair, then C,"(G)is commutative (under convolution). (ii) l f C,"(G)is commutative, then G is unimodular. Proof. (i) Let o be an involutive automorphism of G, identity on K. Then o leaves d g invariant. Then, since each x E G can be written x = k p where o(k) = k, and o(p) = p - l , it follows that f " = for f~ C:(G). For any f,g E CJG) we have

1'

f ( y - ')g(x- ' y ) dy =

g(z)f( z - 'x- ') dz

so that

j*g

= @ * f)".

Since f " * g " = ( f * g ) " , we obtain f * g = g *f forf, g E C,b(G). (ii) Since det(Ad(x)) is bi-invariant under K it suffices by Chapter I, Cor. 1.5 to show IGf(x- '1 dx =

IGf

(x) dx,

f~

CKG).

If g E C:(G) is identically 1 on Supp(f) u Supp(f) this relation amounts to ( g * f ) ( e ) = ( f * g)(e), which is guaranteed by the assumption. Lemma 3.2. Let 4 be a continuous complex-valued function on G, bi-invariant under K . Then 4 is a spherical function if and only if the mapping

is a homomorphism of C:(G) onto C. Proof. For feC,(G) we put L ( f ) = L(fP). We have, by a simple computation, (fb

* g)@= f @* gb.

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3. Elementary Properties of Spherical Functions

409

For a suitably normalized measure dyK on G/K we have

so

Since 4 is bi-invariant and since d y is invariant under the mappings y kyk' it follows that

+

Uf@* 9 ) = JG JG f(Y)g(z,( JK 4 W z ) dk) dY dz.

(2) Moreover,

(3)

U f ) L ( g )=

1J

G G

f(Yls(zWOIW(4dY dz,

Considering (l), the lemma follows immediately.

lG

The norm llfll = If(x)ldx turns the group algebra into a normed vector space. Owing to the additional property 1) f *g1) Illfll 1lg1) the group algebra is a normed algebra. The algebra C,(G) is a closed subalgebra. Theorem 3.3. The continuous homomorphisms of the algebra Ca(G) onto C are the mappings

f where

4

--*

1

f ( x W ( x ) dx,

G

i s a bounded spherical function on

G.

Proof. Let L be a continuous homomorphism of C;(G) onto C.Then the mapping

f

-P

L(f7

is a continuous linear function on the group algebra. Hence there exists a bounded measurable function 4 on G such that

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SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

for all f~ C,(G). Here we may assume that 4 is bi-invariant under K because otherwise it can be replaced by @. Since L is a homomorphism, the relations (2) and (3) imply [by approximation in C,(G x G)] that J ) O k Y ) dk = 4(X)4(Y)

except for a set of ( x , y) E G x G of measure 0. In order to see that 4 is equal to a continuous function almost everywhere select p E C,(G) such that jGp(y)$(y) dy # 0. Then for almost a11 iE G , 4 ( x ) J$Y)P(Y)

dY =

and this last expression is continuous in x . This proves the theorem. Remark. For a similar characterization of arbitrary spherical functions see Exercise B16.

A representation R of G on a locally convex topological vector space I/ will be said to be spherical if I/ contains a nonzero vector fixed under all of R(K). We shall now prove that under the correspondance in Theorem 1.5 the positive definite spherical functions correspond to the irreducible unitary spherical representations of G. Theorem 3.4. Assume Ca(G) commutative. Let 4 f 0 be a positive definite spherical firnction on G and let R be the unitary representation of G associated to 4. Then R is irreducible and spherical. On the other hand, if' R is an irreducible, unitary, spherical representation of G and e a unit vector left fixed by all n(k) (k E K ) then the function (e, n(x)e) is a positive definite spherical function on G. Proof. Let 4 be a positive definite spherical function, let V,, $, and be as defined in $1, No. 1. Let e be the vector in Ej, which corresponds to 4 E I/+. Since 4 is bi-invariant under K it follows that n(k)e = e for all k E K. In order to prove that R is irreducible, consider for each pair a, b E $, the integral R

931

3. Elementary Properties of Spherical Functions

41 1

Since IB(a, b)( I)1u11 llbll there exists a bounded operator P on $3, such that (Pa, b ) = B(a, b) for all a, b E $3., Now, since 4 is spherical it follows from the definition of the scalar product in V, that

1(4L(kx), *>dk

= 4 ( x - % 4 , $)

JK

for all $ E V,. Consequently we have Sx(B(k*)e, b ) dk

for all b E $3,. (4)

This means that Pn(x)e = $(x- ‘)e.

Since the space $3, is generated by the complex linear combinations of the vectors n(x)e ( x E G), it follows that P 2 = P and (5)

P($3+) = Ce.

Let $3’ denote the closure of the sum of all closed subspaces U’ c !ij4 which are invariant under n and which satisfy PU‘ = (0). Then $3’ and its orthogonal complement $3” in 9, are invariant under n. Let U be any closed subspace of $3” invariant under K(U # (0)). Then PU # (0) so by (5), e E U . But then n(x)e E U for all x E G so U = $3&. This proves first that 9, = $3” and second that n is irreducible. On the other hand, let n be an irreducible unitary representation of G on a Hilbert space $3 such that n is spherical. Let e be a unit vector in $3 such that n(k)e = e for all k E K . Before proving that the function (e, n(x)e) is spherical we establish a few facts about K. Let f’ be a continuous function on G with compact support. Then the Hermitian form

satisfies

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SPHERICAL FUNCTIONS AND SPHERICALTRANSFORMS

[Ch. IV

so there is a bounded operator ~ ( f on ) f, such that ( n ( f ) ( a ) ,b) = C(a, b). We write symbolically

4.f) =

1

f(X)4X)

dx.

G

Then the mapping f + n ( f ) is a homomorphism of the group algebra of G into the algebra of bounded operators on f,; in other words, we have a representation of the group algebra on f,. Let % be the subspace of b consisting of all vectors a E f, which are left fixed by each n(k), k E K . Lemma 3.5.

The subspace % is invariant under each operator n ( f ) ,

fE cm. Proof.

If a E % and f E CZ(G) we have for k E K n(k)n(f)a =

f(x)n(kx)a dx = JG

f(x)n(x)a dx = n ( f ) a JG

where we have used f ( k x ) = f ( x ) . Lemma 3.6.

The space % is one-dimensional.

Proof. For f E Ca(G) let A, denote the restriction of n( f) to the Hilbert space %. Let f* denote the function

f * ( x ) = f ( x - '),

x E G.

Then f* E Ca(G) so by Theorem 3.l(ii) (a, n ( f * ) b ) = J G f ( x - ' ) ( n ( x - ')u,6 ) d(x- ').

Hence ( a , A, * b ) = (A,a, b) so the operator A, * is the adjoint of A,. The operators A, therefore constitute a commutative family of normal bounded operators. Consequently they have a common spectral resolution A, = S

P X 4 dE,,

where 1 varies over some (unspecified) space, p , is a complex-valued function, and dEA is a measure whose values are projection operators on %. These operators E(S) commute with each A, and the range E ( S ) % is therefore invariant under each A,.

931

3. Elementary Properties of Spherical Functions

41 3

Now suppose that dim W were >1. Then either all A, are scalar multiples of I or E(S) '3 is for some S different from 0 and %. In both of these cases '3 can be decomposed ill = 91, + 'illz where 92, and aZare nonzero, closed mutually orthogonal subspaces of %, invariant under each A,. Select a , # 0 in '3, and let W,denote the set of vectors n ( f ) a , as f runs through the group algebra. Then %Jl, is not (0)and is invariant under each n(x), X E G . We shall now show that W 1and YI2 are orthogonal. This would imply that the closure of '33, is different from 9, which in turn contradicts the assumed irreducibility of 71. Let f E C,(G) and a2 E %*. Then

so (n(f)a1, az> = ( A , A az>.

This last expression vanishes since cludes the proof.

is invariant under A f Q .This con-

We return now to the proof of Theorem 3.4. It remains to prove that the function 4(x) = (e, n(x)e> is spherical. Let feC:(G). In view of Lemma 3.5 and 3.6 the vector n ( f ) e is a scalar multiple of e and since

it is clear that

Since the mapping f + n ( f ) is a representation of Ca(G) on 9, the mapping

f

-b

/

G

f ( x W 0 dx

is a homomorphism of Ca(G) onto C. Lemma 3.2 now shows that 4 is a spherical function on G. The connection between positive definite spherical functions and spherical representations established in Theorem 3.4 can be made more precise by using unitary equivalence of representations.

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[Ch. IV

Definition. Let G be a locally compact group. Two unitary representations n and n' of G on Hilbert spaces fi and fi' are called unitarily equivalent if there exists a linear mapping A of fi onto qj' preserving scalar products such that n'(x) A = A n(x) for all x E G. 0

0

Theorem 3.7. Assume C:(G) commutative. For each irreducible, uni-

tary, spherical representation n of G on a Hilbert space $5 let gndenote the (one-dimensional) space of vectors in $ which arefxed under each n(k), k E K . Let R denote the set of all unitary equivalence classes w of such representations of G. Then R is in a natural one-to-one correspondence with the set Fp of all positive definite spherical functions 4 on G.This correspondence w -P 4 has the properties (i) If n E w and e is any unit vector in

9W

an,then

n(x)e>.

= (e,

(ii) w contains the representation associated to

4.

Proof. First we note that (e, n(x)e> is independent of the choice of the unit vector e E 'illn and of the choice of n in w. Thus we have a mapping of R into '@. This mapping is onto because

44x1 = +)e+> (x E G) if 4 E Fp, n+ is the representation associated to 4, and e4 any unit vector in 'illne. In order to prove that the mapping is one-to-one it suffices to prove that if ~ E R n, ~ and w if we put $(x) = (e, n(x)e) where e is a unit vector in 'illn, then K + E O . But the desired mapping from a Hilbert space fi (on which n acts) onto $5, is given by (e+9

A:

r

r

i= 1

i= 1

C ain(xi)e -, 1 ain,&xi)e4

where x l , . . .,xr are arbitrary in G and a l , . . . ,a, are arbitrary complex numbers. This proves the theorem. Consider now the Banach space L'(G) of complex-valued integrable functions on G with the norm

llfll =

j If(x)ldx* G

Let L9(G)denote the closed subspace of functions in L'(G) which are bi-invariant under K . With the convolution product L'(G) is a Banach algebra.

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3. Elementary Properties of Spherical Functions

415

Definition, Let f be a function on G, bi-invariant under K. The function 1 defined by

on a subset of the set 0 of all spherical functions 4 on G is called the spherical transform off. According to Theorem 3.3 the continuous homomorphisms of Lb(G) onto C are given by h,: f -+ f ( i p ) , ip being a bounded spherical function. Proposition 3.8. Assume C:(G) commutative. Then the algebra Lq(G) is semisimple; that is, f f 0 E Lb(G) and

f0(4)= 0

(7)

for all bounded

4 E 0,

then lo= 0. Replacing f o by a suitProof. Assume (7) and put f * ( x ) = f(x-). able convolution fo * h (h E Ca(G)) we may assume f o bounded. But then f o * f: is continuous and if f E Lq(G) the convolution (fo* f: * f ) ( x ) exists for all x E G (not just almost all x E G). The linear functional F: Lb(G)+ C defined by F ( f ) = (fo

* f :* . M e )

satisfies F(f*f*) 2 0 and by Schwarz inequality

(8)

I F ( f ' * 9 ) IZ I F(f * f*)F(g * g*),

f,g E LYG).

Letting 4 approximate the delta function on G and putting g = 4q,we deduce from (8)'

IF(f)I25 M F(f*f*), where M is a constant. Using this successively we obtain IF(f)IZ IM F ( f * f * ) 5 M 1 l/ZF(f * f * * f < M l + 1 / 2 + . . . + 1/2" +

IlFll'/2"ll(f * f * ) 2 "11

*f*)1/2 1'2"

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SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Using the formula SUPIB(4)I = lim lgmll’m 4

m- m

(valid for any commutative Banach algebra) we deduce IF(f)lZIM * sup1 (f*f*)-($)l

( M *= const).

4

In particular, F ( f o *fg) = 0, whence fo = 0. Remark. See Exercise C1 for a strengthening of this result.

54. Integral Formulas for Spherical Functions. Connections with Representations 1. The Compact Type

Let (G,K) be a Riemannian symmetric pair, G compact. We shall express the spherical functions on G / K by means of characters of irreducible spherical representations of G. Let 4 be a spherical function on G . Then +(e) = 1, 4(kgk’) = 4(g), and

D4

(1)

=ID4

for each D € D K ( G ) ,I , being a complex number. Let V, be the closed subspace of I ( G / K ) generated by the translated functions g K + 4(xg) (x E G). Let n = n, denote the representation of G on V, given by n(g)f= j’r(e) (g E G) where, as usual, .r(g)xK = gxK. Lemma 4.1. The representation n, is irreducible and consequently

dim V, <

00.

Proof. Let U c V, be a nonzero closed invariant subspace. Then there exists anfE U such that f(o) # 0 (0 is the origin in G / K ) . Since U is complete the integral j K f r ( k ) dk of the U-valued function k + f r ( k )is an element f b of U. Evaluation at a point is a continuous linear functional on U so (by definition) fb(x) = j K f r ( k ) (dk ~ ) whence ,fb # 0. Because of Corollary 2.3, f b is a constant multiple of the spherical function 4 , ( g K ) = $(g). Hence U = V,, proving the irreducibility. The rest follows from Corollary 1.10. Since II is finite-dimensional the operator P = jK n(k) dk is well defined. We have P2 = P and the vectors in PV, are fixed under K , so again by Corollary 2.3, PV, = C#o. The mapping Pn(g)P maps Vd onto C40 and

§41

4. Integral Formulas. Representations

the vector

4o into 4 ( ~ - ' ) 4 Hence ~.

+ ( g - ') = Tr(Pn(g)P) = Tr(n(g)P) = Tr

if

41 7

denotes the character of

I,

n(g)R(k) dk =

R.

s,

X(gk) dk

Theorem 4.2. The spherical junctions on G are precisely the functions of' the jorm

4(g) =

1 K

X ( g - W dk,

where x is the character of' a jinite-dimensional irreducible, spherical representation R of' G . Here 4 is positive dejinite and R is the representation associated to 4. Proof. It remains to prove that if n is a unitary, irreducible finitedimensional representation of G on a Hilbert space Sj such that R is spherical, then the integral above is a spherical function. Let e be a unit vector in Sj which is left fixed by each n(k), k E K . Then if we put $(g) = (e, n(g)e) we know from Theorem 3.4 that $ is spherical: Now put as before

P Then P 2 = P, so

=

s, s,

n(k)dk.

h

Tr(Pn(g)P) = Tr(n(g)P) = Tr n(gk) dk = ~ ( g kdk ) (2) if x denotes the character of n. Now for each x E G, the vector Pn(x)e is left fixed by each n(k), k E K . Using Lemma 3.6 we conclude that PSj = Ce. Let e, = e, e 2 , .. . ,e, be an orthonormal basis of Sj. Since Pei = 0 for i 2 2 it follows that

Tr(Pn(g)P)= ( W e , e> = $ ( g - ' ) . (3) From (2) and (3) it now follows that the integral spherical function.

jK X(g-'k)dk

is a

Remark. For G semisimple a more explicit integral representation is contained in Theorem 4.4 of Chapter V.

2. The Noncompact Type Let X = G / K be a symmetric space of the noncompact type; that is, G is a connected semisimple Lie group with finite center and K is a

418

SPHERICAL FUNCTIONS AND SPHERICALTRANSFORMS

[Ch. IV

+

maximal compact subgroup. Let G = N A K , g = n + a f be corresponding Iwasawa decompositions, and for g E G , let A ( g ) E a, u(g) E K denote the unique elements such that g E N exp A(g)u(g). Let C denote the set of roots of g with respect to a, let 2' denote the subset of positive roots (for the ordering corresponding to n) and let as usual p =f m,a, ma denoting the multiplicity of a. Let a: denote the set of complex-valued linear functions on a.

xmez+

Theorem 4.3. tions

(Harish-Chandra)

As 1 runs through a,* the jirnc-

4 L ( g ) = I K e ( i L + Y ) ( d ( L g )dk)

9

9EG,

exhairst the class of' spherical functions on G. Moreover, two such jiinctions (by and are identical if and only if v = s l for some s in the Weyl group. Proof. From Chapter 11, $5, No. 2 we know that each 4L is a spherical function and = 4sL.Now let $ be any spherical function on G / K . Then D$ = A,$ for each D E D ( G / K ) where the mapping D + AD is a homomorphism of D ( G / K ) into C. Because of Theorem 5.18 of Chapter I1 and Lemma 3.11 of Chapter I11 this homomorphism has the form D + r ( D ) ( i l )for some Iz E a:. But then $ and the spherical function 4Lhave the same system of eigenvalues; by Corollary 2.3 we conclude +L(g) = $(gK). We shall often write +A instead of $. Finally, suppose 9,. = Using again Lemma 3.11 of Chapter 111, we conclude that v and 1 are conjugate under the Weyl group. We shall now prove a certain symmetry property of the spherical function which will play an important role later.

Lemma 4.4.

The spherical jiinction

4Lsatis$es

the identity

WI

4. Integral Formulas. Representations

419

The relations g = n, exp A(g)u(g) = k(g) exp H(g)n, imply k(g) = A(g) = -Zf(g-’). Lemma 5.19 of Chapter I can therefore be written

u(g-’)-’,

s,

F(u(kg- ’)) dk =

s,

F(k)eZP‘A‘ke)) dk.

If we now substitute (6) into (5) and use the formula above, the lemma follows. In particular, we have, putting h = e, (7)

We can now prove an analog of Theorem 3.4 for the case of spherical functions which are not necessarily positive definite. Let IL be a representation of G on a complete locally convex vector space V. Then the restriction I L = ~ RIK is a representation of K on V and for each 6 E we can consider the space V ( 6 ) defined in $1, No. 2. Definition. The representation

IL

of G on V is said to be K-finite if

dim V ( 6 ) < 00 for each 6 E R. The customary terminology is “admissible”; however, we feel that the above term may be more suggestive. Theorem 4.5. Let 4 be a spherical function on X = G/K and I, the closed subspace of b ( X ) generated by the translates of 4 under G. Let T, denote the natural representation of G on I,, T i the contragredient rep, (strong topology). Then resentation of G on the dual space &

(i) T, is quasisimple and K-finite. (ii) T, and T ; are both irreducible; T,(K) and T;(K) admit fixed vectors 4 and 6, respectively (unique up to scalar multiples); and, with a suitable normalization, (8)

4(g

0) =


(iii) Conversely, suppose T is an irreducible, quasisimple, K-Jinite, spherical representation of G on a complete semireflexive locally convex space V. Then for a suitable spherical function 4, T is weakly equivalent to T,. Remarks. (a) As a closed subspace of the semireflexive space b ( X ) , 8, is semireflexive, so T ; is well defined (cf. $1, No. 1). (b) “Weak equivalence” means algebraic equivalence of the representations restricted to appropriate dense invariant subspaces.

420

SPHERICAL FUNCTIONS AND SPHERICALTRANSFORMS

[Ch. IV

(c) There are certain nontrivial implications between the properties irreducible, quasisimple, K-finite (Harish-Chandra [1953J ; Hirai [19683, Appendix; Bore1 [1972]). However, we shall not need them here (although they would reduce a little the assumptions in Theorem 4.5). Proof. (i) Since each f~ 8, is an eigenfunction of each D E D(G/K) the quasisimplicity of T', follows from Corollary 5.31 in Chapter 11. Second, let 1 E a: be such that 4 ( h K ) = 4n(h). Then the translated function @(g) satisfies for each 6 E I?

(T',(@a)P))(hK)=

which by Lemma 4.4 equals

But for g fixed the inner integral, as a function of k, lies in the finitedimensional space Ha. Thus by the continuity of T',(a,) the range T4(cra)8, has finite dimension, so T', is K-finite. (ii) By the Hahn-Banach theorem each OES', can be extended to a continuous linear form on 8 ( X ) , i.e., to a distribution So on X of compact support (Proposition 2.1, Chapter 11). If o is K-invariant we can also take S, to be K-invariant. Then, if k E K, 4 ( g - ' - x) dS,(x) =

S,(@(g)) =

1

@ ( g - ' k . x) dS,,(x),

X

X

which by integrating over K becomes 4(9-

-0)s,4(x)

4 4 x 1 = sAo)6(4T(g)),

where 6 is the delta distribution at

0. Hence

a - 1 = 44Mf),

f E 84,

which proves the existence and uniqueness of the T;(K)-fixed vector and proves formula (8). The irreducibility of T,', follows just as Lemma 4.1 (the proof did not use the compactness of G). For the irreducibility of T; let W c S',be a closed invariant subspace and W o its annihilator in 8,. By the irreducibility of T',, either W o equals 8, or W o = 0. In the first case W = 0; in the second case the semireflexivity of 8, implies W = S',. Hence T; is irreducible. The uniqueness is clear from Corollary 2.3.

541

42 1

4. Integral Formulas. Representations

(iii) We now come to the main part. Let T be as in the converse of the theorem. Let VK = Qa,a V ( 6 ) denote the space of K-finite vectors in V. Let TK denote the restriction of T to K. The operator T,(a,) is a continuous mapping of V onto the finite-dimensional space V(6). Since V" is dense in V, (Corollary 1.4) it follows that TK(aa)Vmis dense in V(6), whence by the finite-dimensionality, V(6)= T,(a,)V". Let u E V". Then the map x + T(x)u of G into V and the map (x, k ) -,T(xk)u of G x K into V are both differentiable, so the mapping x-+

I

JK

a,(k)T(xk)udk = T ( x ) a,(k)T(k)v dk

is also differentiable. Hence TK(aa)Vmc V", so V ( 6 ) c V".

(9) Lemma 4.6. function

Let u E V be fixed under T ( K ) and let

1,:9

+

m(gh)

A E V'. Then

the

(9 E G )

is analytic. Proof. Let w E V" and Aw be the function g -+ (T(g)w, 1). Then if D E D(G)

as we see by writing D as a linear combination of products of the (X E 9). By (9) we have u E V", so by (10) and the quasisimplicity,

x

( Z U g > = W ( g ) d T ( Z ) 4= C z U d if Z E Z(G), cz being a scalar. Thus the function g K -+ &(g) on G / K is an eigenfunction of the operator p ( Z ) E D(G/K) (p is defined in Chapter 11, &I). But among the operators p ( Z ) occurs the Laplace-Beltrami operator on G / K (cf. Exercise A.4 of Chapter 11). Hence 1, is analytic by the ellipticity of the Laplacian (cf. proof of Proposition 2.2). The lemma implies that if u # 0 then dT(D(G))uis dense in V. Otherwise some 1 f 0 in V' would annihilate it; then (10) with w = u, g = e implies that 1, = 0, contradicting the irreducibility of T. Because of the filtration D(G)= Dd(G) by degrees (Chapter 11, &I) we have

LrO

D(G) = @*D,(G), d€K

where D,(G) is the set of elements in D(G) which under Ad&) transform according to 6. Since dT(Ad(k)D) = T ( k )dT(D)T(k-') for k E K,

422

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

u is fixed under T ( K ) the mapping D -+ dT(D)u of D(G) into V intertwines Ad,(K and T. Since every element in D(G) is K-finite [under Ad&)] it follows that dT(D(G))u c V'. More specifically,

D ED(G) and

dT(Dd(G))v c V(6).

But since T,(a,)V = V ( S ) and dT(D(G))u is dense in V it follows that T,(a,)(dT(D(G))u), which equals dT(D,(G))u, is dense in V(6). By the finite-dimensionality we have dT(D,(G))u = V(6). In particular, if 6 , denotes the one-dimensional identity representation of K we have dT(D,(G)) = V(6,), the space of fixed points of T ( K ) in V (D,(G) was defined in Chapter 11, $4). Since U E V(So) was arbitrary this shows that DK(G)acts (under d T ) irreducibly on V(6,). Since (D(G)f) n D,(G) acts trivially on V(S,), dT induces an irreducible representation t of D'(G)/D(G)f n DdG) = D ( G / K ) on V(6,). Since D ( G / K ) is commutative and ?'(do) finite-dimensional, the operators t ( D ) ( D E D(G/K)) have a common eigenvector u # 0. Hence, by the irreducibility, (11)

dim V(d0) = 1.

Fix uo # 0 in V(6,) and select A E V' such that (u,, A) = 1. Let A, denote the linear form w+SK(T(k)w,A)dk on V. Then A,EV', (uo,Ao)=l, and we can define the function 4 on G/K by (12) 4 ( g K ) = m g - %o, A,). We shall prove that 4 is a spherical function. Let f~ C:(G) and as usual Put

T(fb = j,f(g)T(g)V 4 7 Then the vector T ( f ) u o is fixed under T ( K ) , (cf E C).Applying A,, we get *

(13)

J G f ( g ) 4 ( g - l K dg ) =cp

Since T(fl *f2) = T(f1)T(f2) the mapping f -,Jf(g)&g-'K) dg is a homomorphism of C:(G) onto C.But then by Lemma 3.2, the function g +(g-'K), and therefore also 4, is spherical. Let T, be the associated representation of G on 8,. Let I/' c V, 8: c 8, be the subspaces generated (algebraically) by T(g)u, ( g E G ) and T,(g)4 ( g E G), respectively. These subspaces are dense and the mapping -+

(14)

C ai Tbi)Uo C ai T+@iW,ai E C,gi E G , +

i

i

§41

4. Integral Formulas. Representations

423

sets up the desired equivalence. In fact, the map is well defined because of (12); it is one-to-one because if ai4(g; ' x K ) = 0, then


ai T(gi)vo TYxMO) 0, (15) where T' is contragredient to T. But as in part (ii) the semireflexivity of Y implies irreducibility of T", so (15) implies aiT(gi)vo= 0. This concludes the proof of Theorem 4.5. 9

As usual, if 1E a: let A, E a' be determined by B(A,, H) = 1(H) (H E a). Since +,,(x) = $,(x) ( A E a*,, s E W, x E X ) there exists a unique K-invariant function Z -, C $ ~ ( X )on Q such that 4 A A (=~+,(x). ) Denoting by Go the semidirect product K x , p (Q as a translation group being the normal subgroup, the action of K on Q given by Ad&)) we can view p as the quotient Go/K which is a symmetric space of the Euclidean type ([DS], Chapter V). As in Chapter 11, $3, No. 4 let J ( X ) denote the ratio of the volume elements in p and GIK. Theorem 4.7. given by

(16)

Suppose G is complex. Then the function 4z(x) is

4,(Exp X) = J - l/z(X)

/

eiB(Z*Ad(k)X) dk

K

(xE PI-

Moreover, (2, x ) 4 &(x) is a bi-spherical function on Go/K x GIK, in the sense that in each variable it is proportional to a spherical junction.

For the proof let Yz Exp be the function on Q given by the righthand side of (16). Then by Theorem 5.38 of Chapter I1 0

DYz = r(D)(iA)Yz,

D E D(G/K), if A LE a" is conjugate to Z under K. By Corollary 2.3, Yz = 4, = 4z, proving (16). On the other hand, the G,-invariant differential operators on Go/K are the K-invariant differential operators on Q with constant coefficients. The function Z -,4z(x) is a K-invariant eigenfunction of these operators, so the theorem follows. Remarks. 1. We recall that (16) relies on Theorem 5.37 of Chapter 11, which will be proved in the following section [$5, Eq. (29)]. 2. Formula (16) can be used to define ~ Z ( X )for any 2 E pc. By holomorphic continuation we have then 4 k Z (= ~ 4z(x) ) not only for k E K but even for k E K C (the analytic subgroup of Int(gc) corresponding to €'). The function x + 4z(x) is still a spherical function, so for each Z E p" there exists a A E a: such that 4 z = 4, (17) even if Z and A, are not necessarily conjugate under K .

424

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Consider for example an arbitrary nilpotent Z E pc. By Theorem 4.3 of Chapter 111 the function X -,efB(z,x)is harmonic and (17) takes the form (18)

dk = 1,

[;iB(Z.Ad(L)X)

XEp,ZEX

This is of course also a special case of Proposition 2.5. 3. The spherical functions as defined in this chapter are often called elementary spherical functions, zonal spherical functions, or spherical func2 that the group tions of’ class one. More generally, suppose 6 ~ such 6 ( M ) ( M being the centralizer of A in K ) has a nonzero fixed vector. A spherical function of’ class 6 is by definition a joint eigenfunction of D ( G / K )which is K-finite of type 6. It is known that the spherical functions of class 6 are precisely the integrals W g K )=

[

,(iA+p)(A(k-le))F(kM) dk MY

KIM

where F is a K-finite function on KIM of type 6. While this theorem generalizes Theorem 4.3, new methods are required for its proof (cf. Helgason, [19761, $7).

3. The Euclidean Type The notation of $4, No. 2, being preserved we shall now discuss the spherical functions on the Euclidean-type symmetric space p = Go/K. The group Go is often called the Cartan motion group associated with X . As in Chapter I1 let S(p) denote the symmetric algebra over p (with complex coefficients), and if p E S(p) let d(p) denote the corresponding constant-coefficient differential operator on p. As p runs through the set I ( p ) of K-invariants in S(p), the operators d(p) run through the algebra D ( G o / K )of Go invariant differential operators on p = Go/K. Proposition 4.8. @A(

as Lea:; moreover

The spherical junctions on G,/K are the functions y ) = [;iB(A**ky)

= @,

dk

( Y E P)

fi and only i f A and p aye conjugate under W.

The proof is the same as that of Theorem 4.3. Similarly, we have the following result. Proposition 4.9.

G/K.

Theorem 4.5 holds with p = Go/K replacing X =

551

5. Harish-Chandra's Spherical Function Expansion

425

Combining Proposition 4.8 and Theorem 4.7, we have Proposition 4.10. Assume G complex. Then the spherical functions on GIK are given by

q5,(Exp X) = J - 1'2(X)$,(X) ( A E a:) where the $, are the spherical functions on G o / K . Remark. Because of Theorem 5.35, Chapter 11, these spherical functions can also be expressed as finite sums.

95. Harish-Chandra's Spherical Function Expansion 1. The General Case

In this section we shall investigate the spherical function 4, in more detail. We follow the notation of $4, No. 2. Because of Theorem 4.3 we rewrite the spherical transform in the form (1)

f(A) =

j f(x)c$-,(x) dx

( f K-bi-invariant)

G

for all I ~ a for r which the integral is defined. We know from Proposition 3.8 that the spherical transform f + f is injective on LP(G). For the deeper theory of this transform we need information about 4, for beyond Theorem 4.3. Since G = K z K it suffices to know 4, on the Weyl chamber A + = exp(a+). There 4, satisfies the differential equations (2) A(D)4, = r-(D)(iWA, D E WGIK), where A(D) is the radial part of D for the action of K on GIK. According to Proposition 5.23 of Chapter 11, the operators A(D) have coefficients belonging to the ring 9.It is therefore to be hoped that the joint eigenfunctions of the operators A(D), in particular the function 4,, should have expansions on A +. o resembling those for functions in the ring a. It will turn out that while the entire system (2) is needed to show that such an expansion for d n exists, the operator D = Lx alone is sufficient for its actual determination. We begin by proving the important fact that each U E D ( A )can be expressed in terms of the radial parts A(D) ( D E D(G/K)) and elements from the ring W.If p ~ a then z ep denotes as before the function on A given by ep(a)= e p ( ' O g a ) .

426

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Let H(a) denote the set of W-harmonic polynomials on a and H ( A ) the set of corresponding constant-coefficient operators on A o or A . By Chapter 111, Theorem 3.4, we have dim H ( A ) = w , the order of W . We shall now apply Proposition 5.23 of Chapter 11. +

Proposition 5.1. Let t i l r . .., u , be a homogeneous basis o j H ( A ) . Let 11 E D(A). Then there exist unique operators D1, . . . , Dw E D ( G / K ) such that

c ui(e-T(Di)eP). W

(3)

11

=

1

Furthermore, there exist elements g i j E a, D i j E D ( G / K ) such that (4)

11

=

C iii

0

A(Di)

+ 1gijui

i

0

A(Dij).

ij

Proof. The mapping II + 11' = e'pu e - p is an automorphism of D(A) leaving H ( A ) invariant. Thus ui (1 I i Iw) is a basis of H ( A ) , and so by Chapter 111, Theorem 3.4 and Chapter 11, Theorem 5.18, 0

c W

Lf'

=

u; r(DJ

i= 1

for unique elements D i ~ D ( G / K )This . proves (3). In order to prove (4) we may assume I I homogeneous. Let q = deg(u), qi = deg(rr,). Then Di has degree q - q i . Using now Proposition 5.23 of Chapter I1 we have e - T ( D , ) 0 ep = A(Di)

+

hijEij, i

'

where h i j E 9 and E i j € D ( A ) has order I q - qi - 1. Substituting into (3) we get 11 = ui A(DJ + 1 L I ~ (hijEij),

c

0

i

0

i. j

so i

k

where h , E 9 and E k E D ( A ) has order
(6)

A(D)$

Then dim E , I w.

=

x(DW

fbr all D E D ( G / K ) .

5. Harish-Chandra’s Spherical Function Expansion

451

42 7

Fix a. E A + .o. If dim E , > w it is clear by linear algebra that there exists a $ E E,, 4 # 0, such that (ui4)(ao)= 0 (1 I i 5 w). But then by Proposition 5.1, (u4)(ao)= 0 for all ti E D(A). But this gives the desired contradiction 4 = 0 because, by the ellipticity of A(L,) (Chapter 11, Proposition 3.9), 4 is analytic. Because of Corollary 5.20 of Chapter I1 and (2) above the function $ A satisfies the differential equation

A(LA4A = --((A? 1) + ( P , P > W A . (7) From Proposition 3.9 of Chapter I1 we have A(Lx) = ( L A + 2A,) + C m,(coth a - 1)A, (8) lEX+

and (9) Thus the second term on the right in (8) can be regarded as a perturbation term, the principal term being the constant-coefficient operator LA i2A,. Since (LA + 2A,)eiA-” = -((A, A) + ( p , p))e’”-, we are led to search for a solution OAof the perturbed equation A(LX)@A= --((A, 1) + ( p , P ) ) @ A (10) of the perturbed form QA(expH .o) = e(i”-p)(H) C r,(A)e-p(H) ( H E a+). (11) fl€A

Here A has the same meaning as in Chapter 11, 55, No. 2 and the r,(A) are coefficients to be determined. Ignoring questions of convergence temporarily, we substitute the series (1 1) into (10) and equate coefficientsto eiA-P-,. We have

(LA and

+ 2A, + (A,A) + ( p , p))e’”,-,

- ( ( p , p ) - 2(ip, A))e”-P-”

428

[Ch. IV

SPHERICAL FUNCTIONSAND SPHERICAL TRANSFORMS

we write mla1 + + mlcq and put xiForami.rational Putting To = 1, (12) defines r,, by induction on Then T,, function on We now need an estimate for r,,. Let PEA

(miEZ')

p =

m(p) =

m(p).

is

a;.

+ . .. + Za,.

A

For p E A - { 0 } , v E A, s # t E W , let the hyperplanes in a; given by

= Za, q ( s , t ) denote

(13)

Q,,

t) =

[Recall that as for exp(s(1og a)).]

Lemma 5.3. Suppose A E a; does not lie on any hyperplane HE^'. Then there exists a constant KA,H such that

1 r,(A)1 IK,,.,e"H'

also write o,,

and let

fbr all p E A.

Proof. Select constants cl, c2 such that for all p E A,

I(P

and

= { A E air: (w p ) = 2 i ( u A>},

{A: i(sA - tA) = v} (slZ)(H) = P ( H ) = 4s- 'H) so st(lZ) = s(tA); we 7,,(s,

Q,,

c1 E

,'X

+ P, a> - (ia, A> I Icl[m(p) + 13, I

(P9

P ) - 2i(P? A) I 2 CZ m(p)Z.

Then (12) implies

where c = 4c1c;' (dependent on A). Let No be an integer such that

and select K = KA*H such that

Ir,(A)II Ke'("),

(15)

for v E A, m(v) IN o . We shall prove (15) for all v E A by induction on m(v). Let NEZ', N > N o , and suppose (15) holds for all V E A with m(v) < N . Let p E A be such that m(p) = N. Then m

Ir,(A)I

< -

cN-l

1 1 Ke(P-ZkQ)(H)<- KeP(H) aeZ+

k=l

and the lemma is proved. The lemma shows (with 3H for H) that if A f$ Q, ( p E A - {0})then the series ( 1 1) converges absolutely for each H E a'. Being a power series in x 1 = e-'l(H), * . * , x I = e-i'(H),the series converges uniformly in a subchamber ai(H) > c > 0 (1 Ii II), c > 0 being arbitrary. The series

5. Harish-Chandra's Spherical Function Expansion

$51

429

(1 1) can also be differentiated term-by-term and so represents a solution to (10). Remarkably enough, @a is an eigenfunction of all the other operators in A(D(G/K)) as well. Proposition 5.4.

The firnction

is an eigenfuncion of each A(D)

(D E D(G/K)); in fact, A(D)OA = r(D)(il)Oa. Proof. Since D(G/K) is commutative and the map D + A(D) (D E D(G/K))is a homomorphism the operators A(L,) and A(D) commute. Thus the function A(D)@, satisfies (10). But it is clear from Proosition 5.23 of Chapter 11 that A(%, has an expansion of the form

A(D@ a -- e ( i a - M f )

1 r:(l)e-fl(H! flEA

But then the coefficients rf must satisfy the recursion formula (12); consequently they are proportional to the Tfl; that is,

r ; ~ =) r,*w,m,

P E A,

whence A@)@, = r,*(l)Oa.

(16)

In order to evaluate r,*(A) we take the formula for A(D) from Proposition 5.23 of Chapter I1 and apply to the expansion for ad.Because of Lemma 5.3 the term-by-term differentiation is permissible. In the resulting formula e(-id+p)(H)

a@XP H * 0) (A(D)@,)(exp H eo) = r;(l)e(-ia++p)(H)@

we let H -, 00 along a ray in a+. Viewing the expansions of functions in 9 as power-series expansions in x i = e-ai(H)there is no difficulty in passing to the limit termwise. This gives T ( D ) ( i l ) = r,*(l),

proving the proposition. Now if s E W the function Os, will also satisfy (6) (with x(D) = r(D)(il)).Suppose now l E a: is such that I lies on none of the planes so, ( S E W , p~ A - {0}) and that in addition i(sA - o l ) # x for all s # o in W. Then the series Osaare all defined and the exponents i d - p - p (PEA) in the series for QSahave no overlap with the exponents iol p - v ( v E A) in the series for @,,a. Thus there exists a H E a+ such that for s # o~ W, p, V E A ( i s l - p - p ) ( H ) f (iol - p - v ) ( H ) .

430

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

The series for O,,(exp tH - 0) converges uniformly in t for 1 I t < 00. A relation a, O,,(exp tH .i)= 0 (a, constants not all 0) would therefore contradict Exercise D5 of Chapter I. Consequently, the functions OsA(SE W) on A" are linearly independent. Thus, by Corollary 5.2, 4A is a linear combination

L.

Replacing here I by oI ( O E W) we see that c,(I) = c(sr2) where c(I) = c,(I). Recalling now the I which had to be excluded in order that the functions a, should be defined and linearly independent we have obtained the following basic result of Harish-Chandra, where the remarkable c-function first enters. Theorem 5.5. Suppose I E a: is such that i(sI - o I ) $ x for s # in W and that ( p , p ) # 2i(p, sI> for all p E A - {0},s E W. Then

4A=

c(sI)e'"--P ssw

1 r,(sI)e--"

on A +

o

0,

,,€A

where T,,is given by (12) and To = 1. Moreover, the series (17) converges absolutely at each point in A + . o and unifbrmly on each subchamber {a.oEA+.o:cri(loga)> c > 0 (1 I i I 1)). This result shows that on A f . o 4, is a perturbation of the elementary function 1 ,c(sI)eisA-P.It is therefore an important problem to determine the c-function. We now prove a variation of Lemma 5.3 which will be needed later. Let a*+ be the Weyl chamber in a* which corresponds to a+ under the mapping I -+ A,. Lemma 5.6.

Let H E a+. Then there exists a constant K H such that

I r,,(I)I IKHer(")

f i r p E A, I

E

a*

+ i Cl(a*,).

Proof. Consider the radial part A(Lx) in the form (Theorem 3.7 of Chapter 11)

(18)

A(Lx) = 6- '"LA o 8'1' - 6- '/'LA(d1/'),

where d(exp H ) =

fl [3(ea(")- e-=(")11

'"a.

U€E+

5. Harish-Chandra's Spherical Function Expansion

$51

43 I

The functions all2, b - 1 / 2 ,and d = d-1/2LA(d1/2)have expansions on A + of the form 6 - 1/2(expH) = e - p c H )

c b,e-"H), 61/2(expH)

d =

C,,e-"H),

=

fl€A

CEA

1 d,,e-p(H),

do = ( p , p ) ,

PEL

where the coefficients b,, c,, and d , have at most polynomial growth in p. Now (10) and (18) imply that the function Y,(exp H ) = d1I2(expH)O,(exp H

a,(l)e(i"-')(H)

0) =

PEA

where a@) =

r,-v(l)cv

v,~-vEA

satisfies the differential equation LAY, = (-(A, 1) - ( P , P>

(19)

+ dW,*

Using do = ( p , p ) this gives the recursion formula (20)

(

0

9

c

P ) - 2 0 , Wa,(4 =

dvap-vm

0# v.fl-vaA

which is analogous to (12), but more manageable because the coefficients d , on the right do not involve 1. If 1 E a* + i Cl(a*,) then Im((p, 1)) 2 0, so I ( p , p ) - 2i(p, 1) I 2 ( p , p ) 2 c- h ( p ) , where c is a constant (independent of 1 E a* + i Cl(a*,)). It follows that (21)

z

la,(A)l 2 c 4 4 - 1

0 # v, p -

VEA

l&ll%l-v(4l.

Let H E a + ; repeating (with N o 2 c Eve,,( d V J e - v ( Hthe ) ) proof that (15) we deduce from (21) that (14) (22)

la,(1) I ICHeQ("), p E A, 1 E a*

+ i Cl(a*,),

where C HE R only depends on H. But since 0, = d-1'2YAwe have r,(4 =

1

v. p - v e A

~ , - ~bv,( 4

whence by (22)

since b, has at most polynomial growth in v. This proves the lemma.

432

[Ch. IV

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

2. The Complex Case

In the case in which G is complex, Theorem 5.5 can be put into much simpler form. Theorem 5.7. Suppose G is complex. Then

the summations extend over the Weyl group and n(A) = Moreover,

(a,

44 = 7 C ( P ) / W )

(24) and (25) Tr

nrrsz+A).

= the number of' partitions of i p into a

sum of positive roots.

Proof. We know from Proposition 5.15 of Chapter I and Proposition 3.10 of Chapter I1 that with

dIi2(a) = C (det s)esP('oga) S

we have

A(L,)

(27)

=

6 - 1'2 (LA - ( P , P ) )

O

6li2,

Consider now the function F on A . o given by F ( ~= ) 6- l / 2 ( a ) e i a ( l o g a ) . +

Then F has an expansion F(exp H .0) = e(ia -P)(H )

bo = 1,

b,e-fi("), PEA

and because of (27) it satisfies

+ (p, p ) F .

A)

A(WF = -((A,

Thus it must coincide with the function

9a(a

0) =

ma in (1 l), so by Theorem

S- '"(a) C c(sA)eisa(loga),

(a E A +).

S

By analyticity, we have for all a E A,

(28)

C (det s)esP"oga)4a(a S

*

0)

=

c S

c(sA)eiSA(lOgU).

5.5

$51

5. Harish-Chandra's Spherical Function Expansion

433

Replacing here a by us we obtain c(s1) = (det s)c(l). Also, applying the differential operator d(n) on A to (28) and putting a = e we obtain n(p) = c(l)n(il). This proves (23) and (24). Finally, r,(1)3 b, as already observed and

1b,e-'

=

PSA

n

(1

+ e-2" + e-4u + ...),

UEZ+

which proves (25). We can now prove Theorem 5.37 of Chapter 11, that for G complex the radial parts A(D) of D E D ( G / K )are given by A(D) = 6- '"r(D) 0 61'2.

(29)

We note first that both sides of (29) are differential operators on A + . o which agree on all spherical functions. Now let f E 9 ( A + '0) and extend it to a W-invariant function f E 9 ( A . 0). Then the function ?(a . o)

1(det s)eSP('Og

)'

S

has a Fourier integral expansion ?(a . o )

1 (det

s)eSP(lOg')

=

[ ~ ( l ) e i ' ( ' o g d"l).

S

Since the left-hand side is skew under W so is the function F(1). The right-hand side can therefore be written as the average

This gives a decomposition of follows in general. Example.

f into spherical functions 4' so (29)

Let G denote the group SL(2, C ) considered as a real Lie

group. Here A = {q = 0 e-' O):tsR}

and C+ consists of the single root a given by a(log a,) = 2t and a has multiplicity 2. Writing 1 E a$ in the form 1= la ( I E C),Theorem 5.7 reduces to

'"(:

sin(2lt) e") = 1 sinh(2t)'

434

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

$6. The c-Function

1. The Behavior of

4lat 00

The function c(A) has been defined on the set (1)

'$ = { 1 E ~ : A ~ s a , ( ~ E h - O ) , 1 $ z , ( s , t ) , ( V E ~~, SE#W ) } .

This set is the complement in aE of the union of countably many hyperplanes. Each compact subset of a; intersects at most finitely many of these hyperplanes. From [DS] (Chapter VII, Theorem 12.2 and Proposition 12.4) we know that 'a; is a connected open dense subset of a:. It is clearly W-invariant. Lemma 6.1.

The function c(1) is holomorphic on 'a:.

Proof. For 1 ~ ' a Ewe have the equation

4,(h)

(2)

c(s1)Osl(H),

=

H

E a+, h = exp H

ssw

and for each D E D ( A ) ,

(3) Let sl,. . . , S, be the elements of W and let u l , . .., u, be a homogeneous basis of H(A). If loE 'a; and H o E a+ are fixed we shall prove that s i , j s w ) # 0. det({(~j@s,~o)(H~)I~ Otherwise there would be complex numbers a, (SE W) not all 0 such that a,(ujOslo)(Ho)= 0 (1 Ij 5 w). Thus the function f = asOslo on a+ would satisfy (ujf)(H0) = 0 (1 ~j Iw), and by Proposition 5.4 it is an eigenfunction of each A(D) D E D(G/K). But then Proposition 5.1 implies ( D f ) ( H o ) = 0 for each D E D ( A ) . By the analyticity of f this would imply f = 0, contradicting the linear independence of the Oslo. Putting D = u j (1 ~j 5 w) in (3) we can solve the resulting system with respect to c(s1) and deduce the conclusion of the lemma.

Is

2,

Remark. For a more elementary proof of Lemma 6.1 see Exercise B17. Next we relate the c-function to the behavior of 4l at co in A + . If 1= 5 + iq (5, ? € a * ) we put Re 1= 5, Im 1= q. We also denote by a*+ the positive Weyl chamber in a*; i.e., a*+ =

{ I E a * : ( A , a ) > 0 for

aEX+}.

6. The c-Function

063 Lemma 6.2.

435

Let I E ' @ be such that Re(il) E a*+. Then if H E a+ lim e(-iA+P)('H)4A(exp t ~ = )c(1). '++a

Proof. This is a fairly direct consequence of Theorem 5.5. In fact,

(4)

e(-iA+p)

4A= 41) C r,(rZ)e-W

+ C c(sL)ei("--")C r,(sl)e-P.

P

II

s#e

Since Re(il) = -q we know from the proof of [DS] (Theorem 2.22, Chapter VII) that s # e. - q W > s(-q)H, Consequently, if s # e ei(SA-A)('H) --t 0 as +

so the lemma follows by evaluating (4) for tH and letting t + + 00. Remark.

Later (in Theorem 6.14) we shall drop the restriction 1 E 'a:.

For further study of the c-function we shall use Theorem 5.20 of Chapter I to transfer the integration over K in Theorem 4.3 to integration over For the Iwasawa decompositions G = K A N = N A K we have

m.

9 = k ( d exp H(g)n(g) = n l ( g ) exp A ( g ) u ( d

whence (5)

H(g-') = -A(g),

k ( g - ' ) = u@)-'.

Using Eq. (7) of & we I, therefore have = Ai(;[-

p)(H(gk))

dk

and since H ( g k m ) = H ( g k ) ( g E G, k E K , m E M, the centralizer of A in K) we have by Theorem 5.20 of Chapter I

4A(g)= j

n

e(iA-p)(H(gk))

dk M

KIM

=

J:(iA-pI(H(ik(fi)))e-

%(H(fi))

Put g = a E A and note that since ii E k(ii) exp H(ii)N we have (6)

H(ak(ii)) = H(aii) - H(ii) = log a

+ H(a&-')

- H(ii).

We have thus proved the following formula for the spherical function.

436

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Proposition 6.3. Let A E a: and a E A. Then @A(a)

=

e(iA-pP"Ioga)

Here the Haar measure on

s,

e(iA--p)(H(afia"- I ) ) - ( i l + p ) ( H ( f i ) )

fl is normalized such JRe-2~lHIR)I

that

dfi = 1

Each ii E R can be written uniquely ii = exp X where X E en; moreover, X = x,X-, where X - , E g - a . If a, = exp tH, H E a+, then a,iia,-' = e x p ( ~ a e 2x,e-'a'H)X-a). + Hence limf++ma,iia;' = e, so, ignoring questions of convergence, we would expect from Proposition 6.3 and Lemma 6.2,

ca62+

After some preparation we shall prove this formula and, on the basis of it, obtain an even more explicit expression for c(l). 2. The Rank-One Case

In this subsection we assume that G has real rank one. Let a denote the single positive indivisible root. Then ii = g - a + g - 2 a and the mapping ( X , Y) + exp(X + Y) = ii is a diffeomorphism of g-, x g - 2 a onto This is clear from [DS] (Chapter VI, Corollary 4.4 and Chapter IX, Lemma 3.9) ((t, z ) is determined by ii) and also from the upcoming Lemma 6.7. From [DS](Chapter IX,Theorem 3.8) we have e~W(fi)) = [(I + c l ~ ~ 2 )+2 4 4 y 1 2 ] ( 1 / 4 ) h + 2 m 2 m ) . (8)

m.

Here ma and m2, are the multiplicities of a and 2a, respectively, c-' = 4(m, + 4m2,) and 1212= - ( Z , OZ), ( , ) being the Killing form. Theorem 6.4. (C of real rank one). The c-function extends from 'a: to the meromorphic function

where a. = a/(a, a ) and c0 =

r(&, + m2, + 1))2(W ) m . +mzm.

Moreover, formula (7) holds, the integral being absolutely convergent i f and only i f Re(i;l) E a*+.

061

6. The c-Function

437

Proof. It is clear from Theorem 1.14, Chapter I and [DS] (Chapter VI, Corollary 4.4) that exp*(dH) is a constant multiple of the Euclidean measure dX dY. Let z = x i y e C be determined by 1 = zp. Then we have for Re(iz) > 0, P = %mu 2m2u),c1 = const.,

+ +

dfi

J:-(ii+p)(H(fii)

=

1..Lm + [(l

c1

= c2

C ~ X I+~4cl ) ~Y 1 2 ] - P ( i r + 1 ) ddXY

jom Jom[(l + r)2 +

s]-~(iz+l)

r ( 1 / 2 ) m u - 1 S( 1 / 2 ) m z U - 1 dr ds,

where c2 is a constant. Here we first evaluate the integral over s by introducing the variable t by s = (1 + r)2t. Then we get (10)

I _ e l i i + u ) ( i ( ( f i ) ) dfi

N

=

c2 Jomr(1/2)mm-1(1

+

r ) - 2 p ( i z + I)+mzu

dr J0K't(1/2)m2m-1(1 + t)-P(iz+

1)

dt

and the convergence statement is now obvious. Using the formula

we obtain (with a new constant c3) J:-(ii+n)(~(li)

dfi = c3

U W + P, ao> - m2u))l-((4

ao))

r(f(d.+ p, ao))r((il+ p, ao> - m d ' which by the duplication formula 1-(2~)= 22z-l n - lI2r(Z)r(z + 9) (12) and the given normalization of dfi reduces to the right-hand side of (9), the constants being determined by c( - ip) = 1. It remains to prove (7) for Re(iz) = - y > 0. Select E (0 < F. < 1) so small that 1 + ~y > 0 and put a = exp t H ( t > 0, HE^') in Proposition 6.3. By (8), 0 I p(H(afia-')) Ip(H(fi)), so the integrand in Proposition 6.3 is majorized by e - ( y + l)p(H(ruSs- ' ) ) e ( l)p(H(ii)) ~< - e ( - ~ + ~ ~ ) p ( H ( . f i . - ' ) ) e ( ~ I)/NH(fi)) < e ( - ~+ a ~ ) p ( H ( f i ) -) ~I)p(H(W (~ = e(ey- l)p(H(fi)) which as we saw is integrable. Combining Lemma 6.2 and Proposition 6.3 we can let t -, + a under the integral sign and derive (7). This proves Theorem 6.4.

438

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

3. Properties of H(A)

For the justification of (7) in general we need more information about H($.Given the positive Weyl chamber a' let + a denote the dual cone defined by + a = {Hea:B(H,H') > 0 for all H ' E ~ + } , and let 2 and 'a denote their respective closures. We know from [DS] (Lemma 2.20, Chapter VII) that a' c 'a. Also, ' a = { ~ 1a j A a , # O : a j t O 1, (13) j= 1

air.. . , a, being the simple restricted roots. Thus 'a =

Lemma 6.5.

5 - (0).

Let a E A'. Then

log a - H(ak) E 5

for each k E K .

Proof. Extend a as usual to a Cartan subalgebra If of gc (C indicating complexification) let bR c bC be the subset where all roots a E A = A(gc, If) are real and consider an ordering of the dual of bR compatible with our ordering of the dual a* of a ([DS], Chapter VI, 53 and Chapter IX, 51). Let A' denote the set of positive roots. For the proof of Lemma 6.5 we may assume that G is the analytic subgroup of the simply connected complex group GC.In fact, H(akz) = H(ak) for all elements z of the center of G . Let z/ be the analytic subgroup of GC with Lie algebra u = f + ip. Then U is compact, so we are automatically preserving our assumption that K is compact. Our choice of G permits us to identify finite-dimensional representations 7c of G on a complex vector space V with such representations of g, GC,gc, U, andu. We can always introduce a norm I I on I/ such that a(u) is unitary for each U E U . Now we fix such an irreducible 7c and let A denote its highest weight (cf. Chapter V, 61).Every other weight A' then has the form

A ' = A - m 1fi 1

_ . . . - mp&

(mi E Z',

piE A').

By the compatibility of the orderings we have A(H) 2 A(H) for H E a'. Since det a(g) = 1 for all g in the commutator group of GC,hence for all g E GC we know that a ( H ) is not only self-adjoint but has trace 0. Thus its eigenvalues are real and add up to 0; hence A(H) 2 0 for H E a'. Let u E V be a unit vector of weight A. Let a E A + . Since e"(loga)is the maximal eigenvalue of n(a) we have

(14)

In(ak)uI = I n(a)a(k)uI I eh(lo*a ) In(k)uI

= eA(loga).

961

6. The c-Function

439

On the other hand, ak = k , exp H(ak)n, and 7c(nl)u = u. Thus

Iz(ak)uI = eA(H(ak)),

(15) so by (14)

aEA+,k E K. A(1og a - H(ak)) 2 0, (16) According to Theorem 4.1 of Chapter V we can for each i (1 I i I 2) choose 7c = 7ci such that its highest weight A = Ai satisfies

Ai(If n f)

(17)

= 0,

Ai(Auj) = ciaij,

where cj > 0 and A,, as in (13). Writing

we deduce from (17) and (16) that bj 2 0, and this proves the lemma. Corollary 6.6.

Let fi E ~ ( f i E)

and a E A'. Then

5,

~ ( i i-) H(afia-') E

5.

The second relation is an immediate consequence of (6) and Lemma 6.5. The first relation follows by taking a = a, = exp t H in the second one and letting r + co.

4. Integrals over Nilpotent Groups As before let M and M' denote the centralizer and normalizer, respectively, of A in K and W = M ' / M the Weyl group. Fix S E W and select a representative rn, E M'. Consider the groups

N s = m,Nm,-',

= rn,mm,-l,

m,= m n NS-'

whose Lie algebras, respectively, are given by Its =

c

s-'p>o

gp,

iis=

c

s-'/?<0

gp,

fi, =

c

po

gp,

where g = go + g, is the usual root-space decomposition of g (cf. [DS], Chapter IX, $1). In order to study the integral (7) it is convenient (for the sake of induction proofs) to consider the more general integral

where dfi, is a Haar measure on N,. We need some simple lemmas about nilpotent groups.

440

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Lemma 6.7. Let N be a connected nilpotent Lie group with Lie algebra n. Then (n, exp) is a covering space of' N . If' N is simply connected, the m a p exp: n -+ N is a difleomorphism. Proof. We know from [DS] (Chapter VI, Corollary 4.4) that

exp: n + N is a regular mapping of n onto N . Let c denote the center of n and put n, = {X E c: exp X = e } . Clearly exp(X

+ Z) = e x p X e x p Z = e x p X

if X E n , ZEn,,

and since exp, I c = exp,, n, is a discrete subgroup of n. On the other hand, if X, Y E n such that exp X = exp Y then eadX= eadr, so by [DS] (Chapter VI, Lemma 4.3, ad X = ad Y. Thus Y = X Z, 2 E c, so exp X = exp X exp Z, and thus Z E nc. Thus

+

exp-'(exp X ) = {X + z : Z ~ n , } and the lemma follows. Lemma 6.8. Let N be a connected, simply connected nilpotent Lie group with Lie algebra n, n(') a strictly decreasing sequence of' ideals of n such that [n, n")] c n""), n, and n2 two mutually complementary subspaces of n such that nW =

n, n n(') + n, n n'')

( i = 0,1, . . . ; n") = n).

Then the mapping +:(X, Y ) - + e x p X e x p Y is an analytic difleomorphism of'n, x n2 onto N . Proof. Let s be the biggest index such that n(') # 0. Then n(') belongs to the center of n and the corresponding analytic subgroup N(') of N is, by Lemma 6.7, a closed and simply connected central subgroup of N ; the factor group N/N(') is simply connected. We shall first prove, by induction on s, that the mapping is a bijection. By the induction hypothesis we may assume this holds for the complementary subspaces n,/(nl n d"))and n2/(n2 n n")) of the Lie algebra n/n(') of N/N('). This implies that if n; and n; are complementary subspaces of n, n n'") and n, n n(') in n, and n2, respectively, then the mapping (X, Y) -+ exp X exp Y is a bijection of n; x n; onto the set N' = exp n', exp n;. A second implication is that the mapping (n', z ) -+ n'z is a bijection of N' x N(') onto N . But N(') being a vector group, the mapping (X, Y) -, exp X exp Y is a bijection of (n, n d')) x (n2 n n")) onto N('). Putting these statements together, we find that is a bijection.

+

+

@I

6. The c-Function

441

Finally, 4 is regular. For this let X o # 0 in n,, Yo # 0 in n2. Then d4(*,Y,(X,, Yo) = d4(X,Y)(X,,0) + d4(X,Y)(O,Yo). Since d 4 ( x , y ) ( X o0), is the tangent vector to the curve t

+ exp(X

+ t X , ) exp Y

at t = 0, we have from [DS] (Chapter 11, Theorem 1.7) and the relation dR, = dL, Ad(x-') 0

Similarly,

If d 4 0 , y , ( X , , Yo) = 0, we thus conclude

Let i be the largest index such that X , , Yo E di).Then (21) implies x0 - yo E n('+') = n l + n2 ,+'+I), so we get the contradiction X , , Yo E nCi+').

5. The Weyl Group Acting on the Root System Next we need some results about the Weyl group W.As before, Z denotes the set of roots, Co the set of indivisible roots ([DS], Chapter IX, §2), Z+ the set of positive roots, Z- = -C+, and Z t = C, n Z+,

=

C, n C-.

If a E Ci put (a) = C+ n {a, 2a) (i.e., the set of positive roots proportional to a), and if B E C; put (8) = Z- n {p, 28). By [DS] (Chapter X, Lemma 3.1 1) we have (22)

s,,(Zo+ - {a,}) = C,+- {ai)

if a i is simple.

C = C; n s-'Z; We shall now generalize this result. For s E W let : and let n(s) denote the cardinality of the set .:C Note that : C consists of the elements in C; which separate a+ and s-'a+.

442

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Lemma 6.9. Let S E W , let a i E C be a simple root, and put si = sZi. Then (i) si(C: - {ai}) = ELi - {ai}. (ii) a, is contained in exactly one of the sets Z ,: CAi. (iii) n(s) - 1 = n(ssi) if a, E Z , : n(s) + 1 = n(ssi) if a, # Z : . Proof. (i) If aEC: - {ai} then by (22), s i a E Z + . Since ssisia = s a ~ C -we have s , a ~ Z & . Also, sia # ai because a # -ai = sicti. Thus we have si(Z: - { a i } ) c Xi, - {ai}. Replacing here s by SS, we get

si(CL, - {ail) c : C - {Ni}. This proves (i). , a, E Z i t . Then sa, E Z;, --ai = ssi a, E C;, which : (ii) Suppose a, E Z is a contradiction. The possibility ai #,:C ai # CL,leads to a similar contradiction. Thus if aiEZ; we have ai#C:,, so ELi = ZLi - {a,}, whence n(s) - 1 = n(ssi) by (i). Similarly, if a i # C : then a i E C A i and (i) implies n(s) + 1 = n(ss,). This proves (ii), (iii) and the lemma. Each S E W can be written s = sa, “ . s a p where each ai is a simple root. This expression is said to be reduced if p is as small as possible. This minimal p is called the length I(s) of s (cf. [DS],Chapter VII, 57, where such a notion is considered for the affine Weyl group). Proposition 6.10. W e have

n(s) = l(s)

for

SE

W.

Proof. Let k = n(s). If k = 0 then s = e so l(s) = 0 = n(s). Proceeding by induction on k we assume n(s) = l(s) holds for S E W , n(s) < k - 1. Let S E W, n(s) = k > 0. Then s # e so saiEZ- for some simple root ai; that is, ai E . : Z Let sr = ssi. By 6.9(iii) above n(sr)= n(s) - 1 = k - 1. By the inductive assumption, n(sr) = l(s’). Since s = s’s, and I@’) = k - 1 we get l(s) I k = n(s). Thus it remains to prove n(s) II@). Writing p = I(s), s = s1 . sp we have by (iii) n(os,) s n(o) + 1 for each c r W. ~ In particular, n(s1 . . * sp) In(s1 . sp- 1) + 1 I . . . In(sl) + p - 1 = p by (22); thus n(s) Il(s) as desired.

Corollary 6.11. Let s # e in W and let s = s,, sLIpbe a reduced expression. Then B1 = -s-’a1 is positive. More generally, if (23) s ( ~= ) s a q + I ...s a p , B q + 1 = -s(‘)-’ aq+ 1 ( = S a p * * * saq + 2 aq+ 1)

where 0 5 q 5 p - 1, then (24)

=

{Bl,

’.* I

BPI.

@I

443

6. The c-Function

Moreover, s-

. . . sppand

= spl

s p p . - - s p j j ? i> 0

1 I i c j Ip.

for

Proof. We have fll = s-lsaIa1 = sap- - - sa2a1 and the element s a P . . - s a has 2 length p - 1. Thus by Proposition 6.10 n(s-'s,,) = p - 1, whence by Lemma 6.9(iii), a1 E Z -: I , so /I1 > 0. Since the expression (23) for s ( ~ )is reduced we have, more generally, & + l > 0. We also have by (22), since s - l = ( S ( ~ ) ) - ' S ~ ~ , s-'(C;

- {-al})

= ( s ( l ) ) - l ( ~ ;- { - a 1 })*9

that is, s-%;

- { B l } = (s(1))-12;

- {-PI}.

Intersecting with C l we obtain the disjoint union Z: = Z,:l, u

(25)

{/31>,

and now (24) follows by iteration. Next, the relation &+ = sap. . sa4 a,+ implies Spq+ 1 - Sap ' ' * S a 4 + 2 S a q + i S a q + 2 * * * S a p , whence -1 sp1sp2 * * * spp - sap* * * sa2sa, = s . +

Finally, sp,

. .spjBi *

= Sap * * * ( s a p *

- saj- ISaj-2

* *

*

Saj+

ISajSaj+

I * * *

sap)sap *

.

*

Sat+ lai

*sa*+1ai

is positive, as we see by using the first part of the proof on the reduced expression oi, = sa,sa,+ . . . saj- This proves the corollary. Remark. By (24) the planes fl, = 0,. . . , p1 = 0 are exactly the root planes separating a + and s-'a+. Consider a line segment from a point in a+ to a point in s- la" such that its intersections with the planes Bp = 0, . . .,p1 = 0 are all different. The relations s-

l

= spl

/3i(sp,...spp(H))> 0

* * *

spp,

for H Ea+, i < j,

express geometrically the passage from a+ to s-'a+ (cf. [DS],Chapter VII, Lemma 7.3).

444

[Ch. IV

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

6. The Rank-One Reduction. The Product Formula of G indikin-Karpelevic With s # e in W as in Corollary 6.11 we write now B for bI and consider as in [DS] (Chapter IX, $2) the subalgebra g, of g generated by the root spaces g, and g-,. Let G , c G denote the analytic subgroup corresponding to go,. Then G , is semisimple and if f, = f n g(,), pa = p n g(, then g(,, = f, + p, is a Cartan decomposition. The space a, = RA, is a maximal abelian subspace of ps. Let K , and A(,) be the analytic subgroups of G , corresponding to f, and a,, respectively, and let G , = K , A ( P ) N s be the Iwasawa decomposition corresponding to R'A, as a positive Weyl chamber in a,. The group N , has Lie algebra n, = g, g., We put N, = ON,, fi, = On,, and let M , denote the centralizer of A(,, in K , . Note that the decompositions G = K A N and G , = K , A ( , ) N P are compatible. Because of (25) we have the direct vector space decomposition of subalgebras

+

+

Ti, = iis. fi,,

(26)

where for simplicity we write s' for d'), /? for PI. If y1 < y2 < . are the elements of C+ in increasing order and we write n") = fis n j>i. 9-n) then the assumptions of Lemma 6.8 are satisfied with the subspaces n, = ii,,, n2 = h,. Hence we obtain the global decomposition

(1

the product maps being diffeomorphisms. Moreover, the Haar measures dii,, dii,.. and d i p , suitably normalized, match under the product maps.

This is a simple consequence of Chapter I, Lemma 1.11, because (19) and (20) show that the product map has Jacobian of determinant one. Similarly, we use Lemma 6.8 on the ideals

in n(")-'. Then we obtain with n, = n'"')-'n ii,n, = n(")-' n n N(S')- - ( N W - ~ N ) ( N ( s ' ) - ' N ) = N,,(N(S')-I N ) (28) in the sense that the product map is a diffeomorphism under which the Haar measures correspond. Thus we have the manifold identification

m,,

(29) = N(s')-'/(N(s')-' N ) , under which the Haar measure on the left corresponds to the invariant measure on the right. We shall now use the decompositions (27) and

6 . The c-Function

063

445

(28) on the integral c,(A) in (18). Let af denote the set of I ~ a for z which (18) converges absolutely. If A E a: we have by (27)

c,(n) =

1%

e-(il+P)(H(fis))

=s

d n, -

e-(il+P)(H(fi@is,))

dfi dfi , =

RgN,,

l l s

s

e - ( i l + p ) ( W 4 g n g f i s , ) )d n, - dii,,,

Npvd

where in accordance with the decomposition G , = K , A ( , ) N , we have written ii, = kpupn,. Since s'fl > 0 we have N , c N(")-', so by (28) we have a unique decomposition n,ii,. n i = (ii,.)*n.

(30)

m,.

The mapping ii,. + (ii,.)* of onto itself corresponds via (29) to the diffeomorphism of the quotient space (29) induced by the inner automorphism n' + n,n'ni of N @ ' ) - 'leaving N(,')-I n N invariant. By unipotency of Ad(n,) this diffeomorphism preserves the quotient measure and thus the map ii,. -+ (&)* is measure preserving. Therefore, since

(31)

H(a,n, ii,,) = H(a,(ii,.)*),

we can remove n, in the last integral. Thus cs(4 =

s

e-(iA + p ) ( H ( a g f i , . ) ) dii,

dii,, ,

RgR..

which, since H(ii,) = log a,, equals

s,,.

e-(il+p)~H(4gis,4~

(il+p)(H(fig))

d n, - dii,..

However, by Chapter I, 55 (2), we have

so we obtain n

(33) Lemma 6.12. The restriction of 2v tion f o r the rank-one space G p / K p . Proof.

+ p t o RA, equals pa, the p-firnc-

By the definition of v we have 2v = (s')-'p - p so

(2v

+ P>(A,) = P(A,,).

446

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Since a 1 is simple, s,, permutes X+ - (a1) so if 1

pa, = -

C

m,y

2 yE(ai) ( P - P a , ) ( A a , ) = s a I ( P - P a l ) ( A a , ) = (P - P a , ) ( - A a , ) *

Consequently, (34) P(Aa,) = P a , ( A a , ) * Since pa,(Aa,)= ps(AB)the lemma is proved. We can now state the basic result for the integral c,(A). Theorem 6.13. Let s E W a n d let

(35)

c,+n s-'&

=

{PI,.. .,a,},

Ns= exp

L+:'r-

Then the integral

(36)

cs(n) =

e - (il+ p ) ( H ( k ) )dfi

S

JN'

converges absolutely ifand only

(37) (Re(iA), P ) > 0 and ifdfis is suitably normalized,

if for

fl E C+ n s-lC-,

where c, is the c-jiinction for the rank-one space G J K , and restriction of A t o a, = RA,.

A, denotes the

Proof. Using Lemma 6.12, formula (33) reads cs(A) = cs.(A)cs(lB)as, where as is a constant. The product formula (38) now follows by iteration. The convergence statement follows by the Fubini theorem if we take the analogous statement of Theorem 6.4 into account. If we substitute into (38) the value of c,(A,) given by Theorem 6.4 we obtain a product formula for c,(A). If ( , ), is the Killing form for G, we shall encounter the expression

#I

6. The c-Function

447

where a. = a/(a, a ) and the constant a, is given by a, = 2(1/2)m.+m.r(&m a + z a (40)

fl

a E r,+ns- 1x<

+ 1)).

Thus c,(A) extends to a meromorphic function on a:. We still must justify (7). Note that iJ = W,,if s* is the Weyl group element which maps a+ to - a f . Thus by (37), the integral

converges absolutely for Re(iA) = - u E a*+. Proceeding as in the rankone case, select E (0 < E < 1) such that p + E V E a*+. If a = exptH ( H E a + , t > 0) we see from Corollary 6.6 that the integrand in Proposition 6.3 is majorized by (41) e- (rl+p)(H(Ga- l))e(rl-~)(H(fiN < -e)(-rlW(afia- L ) ) e ( q - ~ ) ( H ( W - e ( l - ~ ) ( - g ) ( H ( f i ) ) ~ ( r l - p ) ( H ( % )= < ) e(~q-~)(H(fi)) 9

which is integrable. This justifies our heuristic derivation of (7) and we have obtained the following result. Theorem 6.14. The c-jiunction for the semisimple group G is given by the absolutely convergent integral

the Haar measure dii being normalized by Also, c(A) is given by the formula

, f exp( ~ -2p(H(ii)) dii = 1.

where a. = a/(a, a ) and the constant co is given by c( - ip) = 1. Finally, i f H E a’, Re(iA) E a*+, lim - i l + p ) ( W 4&XP t H ) = 4 ) (44) f++m

Remarks. .1. Formula (43) gives an extension of c(A) from a holomorphic function on the set ‘a: to a meromorphic function on a:. 2. If G is complex, we have for each a, ma = 2, m2. = 0. Using T(x + 1) = xT(x), the duplication formula (12), and (43), we recover the formula c(A) = n(p)/n(iA)from Theorem 5.7.

448

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

3. Suppose G has all its Cartan subgroups conjugate. By [DS] (Chapter IX, Theorem 6.1) this happens if and only if rank(G) = rank(K) + rank(G/K); equivalently, if and only if all roots a E Z have even multiplicity. We also know from [DS] (Chapter X, Exercise F4) that in this case m2. = 0 for all a. Using again the duplication formula (21) we deduce the following result. Corollary 6.1 5. Then

Suppose G has all its Cartan subgroups conjugate.

44 = d ( P , a o ) ) / d ( 4 ao)) where q(x) is the polynomial

of degree $ dim N . 4. We shall see later that every detail in the formula for the c-function is conceptually significant: its behavior at infinity is significant for the spherical transform, the location of its zeros significant for the proof of the Paley-Wiener theorem in the next section, the numerator enters into the theory of conical distributions and intertwining operators (cf. Helgason [1970al) and the denominator decides the irreducibility of the eigenspace representations for G / K and the main properties of the Poisson transform (cf. Chapter 11, @ No.I1). , In view of these examples, the c-function gives a graphic illustration of the gamma function’s appearance “in nature.”

57. The Paley-Wiener Theorem and the Inversion Formula for the Spherical Transform According to the classic Piley-Wiener Theorem 2.9) the Fourier-Laplace transform

theorem

JRn

(Introduction,

f(x)e-i(”*5’ dx, f ( x ) -,f * ( ( ) = (1) where ( x , C) = x , c l . . . + x.5. (x E R”,c E C”) maps gA(R“) onto the space P ( C ” ) of entire functions F of exponential type A, that is, entire functions satisfying for each N E Z+ the inequality

+

(2)

suplF(c)l(l 5

+ lcl)Ne-Al’mcl<

00.

$71

7. Paley-Wiener Theorem. Inversion Formula

449

We shall now establish an analogous result for the spherical transform n

(3)

f ( l )= J f ( g ) 4 - I ( g ) dg

( f K-bi-invariant)

c

for a connected, noncompact, semisimple Lie group G. In other words, we shall give an intrinsic characterization of the functions f(1)as f runs through the space 9P(G) of functions in 9 ( G ) which are bi-invariant under K . For this we make extensive use of the results in @5, 6. In general if S is a function space on G (say S = b(G), Lp(G), etc.) Sb denotes the subspace of functions in S which are bi-invariant under K . 1. Normalization of Measures

It is convenient now to establish some conventions about the normalization of certain invariant measures. The groups G, K , A, N , M , and their Lie algebras g, f, a, n, fi, m being as in §&I-6, let as usual 1 = dim a. The Killing form on g, denoted by B or ( , ), induces Euclidean measures on A, a, and a*; multiplying these by the factor (2~)-(''~)',we obtain invariant measures da, d H , and d 1 and the inversion formula for the Euclidean Fourier transform

m,

f ' * ( ~ )= S,f(.)e-iA(loga)da ,

(4)

l~a*,

holds without any multiplicative constant, f ( a ) = ~ ~ * ( l ) e i A (dl o l g a ). f € W ) .

(5)

1

These choices of da, d H , and d l are called regular normalizations. We normalize the Haar measures dk and dm on the compact groups K and M , respectively, such that the total measure is 1. The Haar measures on the nilpotent groups N , N are normalized such that

The Haar measure dg is then normalized such that

l;

j ( g ) d g = /KANf(kan)e2P(10ga' dk da dn,

f

E 9(G),

(cf. Chapter I, $5). The measures dg and dk define the invariant measure d x = dg, on X = G/K (Chapter I, Theorem 1.9).

450

SPHERICAL FUNCTIONSAND SPHERICAL TRANSFORMS

[Ch. IV

2. The Image of g q ( G ) under the Spherical Transform. The Paley-Wiener Theorem As before we Put

(6)

~ , ( a )= ep(log4)/ / ( a n ) dn,

f ' E kaq(G).

Then we know from Chapter 11, $5, Eq. (37), that F , is W-invariant and

f(A)

(7)

=

1

F , ( U ) ~ - ' " ' ' " ~da. ~)

A

Let R > 0 be such that the function g K + f ' ( g ) on X = G / K vanishes outside the ball BR(0).Then by [DS] (Chapter VI, Exercise B2 (iv)) (8) F,(a) = 0 if d(o, a - 0) > R , d denoting distance. But then (7) and (2) imply that for each N there exists a constant C Nfor which

(9)

~ f ( n ) lI cN(l+ IAI)-NeRllmAI,

E Z',

IZEa$.

Furthermore, f is W-invariant (since F , is). Let X R ( a E ) denote the set of entire functions on a: satisfying (9) and as before put X ( a $ ) = X R ( a $ ) . We write &?,(a$) and X k ( a $ ) for the respective subspaces of W-invariant elements.

UR,O

Theorem 7.1. The spherical transjorm f + f is a bijection of' @ ( G ) onto Xw(aE). Moreover, the jirnction g K + f(9) has support in the closed ball BR(o)- if'and only gf E X$(az). Proof. It remains to prove the surjectivity. For this we start with a simple estimate of the c-function. Proposition 7.2.

For suitable constants C1 and C 1 ,

Ic(A)l-' IC ,

+ CzIIz(p

if Re(il)ECl(a*,).

Here p = $ dim N . Proof. By a well-known property of the Gamma function (see, e.g., Titchmarsh [1939], Chapter IV),

(10)

lim ' ( z + a) e-alogz = 1,

lz 1 -t a

'(z)

largzl I II - 6,

where U E Cis any constant, log is the principal value of the logarithm and 6 > 0. Putting z , = (iA, ao) and using Eq. (12) of 56, we have

c(A)-

=

c;'

n

a€&$

Ca(Za),

471

7. Paley-Wiener Theorem. Inversion Formula

45 1

where

It follows from (10) that lim ~ , ( ~ ) 1 ~ 1 - ( 1 / 2 ) ( m . + m z . ) = 2n1/2 14-m

in the half plane Re z 2 0. Since Re((& a,,)) 2 0 for each a EX', proposition follows from the last formula.

the

Next, let F be a W-invariant holomorphic function on a: satisfying (9); i.e., for each N E Z', We write the expansion (17) of 45, in the form 4,(exp H ) =

(12)

1 $,(A,

H),

1 E 'a;, H Ea+,

,€A

where (13)

$,(A, H) =

c(sA)r,,(sA)e(isA"-P-p)(H), A E 'a:, H E a'. saw

Motivated by the inversion formula [Theorem 7.5(i)] we consider the K bi-invariant function

f'b)=

(14)

I

F ( 4 4 , ( g ) I c ( 4 I - dx,

9 E G,

a*

the convergence being assured by Proposition 7.2 and (1 1). Using Theorem 4.3 we see that for each D E D ( G ) , ( D 4 J ( g ) is bounded by a polynomial in A, uniformly for g varying in a compact set. It follows that f' E b(G). We wish to prove f ( g ) = 0 for g K # BR(o),or, equivalently, f(expH) = 0

for H E a f , (HI > R.

For this we use the expansion (12) and have for H E a', h = exp H (15) f(h) =

J F(4 1 a*

H ) I c ( 4 I- d l

c

=

1 p(A)$,(A,

H ) Ic(4 I -

PEA

the term-by-term integration being justified by Lemma 5.6. Note also that for A E a* (16)

lc(A)12

=

c(A)c(-A)

=

c(sA)c(-sA)

(SE W ) ,

and the integrand in the last integral is well-defined.

452

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

The following result is crucial both for Theorem 7.1 and for the proof of Theorem 7.5.

Then f o r each p E A

Theorem 7.3. Let F E XE(a:).

I*

F(A)$#(A, H ) ~ C ( A )d~A -=~ 0

i f ' H E a ' , [HI > R . Proof.

Because of (13), (16), and the W-invariance of F it suffices to

prove

I*

F(-&(5)-'e-ir(H)r P ( - 5 ) d t

(17)

=0

for H E a+, [ H I > R . To do this we shift the integration in (17) into the complex space a: in such a way that the singularities of the functions c(A)- and I-,( - A) are simultaneously avoided, Cauchy's theorem applicable, resulting in estimates for (17). By the recursive definition of the rational function A -,r,,(-A) it is clear that it is holomorphic outside the set where ( v , v ) = 2 i ( v , -A) for some v E A - {0}, that is, outside the set

'

(18)

{ A E a: : ( v , A)

= $(v, v )

for some v E A - {O}}.

On the other hand, the function A + c(A)-' is holomorphic on the complement in a: of the set (19)

{ A E ~ E : ( A , ~ , ) E + ~-((0)) Z + for some aEZ,'}

Fortunately, both sets (18) and (19) are contained in the set {A~a::i(v,il) < 0 for some V E A- {O}}.

(20)

The domain of integmion in (17) will now be shifted away from this set. Consider any A E a: for which Re(iA) E a*+. This means that if il= 5 + iq (5, q ~ a * and ) A , = A , iA, then - A , E ~ + .Let H 1 , .. . , H I be the basis of a dual to the basis a l , ..., a1 (of simple roots). For A ~ a z we write A , = AjHj ( A j = t j + iqjc C)and identify a function 4 on a subset of a: with a function on a subset of C' by the convention 4(A)= &Al,. . . ,A,). All ti, q k are real and q k < 0 (1 I k I0. The function

+

cj

G(A) = G(Al, . . . , A l ) is holomorphic at each point A = (tl

+ is'q,,.

. .,

=

F ( - A ) C ( A ) - ' ~ - ' ' ( ~ ) TP( - 4

+ is1ql)

(0Is1 I1,. . . , O IsI I1)

+

because condition (20) is violated. In fact, if V E A- {0}, v = nlal + nlal then ( v , A) = n,{rj + isjqj) has a nonpositive imaginary part.

cj

- . a

7. Paley-Wiener Theorem. Inversion Formula

571

453

If the function G(A) behaves suitably at infinity we can therefore conclude from Cauchy's theorem that r

r

In order to estimate G we look at the recursion formula (12) of 55 for I-,, and observe that, by induction on rn(p),

r,(-(t + iq)) = p,(5 + iv)(q,(5 + iq))-', where p,, and q, are polynomials and 4,,(5

+ iq) =

n

O # v,p -YE A

( ( v , v> - 2(v, q )

+ 2i
Consequently, we have for a constant c,

1 y - 4 5 + iqNl I c,lp,(t

(22)

+ iq)l,

( € a * , -?€a*,.

The estimates (11) and (22) together with Proposition 7.2 show that the application (21) of Cauchy's theorem is legitimate, so for each - r ] ~ a * , ,

.J

F( - <)c(()- le-it(H'r,(- 5 ) d<

=

1.

F ( - ( - iq)c(t + iq)-'e-'r(H)+'J(")r(-.( - ir]) d5.

Choosing the integer N in ( 1 1) equal to 1+ 1

+ deg(p,) + P,

we have

(23)

I Q ( H )1 I

CeRl"le"'H) for all H E a + , -q

E a*+,

C being a constant. Now fix HE^+ with IHI > R. Let A , = - i H ( t > 0). Then by (23)

IQ(H)I I Cef l H I ( R -

r]

be such that

IHI).

Letting t --t +a, we obtain Q ( H ) = 0, and the theorem is proved. Now ( 1 5 ) shows that

(24)

f(g) = 0

if gK #BR(o).

To conclude the proof of Theorem 7.1 it remains to prove that f is a constant multiple of F. For this we anticipate Theorem 7 3 i ) below. This result [and (14)] imply that the function $(A) = ?(A) - cF(L) satisfies n

454

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Integrating this against any h E Qh(G), we obtain

s,.

$ ( I ) ~ ( - I ) I c ( I ) I -dI ~ = 0. (26) By Theorem 4.3, 4A= +,, implies that I and p are W-conjugate; hence the functions h [h E g'(G)] separate points in the space a*/W. They vanish at 00 and form an algebra of functions on a*/W closed under complex conjugation because of the formulas (hi * h2)"(I) = hi(I)hz(A),

(27) (28)

(h*)I(I) = (&I))-

if h*(g) = h(g- l),

the latter formula coming from Eq. (7) of $4. Now the Stone-Weierstrass theorem on a*/W, formula (26), and the W-invariance of $ imply that $ = 0 on a*, and hence on a: by holomorphy. This concludes the proof of Theorem 7.1 [modulo Theorem 7,5(i)]. Corollary 7.4. Let g W ( A ) denote the subspace of W-invariants in 9 ( A ) . Then f + F , is a linear homeomorphism of @(G) onto g W ( A ) . Moreover, * denoting convolution on G and on A ,

(29) F,,*,z = F,, * b z , In fact, because of (6) and (7) the map J'+ F , maps @(G) continuously into g W ( A ) . Because of (7), Theorem 7.1, and the PaleyWiener theorem for the Euclidean space A we see that the map is injective and that the image equals gW(A).Because of Lemma 1.13, gh(G) and QW(A) are LF-spaces (cf. Proposition 1.12 and preceding remarks). Being a one-to-one continuous mapping of one LF-space onto another, the map f + F , is a homeomorphism. 3. The Inversion Formula

Preserving the notation above, we shall now prove the inversion formula and the Plancherel formula for the spherical transform. Theorem 7.5.

The spherical transform satisfies

(i) c f ( g ) = j a * f ( W n ( g ) I c(I)I-' d l , where c is a constant independent off ;

E G,

.f E =@(GI,

(ii) cjGIf(g)12dg = ~ ~ * I J ( I ) I ~ I C ( I ) I - ~ ~ I , and the image Qh(G)- is dense in LZ(a*/W,Ic(~)(-~ dl). Here the normalizations of d I and dg can be arbitrary.

971

7. Paley- Wiener Theorem. Inversion Formula Proof. (i) For $ E 9 ( X ) define

E

455

@ ( G ) by

Then T is a distribution because it is composed of the maps $ E %X) JE

+

$'

9'(G) + F,

9EW

)

+

E %G)7 E9(A),

j-p*(l)lc(l)l-2 d l E c7

the first two being obviously continuous, and the third being continuous because of (4)and Proposition 7.2. We shall now prove that the distribution T has support at the origin o = {K} in X. For this fix a function f o e 9 ( A ) with support in the unit ball B , ( e ) - such that in addition the function F ( I ) = f g ( l ) satisfies F(0) = 1, F 2 0 on a*, F is radial (in particular, W-invariant). Let $ E 9 ( X ) , f = $'. By dominated convergence, and the Fubini theorem

f(l)l~(l)l-~ dl

T($) =

=

=

6.

f(A)F(~l)lc(A)l-~d l

)

lim [~JGf(s)$-ds) dg F(E41c(4r2 d l &+

=

lim E-0

a*

0

lim &+O

s,

$(x)h,(x) dx,

where

(32)

h,(gk) = ~ w ) 4 - i ( s ) l c ( A ) l - 2

dl.

Thus, viewing h, as a distribution on X,

(33)

T($) = lim h,($). &+O

Because of (2), (15), and Theorem 7.3 h, has support in the closed ball B,(o)-. Hence supp(T) c (0).

456

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Next we show that T can be extended to a measure. For this it suffices to prove that the L'-norm llhell remains bounded as E + 0 because then by (33).

I T ( $ )I

I lim SUP L+O

s,

I J/(x)h,(x) I d x

I lim SUP l l $ l ~ ~ ~ ~ ~ l I l mconst l l ~ l$(dl, ~lll e-0

11

)Im

denoting the uniform norm. Now

But using Proposition 7.2, and the boundedness 1 $A(g)I I C for

A E a*,

gK E BAo)

*.J

Jo*

so

On the other hand, X being a Riemannian manifold, ') voI(B,( 0 ) ) = o(cdim

for

E

small.

These inequalities show that Ilh,ll is bounded as E + 0 so T can be extended to a measure. We have now proved that T ( $ ) = c$(o) where c = const; in other words, (34)

f(l)l~(l)l-~d l ,

cf(e) =

f E 9'(G).

0'

It is clear from (27) and (28) that c # 0. To prove Theorem 7.5(i) for each g E G consider the function

7. Paley-Wiener Theorem. Inversion Formula

071

457

Then h E @(G) and using Proposition 2.2 we have

=

4- As - ')f(4,

so by Eq. (7) of &I,

i;<4= 4n(S)f(O Now (i) follows by using (34) on the function h. To derive (ii) we use (34) on the function g 1 -+ f(glg)f'(s) dg, taking (27) and (28) into account. The last statement of the theorem is clear from Theorem 7.1. Remarks. 1. The reader will have noticed that Theorems 7.1 and 7.5 are curiously intermingled. Both theorems use Theorem 7.3, which in turn relies on properties of the expansion for 4nand the c-function. 2. With specific normalizations of dlZ and dg the constant c in Theorem 7.5 can be determined (see Exercise C4). 3. Theorems 7.1 and 7.5 deal with functions on G , bi-invariant under K , that is, with functions f on X = G / K that are K-invariant: f ( k . x ) = f ( x ) . For general analysis on X it is of course essential to drop this invariance condition and to consider arbitrary functions F on X . Writing A(gk, k M ) = A(k-'g) ( k E K, g E G), we define the Fourier transform F of F by

F(x)e'-i"+p"A("*b))dx,

A E a:,

bEB

=

K/M.

This was introduced in the author's paper [1965b]; a Plancherel- and a Paley-Wiener-type theorem were given there and by the author ([1970a] and [1973a]). See the Introduction (Theorem 4.2) for the simplest case. 4. With the density function (35)

we have with suitable normalizations

458

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

If 6, is the density function 6 for the rank-one space G a / K , (c(EZO+) formula (35) can be written

where H , is the orthogonal projection of H on the space a, = RA,. In view of the Plancherel formula (with d 1 properly normalized)

l+

If

(exp H ) I2W) dH =

j I A4 IZ

I c ( 4 I - CiA

( fEWG))

0%

the product formula (36) makes the Gindikin-KarpeleviE product formula

c(A) = co

n c,(A,)

01 E

xA

(co = const.)

look quite reasonable; it is an intriguing question whether it can be proved by such simple considerations.

58. The Bounded Spherical Functions 1. Generalities

According to Theorem 3.3 the continuous homomorphisms of the convolution algebra L'(G) ( = L'(G)h) into C are the maps

where 4 is a bounded spherical function on G . When G is connected, noncompact semisimple each spherical function $ has the form $ A (A E a;) and we are thus led to the question (2)

For which 1 E a; is

4Abounded?

By Banach algebra theory the kernels of the maps (1) are the regular maximal ideals of LP(G),so an answer to (2) amounts to a parametrization of the maximal ideal space of Lb(G). Theorem 8.1. The spherical function 4A is bounded ij' and only 1 belongs to the tube a* + iC(p), C(p) being the convex hull of the points sp ( s E W).

Let Co(p)denote the interior of C(p).

8. The Bounded Spherical Functions

681 Corollary 8.2.

459

The spherical transforms JG

+ iCo(p).

are holomorphic in the open tube a*

In fact, the integral of f ( I ) over any plane closed curve in the tube + iCo(p) is 0, 4 - Abeing bounded and holomorphic in I . By Morera's theorem, f is holomorphic. The proof of Theorem 8.1 will be based on a few lemmas. The first is a geometric result which is only needed for the point H = A,, but the proof works for any H E p . a*

2. Convex Hulls

Consider the positive Weyl chamber a + and as in $6. No. 3 the dual cone + a = {HEa:(H,H')>OforallH'~a+}. Let a- = - a + , - a = -(+a) and let C + , C - , + C , and - C denote the closures of a+, a-, +a, and -a, respectively. For H E a let C ( H ) denote the convex hull of the orbit { s H : s E W } . Lemma 8.3. Let H E C + . Then

u

+ - C ) } = sn s(H + -C). ew n C ( H ) = C+ n(H + -C).

C(H) =

(i)

s{C+ n ( H

S E W

(ii) C f

Proof. We first prove (ii). Given I E a* such that Al E C + , consider the minimal a E R such that the half space I - a I 0 contains C(H). I f H o E C ( H ) satisfies I ( H o ) = a and if we write H o = a,(sH) (a, 2 0, as = 1) then some sH satisfies I(sH) = a. Hence Il(sH-H')>O for all H' E C ( H ) . Since I ( H - s H ) 2 0 ([DS], Chapter VII, Theorem 2.22) we obtain by addition A(H - H') 2 0. Thus H' - H E - C ; that is,

csEw

cs

C(H) c H + -C. (3) On the other hand, suppose H' E C + n ( H + - C ) . To prove H' E C ( H ) it suffices to prove that for each I E a*, there exists a o E W such that I ( o H ) 2 I(H') [so H' and C ( H ) cannot be separated by a hyperplane]. For this we select CT E W such that o-'Al E C + . Then since the relations

H

-

H'

E

+ C , H' - o-'H

we have ( a - ' A l , H (ii).

-

E

+C

imply H - o - ' H '

E

+C

o - ' H ' ) 2 0 ; that is, A(oH) 2 A(H'). This proves

460

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Let C ( H ) , C1, and C2 denote the three terms in (i). By (ii), C ( H ) c C,; by (ii) and (3) we have for any s, CT E W,

+ - C ) ) = sC+ n C ( H ) c C ( H ) c a(H + -C), C1 c C1. Finally, if H' E C2 then sH'E H + -C for all s E W.

s(C+ n (H

whence Choosing s such that s H E C+,(ii) implies sH' E C ( H ) so H' E C(H).Thus C2 c C ( H ) and the lemma is proved. We prove now the first half of Theorem 8.1. Lemma 8.4.

4Ais bounded

if

1 E a*

+ iC(p).

+ iq (5, q E a*). Because of the inand because of Lemma 8.3(i) we may assume

Proof. As usual we write A = 5

variance

&=

- A , E C+ n (A,

+ -C).

Let Haar measures be normalized as in $7. No. 1. I f f € Lb(G),we have by the Fubini theorem JGf(g) dg = /ANf(an)e2p(10ga) da dn,

the function

exists for almost all a E A, and

(4) Since - A , E C+ we have - s q ( H ) I - q ( H ) for H E a + , s E W, so if A + = exp(a + 1,

n

since p + q is nonnegative on a+. Using (4) and the Weyl group invariance of F , we have, if I W I denotes the order of W,

083

8. The Bounded Spherical Functions

46 1

Thus by ( 5 )

so

4i,,, and therefore 4A,is bounded.

Lemma 8.5.

r]

Suppose

+ p is nonnegative on a+. Proof.

4a

is bounded and that

- A , E ~ + . Then

Let H E a'. By the assumption, Re(iA) E a*, so by Theorem

6.14

lim e(- id + p ) ( t H )4&XP t H ) 1-+m

exists and is #O. Since 41 is bounded we conclude by taking absolute values, (r] + p ) ( H ) 2 0. This proves the lemma. In order to generalize this lemma to the case - A , € ? we use the concept of a boundary component of the symmetric space. This tool was introduced by Harish-Chandra [1958bI in a different context.

3. Boundary Components Fix an element H o E a+ and let XO =

{ a E X:a(Ho) = O}.

Let g' denote the subalgebra of g generated by the root subspaces go! (a E 2)).

462

[Ch. IV

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

Lemma 8.6. tion

The algebra g' is semisimple, has a Cartan decomposi-

g' = f'

+ Q'

where t' = g' n f,

Q' = g' n Q,

and the space a' = g' n a is a maximal abelian subspace of' more, in the orthogonal decomposition

Q'.

Further-

a = a ' + a ,0

(6)

with respect to the Killing form of g we have

(7)

a' =

c RA,,

a' =

{H E a : a ( H ) = 0 for a E Co}.

zsro

Proof. The proof is analogous to that of Proposition 2.1, Chapter IX in [DS]. Let g' denote the complexification of g and extend a to a maximal abelian subalgebra t, of g; then If is a Cartan subalgebra of g' ([DS], Chapter VI, 53). Then

where p is a root of (gc, hc) and p denotes the restriction Pla. The subalgebra ij of gc generated by (s')~ ( p E Xo) is semisimple and has the subspace 6 = & e E ~ CH, as a Cartan subalgebra ([DS], Chapter 111, Exercise Bl). The real form ij n g, which equals g', is also semisimple. Invariant under the Cartan involution 0, g' has the indicated Cartan deRA, equals composition ([DS], Chapter IX, Lemma 2.2). The space 6 n Q' and is thus a maximal abelian subspace of Q'. This space equals gf n a because

cDlsr0

ij n p'

c a n g' = a n Q' c

ij n Q'.

This proves (7) and the lemma. The restrictions a I a' (a E C ' ) with respect to a'. The subset (9)

form the set C' of restricted roots of g'

(a')+ = { H E a' : a ( H ) > 0 for a E C ' n X+} = a' n a+

is a maximal connected subset of (a')', hence a Weyl chamber in a'; let (C')' denote the corresponding set of positive roots. We have the compatible Iwasawa decompositions (10)

where

g=f

+ a + n,

g' = f'

+ a' + n',

8. The Bounded Spherical Functions

§81

463

Moreover, if G', K ' , A', N' are the corresponding subgroups of G we have

K'

(12)

=

G' n K ,

N' = G' n N ;

A' = G' n A ,

in fact, the inclusion c is obvious in each case, so (12) follows from G'= K'A'N'. The symmetric space X ' = G'/K' is called a boundary component o f X = G / K . Since p' is clearly a Lie triple system, X ' is a totally geodesic submanifold of X (cf. [DS], Chapter IV, $7). Lemma 8.7.

Let W' denote the subgroup of W fixing each point in

a' and I W'I its order. Then

(i) The mapping

is the orthogonal projection of a onto a'.

(ii) Zf'H

E a+,

then

a(n'(H)) > 0

jbr each a E 1 ' - Co n C+.

Proof. First, if L Y ECo we have a(n'H) = 0 so n'H E a'. Second, if A' for each s E W ' so

S,E E

W' so s,(n'H) = n'H; thus

ao is arbitrary we have sA' = A'

1 B(H - noH, A') = B ( H , A') - 7 B(sH, A') = 0,

Iw I

1

SSW'

proving (i). For part (ii) let n' : a + a' be the orthogonal projection and let H Since

a(H - n'H) = 0 for LY

E

E a+.

Co,

we have (13)

a(n'H) > 0 for a E 1 ' n C+.

Let B' denote the Killing form of g' and, given the restriction Cr = a(a' (a E C '), let A , E a' be determined by B'(H, AE)= a(H) ( H E a'). Then if X , E ga we have E X , , ex,] = B(x,,e x , ) A , = B'(x,,ex,)A, (cf. [DS], Chapter IX, 51). It follows that (14)

A,

=

c,Aa,

C,

> 0.

464

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

As noted in 56, No. 3,

(15) simple

+

But if Ei E (C')' is simple, ai itself is simple. In fact, if cli = j y, p, y E C+,then 0 = ai(Ho) = p(Ho) + r ( H o ) so p, y E Co and & = fl + 7, contradicting the simplicity. By (13), d(H)E (a')', so by (14) and (15)

d(H)=

(16)

C biAu,

(bi 2 0).

Ui E XO

simple

Now H = d(H)+ x'(H). Suppose P E C+ - X' n C + is simple. Then by (16) and [DS] (Chapter 111, Lemma 5.6), we have P(n'(H)) I 0 so O(n'(H)) = p ( H ) - /?(n'(H))> 0. Finally, if a E C+ - C ' n C+ is arbitrary we write o! n i p i (ni > 0, pi simple). Then since a(H') 0 and Pi(Ho) 2 0 for all i, we have pi E C+ - Z' n C+ for some i). Writing

=xi

=-

we therefore conclude that

and the lemma is proved. Remark. By general theory ([DS], Chapter VII, Theorem 2.15) W' is generated by the s, for which a E C vanishes identically on a'; that is, a E 72'. Thus the restriction W'la' constitutes the Weyl group of X' = G'/K'. In order to make use of Theorem 6.13 later we insert here the following lemma. Lemma 8.8. There exists an s E W such that Z+ - ' X n Z' = Z+ n s- 'Ed.

Proof. Let bl,.. .,pp, /3p+l,. . . ,j l be the simple roots such that B1(H') = . . . = p,(H0) = 0 but @+, l(Ho) > 0,. . . ,pl(Ho) > 0. We change H ' slightly to a regular element H E satisfying

(17) fli(H') < 0, -

p p + l(He) > 0, * * fip(He)< 0; We claim now that if 1 H E - H ' I is sufficiently small,

(18)

{ p E c+:p(H0) > O}

=

9

bi(HE)> 0.

{ p E c+:p ( H E )> 0).

§81

8. The Bounded Spherical Functions

465

The set on the left is contained in the set on the right if Hsis sufficiently close to Ho. On the other hand, suppose p E C+,p(He) > 0. Writing /3 = n i p i (ni E 2') we see from (17) that n, > 0 for some i > p. But then p(Ho) = ni(Pi(Ho)> 0, so p belongs to the left-hand side of (18). Thus

If=

cp+

C+-ConC+ = { p ~ C + : p ( H ~ ) > o } ,

which equals C+ n s-'Z- if Heinto - a + .

s

is the Weyl group element which sends

Lemma 8.9. Suppose + A is bounded and that - A , € boundary of a+). Then q p is nonnegative on a+.

+

Proof.

a? - a +

(the

Let H o = - A , and construct g', etc., as above. Put A" = x+ - Co n Z+ A' = xon Z+,

(19)

iif =

c

g-a,

A" =

c

g-a,

asA"

asb'

and let R and N" denote the corresponding analytic (nilpotent) subgroups of G'. Let 2p', 2p" denote the sums of the roots in A' and A", respectively. Because of Lemma 6.8 and Eqs. (19) and ( 2 0 ) of @ we can select Haar measures du, dw on R and R", respectively, such that the product measure dv x dw corresponds to dii under the diffeomorphism (u, w ) + uw of ml x mll onto N. Let us use this change of variables in the formula for ~ $ ~ (ina )Proposition 6.3, and let a E A' = exp(aO). Putting p = il - p, v = -(U + p), we have, since a centralizes R, ep(H(aRa

- p('oga)

ss

-'))ev(H(n)) dfi

ep(H(vawa-

du dw,

'))v(H(UW))

1' R,,

Here we decompose u according to G' = K'A'N', o = k'a'n', and use the fact that G' normalizes iy" [if a E Z, a(Ho) > 0, then a E A ] . Then the last expression becomes p ( H ( a a ' W a - l))ev(H(a'w))du dw

e)l(lOPG)s

1,

a-

466

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

where we have used the Jacobian formula d(a'w(a')- ')/dw = e-2p"(10ga'). Since ,u + v + 2p" = -2p' this proves the formula (20)

41(a)= c e W - p ) ( l o g a )

s,.

e(iA-p)(H(awa- ' ) ) - ( i A + p ) W ( w I )dw

(a E A')

where the constant c equals jp e-2p'(H'")) do. Now let H E a be such that (21)

a(H) = 0 for a E A ,

a(H) > 0 for a E A ,

+

and put a = exp t H ( t > 0) in (20). Select E (0 c E c 1) such that p cq E a*+. Then [cf. Eq. (41) of $663 the integrand in (20) is majorized by e(eq-p)(H(w)), which is integrable on P (Theorem 6.13 and Lemma 8.8). Therefore (22)

lim

e(-il+p)(fH)4A(eXP

tH) =

1-+m

, - ( i l + p ) ( H ( w ) ) dw,

JN,,

which by Theorem 6.13 and Eq. (39) of §6 is #O. Taking absolute values and using the boundedness of 4Awe deduce that (23)

(q

+ p)(H) 2 0

if H satisfies (21).

Finally, let H E a+ be arbitrary. We use p ( H - sH)2 0 (s E W), the orthogonality q(H - no(H)) = - ( H o , H - .'(El)) = 0, and the fact that no(H) satisfies (21) because of Lemma 8,7(ii). Then, by (23)

+ p ) ( H ) = (? + p)(n'(H)) + (q + p)(H - .O(H)) = (q

+ p)(nO(H))+ p(H - .'(If)

2 0.

This proves the lemma. Now Theorem 8.1 follows easily. Suppose 4Ais bounded. Select S E W such that A-,,,E C + . Since 4sA= 4A, Lemmas 8.7 and 8.9 imply A,,,, E +C;i.e., A-s,, E C+ n ( A , + -C), so by Lemma 8.3, q E C(p). We restate (22) as a corollary generalizing the limit relation of Theorem 6.14. Corollary 8.10. Let Re(il)ECl(a*+), put Ho = ARe(il)and defne ao and s as in Lemmas 8.6 and 8.8. Then if H E a' n a+, ljm

e(-iA++p)(W

4A@P t H ) = c s ( 4 9

I'+a,

provided the measure on N is appropriately normalized.

991

9. The Spherical Transform on p, the Euclidean Type

467

89. The Spherical Transform on p, the Euclidean Type

We consider now the symmetric space G o / K = p of the Euclidean type and the corresponding spherical functions

as given by Proposition 4.8. The corresponding spherical transjorm is given by

f being a K-invariant function on p and L E a: being such that the integral exists. We shall now establish the analogs of Theorems 7.1 and 7.5 for this case. According to Chapter I, Theorem 5.17, we have for F E .9(p),

where d Y and d H are the Euclidean measures, dk the normalized Haar measure, and

(3)

6 ( H ) = co

1 n I,

C~(H)"~

H E a,

asX+

co being a constant.

We consider now the function spaces Q(p), 9 ( p ) , Q(a*), 9'(a*) and the subspaces 9,(p), Y,(p), gw(a*), Y w ( a * ) of K-invariants and, respectively, W-invariants (cf. Chapter 11, $5, No. 2). Theorem 9.1. onto Y,(a*) and

The spherical transform f

(4)

(5)

where c is a constant independent off.

f

is a bijection of 9&)

468 Proof.

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

I f f * is the Euclidean Fourier transform

we see that j"*(Al)=?(A), so (4) and ( 5 ) follow immediately from the inversion formula and the Plancherel formula for the Euclidean Fourier transform if we take (2) into account. For the surjectivity in Theorem 9.1 use Corollary 5.11 of Chapter I1 as well as the fact that F + F* maps Y ( p ) onto itself. Next we would like to characterize the image of the space !2&) under the spherical transform. Let q be the orthogonal complement of a in p and put

(7)

f(H)=

1

f ( H + Q)dQ,

f

E QK(P),

q

dQ being the Euclidean measure. Then (1) can be written (8)

f(4 =

1

e-"")f(H) dH;

0

this formula is the Euclidean analog of Eq. (7) of 57. If f has support in the ball I YI 5 R, then by Eq. (2) of $7, 3. XE(aE), f clearly being W-invariant. Theorem 9.2. The spherical transform f + f on p is a bijection of Q K ( p ) onto .Ww(az). The function f has support in the ball I Y I I R i f and only iff E .W $(a:).

Although this theorem is an analog to Theorem 7.1 we cannot imitate the proof of that result, not having a series expansion of @ A analogous to that for 4A.If on the other hand we want to proceed as for Theorem 9.1 and make use of the relation (9) we are led to asking for the following result. Theorem 9.3. Let 4 be a W-invariant entire function of exponential type R on 8. Then 4 extends uniquely to a K-invariant entire function Q, of exponential type R on the complexijcation pc of p.

Although this result has a certain formal resemblance to the extension theorems in Chapter 11, $5. No. 2 (Theorem 5.8 and Corollary 5-11), it requires entirely different tools, partly because Int(F)(ac) # pc, in contrast to Int(f)(a) = p (Int denotes an adjoint group).

9. The Spherical Transform on p, the Euclidean Type

091

469

As in Chapter 111, $3 let I ( @ ) denote the algebra of W-invariants in the symmetric algebra S(aE) and let j l , . . .,j , be homogeneous algebraically independent generators of I(aE). We know from Corollary 3.12 (loc. cit.) that the mapping

(10) j : H + (jl(H), . . . ,j , ( H ) ) induces a bijection of ac/W onto C'.

(HE a')

Lemma 9.4. The mapping j : 0' -,C' is proper (that is, the inverse image of any compact set is compact). Proof. Let n = dim p. For any k > 0 there exists a positive number rk such that for any monic polynomial p(A) = 1" + ci 1'where ciE C the inequalities lcil Ik (0 Ii I n - 1) imply lpl Irk for any root p of the equation p(1) = 0. (In fact if lpl 2 1 we would have lpl" IknJpI"-', i.e., lpl Ikn, so we can in any case take rk = kn + 1.) Now let H E a' and consider the characteristic polynomial

x:-

det(A1 - (ad H)'lp') E A" + U"-~(H)A"-' + (11) + aI(H)A1, the roots of which are 0, a(H)? (a E C). There exist unique polynomials p o - l,. . . ,pl such that ai = p i ( j l , .. . ,j,). This means that

(12) Let D

ai(H) = pi(j(H)), c

H E a'.

C' be any compact set. Let

k

= S~PlPi(Ol* {ED,

all i

Then by (12), Iai(H)I Ik for all H E j - ' ( D ) . But then by the above

I a(H)I Ir:/'

for all H E j - '(D).

This means that the closed set j - '(D)is bounded, hence compact, and so j is proper. Lemma 9.5. Each W-invariant holomorphic function q5 on a' extends uniquely to a K-invariant holomorphic function 0 on p'. Proof.

(13)

Let

+(H)= 1aa,, .... a,&' . .

*

H

= (h1, *

W

be the Taylor expansion of

4, which we write as

*

9

hJ E a',

470

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. 1V

where Am(H) = a1

1

+ ... + a r = m

GI,

..., a l h t l

* * *

Kl-

Each A,,, is W-invariant and can therefore be expressed Am =

C bp ,..... p,i?

. .jP, *

(8)

c’,

where pi deg(ji) = m. Since the mapping j : ac + C’ is surjective we conclude from the absolute convergence of (14) that the series

where 5 = (tl,.. . ,5’) E C’and

w<) = 1 bp,, B

...,pltfl

*

* < ~ 1

(1Pi deg(ji) = m i

converges absolutely. We shall now prove that (15) converges uniformly on each compact subset D c C’.[Since (15) is not a power series this is not evident.] By Lemma 9.4 the set b = j-’(D) is compact. The series (13) is a power series so it converges uniformly on the compact set b; so does the series (14). But B,(j(H)) = Am(H),so we conclude that (15) converges uniformly for E j(b)= D . This proves that +* is holomorphic on C’.Now each j i extends to a K-invariant polynomial Ji on pc (Chapter 11, Corollary 5.12). The function @ ( Z ) = 4*(Jl(Z), . . . ,J,(Z)) has the property required in the lemma. In order to prove Theorem 9.3 let gc = F + pc be the complexification of the Cartan decomposition g = f + p . Let GC be the simply connected Lie group with Lie algebra gc, K C , G, and U the analytic subgroups corresponding to F,g and u = f + ip. For X E pc we write X = X’ iX” (X’, X” E p). Let 4 E XR(aC) be W-invariant and 0 the extension to pc given by Lemma 9.5. If Z E pc the mapping k + @(k . Z ) - @(Z) is holomorphic on KC and vanishes for k E K ; hence it vanishes on KC so (D is Kc-invariant. In order to deduce from the inequality

<

+

(16)

I ~ ( H+’ i w ) I ~ CeRIwI,

H’, H

I@(x‘+ ~x”)I ICeRlx”l,

x’,X” E p ,

E

a,

an inequality (17)

by means of the Kc-invariance of @ it would suffice to show that the infimum of IZI as Z runs through the orbit K C -(X’ + iXf‘) is reached for some Z = H’ + iH” in aC; in fact then

1@(xf+ ~ x ’ =~ I) ~I ( H+’ ~ H ” )II CeRIH”II

C ~ I ~ ” ~ .

991

9. The Spherical Transform on p, the Euclidean Type

47 1

Let g t denote the Lie algebra gc considered as a real Lie algebra and BR its Killing form. Then BR = 2B on g x g. We have the Cartan decompositions

+ iu,

gf = u

GC = U exp(iu)

and we let q denote the corresponding Cartan involution of g: and GC. If Z = Z iZ” ( Z , Z € p ) then q ( Z ) = - Z + iz” so

+

z

”

= -

fi(Z

+ &q).

Using the invariance of BR under Ad(KC) and q, and the relation BR(iZ,iZ)= - B R ( Z , 2) we obtain for k E KC (18)

I(k*Z)”12= - $ { B R ( Z , Z ) + B R ( q ( k . Z ) ,k * Z ) } .

Now let X € p c be regular (and hence semisimple). Then the orbit KC - X is closed (Chapter 111, Theorem 4.21). Since BR(q(Z),2) S 0 it is clear that the function k + -BR(q(k . X ) , k . X ) has a minimum on KC. By (18) the same is the case for the function k + I(k . X)”I2 on KC. I f k , E K C is a minimum point we have by (18)

(i

Lo

BR(q((exprTk,) . X ) , exp t T k , . X )

for each T E kC. Using qz = 1 it follows that

=0

BR(q(k, . X ) , [F,k, * X I ) = 0. But BR is invariant and nondegenerate on fC so this implies [q(k, * X ) , k, * X ] = 0.

(19)

This shows that the elements (k,.X)’ and (k;X)” commute; since they both belong to p there exists a maximal abelian subspace b of p containing them both. Since b is conjugate to a under K there exists an element k , E K such that the vector H = k,(k,. X ) is in a‘. Then

H” = ( k o k , X)” = ko . ((k, . X)”), *

so

1

(20) This proves that if sion Q, satisfies

~ = 1min I ( k . X)”l. ksKC

I$

E &‘$(aC)

then its K-invariant holomorphic exten-

CeRIX”I (21) IQ,(X)I < for all X E pc which are regular and semisimple, C being a constant.\But such X are dense in pc so (21) holds on all of pc by continuity. We have

472

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

also by the K-invariance of constant CN such that (22)

(D

on p that for each N

I(D(X’)I IC N ( 1

E

[Ch. IV

Z + there exists a

+ lx’l)-N, X ’ E p .

Let E > 0 and let F be an arbitrary entire function on pc of exponential type E [cf. (2) of $73. Then (DF is entire of exponential type R + E and so, as recalled at the beginning of $7, the Fourier transform (OF)- = 6 * F has support in the ball Cl(BR+t(0)).Since F can be arbitrary in 9&(,,(p) it follows that 6 [which by (22) is smooth] vanishes for Y Ep, 1 Y I > R . Hence (D = XR(pC) so Theorem 9.3 is proved. Now Theorem 9.2 follows immediately: If $€.#;(a*) we define 4 E *&(aC) by $(A,) = $(A) and define the K-invariant (D E ZR(pC) as above. Then

@(X) =

/

f ( Y ) e - i B ‘ X , rdY, )

P

where f’E B,(p) and Y(A) = @(A,) =

J-Kw4) dk = m,

as claimed. Taking (8) into account we obtain from Theorem 9.2 the following analog of Corollary 7.4. Corollary 9.6. The mapping f 9 d P ) onto Qw(a).

-,f

is a linear homeomorphism of’

$10. Convexity Theorems

Let p, denote the space of all n x n complex Hermitian matrices and let a, c p, denote the subspace of all diagonal Hermitian matrices. Consider the projection Q : p, -, a, defined by ( Q Y ) i j= d i j xj; in other words, QY is the diagonal in Y. If Y Ep, let A1 2 A, 2 - 2 A,, be the eigenvalues of Y and H E a, the diagonal matrix with entries A1,. . . ,A,, i.e., H , = dijAi. Furthermore, if cr is any permutation of { 1, 2, . . .,n } let OH be the diagonal matrix with diagonal i.e., OH)^^ = 6iiAu-Ici,.Let C ( H ) denote the convex hull of the set of aH as cr runs over the permutation group G,, and let O ( H ) denote the set of all n x n Hermitian matrices with the eigenvalues A l l . . . ,A,. In [1954a], Horn proved the following result.

§lo1

10. Convexity Theorems

473

Theorem 10.1

I

Q(O(H))= C ( H ) . We shall now state and prove Kostant’s generalization of this result to the tangent space p to a symmetric space X = G/K of the noncompact type. As usual let a c p be a maximal abelian subspace and let Q : p + a denote the orthogonal projection (with respect to the Killing form). For Y E p put k . Y = Ad(k)Y (k E Y) and as in 58 let, for H E a, C ( H ) denote the convex hull of the orbit W . H. Theorem 10.2. For each

H E a,

Q(K * H ) = C(H).

For the proof we shall use the general notation from this chapter (especially &8,9). We begin by proving Q(K * H ) c C(H).

(1)

If this were false there would exist an element Z E K . H such that Q ( Z ) $ C(H). By the convexity, Q ( Z ) and C ( H ) can be separated by a hyperplane, so there exists an H, E a such that

(Ho,Q ( z ) ) > ( H o , 4H)),

(2)

CT E

W.

By continuity we can take H o regular. By the compactness of the orbit K H the function

Y + (Ho, Q(Y)> = (Ho, Y> on K . H has a maximum value at a point Z , . Then

(g

(H,,Ad(exp l T ) ( Z , ) ) )

=0

for T E €,

I=,

so (H,,[t, Z , ] ) = 0. Hence [H,,Z,] = 0, and so, by the regularity of H,, Z o E a. But K . H n a = W . H ( [ D S ] , Chapter VII, Proposition 2.2), so Z o E W .H . But then the maximality of 2, contradicts (2). For the converse of (1) we prove two lemmas. If HI, H 2E a, {HI, H2} denotes the segment H I - t ( H , - H2)(0 5 t I 1) and if a E X, b c a, we put (3) Lemma 10.3. Let b c a be a convex subset in the half-space a 2 0. Then {b, sub} is convex.

474

SPHERICAL FUNCTIONSAND SPHERICAL TRANSFORMS

[Ch. IV

+

Proof. Let H,,H 2 E {by s,b}; if r , , r2 2 0, rl r2 = 1, we must show = r,Hl r , H , E {b, sub}. Now Hi E { A i ,s,Ai}, i = 1, 2, where A , , A 2 E b; thus, since s,(H) = H - a(H)A:, where A: = 2Ad(a, a), we have

+

H

H i= A i- t,Ah

(4)

where 0 It i Ia(Ai).

Hence H = rl(Al - t,Ak)

+ r2(A2 - t2Ak),

which belongs to the segment {rlAl + r 2 A 2 , s,(r,A, + r 2 A 2 ) } since rlt, r 2 t 2 Ia(rlA, r 2 A 2 ) .This proves the lemma. A sequence y l , . . . ,y, E CJ (the positive indivisible roots) will be called a strongly positive sequence (of length n ) if for each H E a + we have

+

(5)

+

yi(syjsyj- I

. .s y , ( H ) ) 2 0,

for all 1 Ij < i In.

Corollary 6.11 (and the subsequent remark) shows how each o # e in W can be written as a product o = s y n .-.syl where yl,. . . ,y, is a strongly positive sequence. Lemma 10.4. Let b c a be any subset such that for any A E and ~ any root a E C the segment { A , s,A} lies in b. Then for each H E b the convex hull C ( H ) of W. H also belongs to b. Proof. Since saA E b for each A E ~ we , see that b is W-invariant. Thus it suffices to show C ( H ) c b for each H E b n C + where C+ is the closure of a+. Let H E b n C+,let o,, . ..,on+ E W, and c(ol,. . . ,on+,) be the convex hull of H,a,H, . . .,a,+ , H . We shall prove

(6)

4 ~ 1 * ,*

* 9

an+ 1) c a

by induction assuming c(sl,. . . ,s,) c b for any sl,. . . ,s, E W. For this it suffices to prove for any ol,. . . ,onE W,

. . on,sy,sy, - . . . syl) c b by induction on m, y,, . . . ,y, being any strongly positive sequence. With the definition sy,. . . syl= e if m = 0, (7) holds for m = 0; assuming it to hold for m we shall prove it for a strongly positive sequence yl,. . . , y,+ 1. Put H, = sYm...sY1(H) and y = y,,,+,. Since H E C+ we have y(H,,,) 2 0 and y(H) 2 0. Given j (1 Ij I n) we have y(o,(H)) 2 0, where o; is one of the elements oj or syoj.By induction, the set (7)

4 ~ 1 ,

* 9

1

b, = c(a;, .. . ,o;, sy,

* * *

SYl)c

b.

Also b, lies in the half space y 2 0, so by Lemma 10.3, {bl, syb1} is convex. By assumption, A E by a E C implies {A, saA} c b. Thus

0101

10. Convexity Theorems

475

{bl, sybl}c b. Since aj is one of the elements a) or sYa>we have ajH c {dl, s,d,} for each j . Also H m + 1 = sy(Hm)

E {bl,

sybl},

so by the convexity of {blysybl} C ( ~ l r . . . , a ~ , s y , + I "= ' s {bl,sybl} y,) = b.

This proves (7) for all m. Thus (6) holds for all al,.. . ,an+ E W and the lemma is proved. Returning now to the proof of Theorem 10.2 we put c = Q(K H). It remains to prove C ( H ) c c. Because of Lemma 10.4 it suffices to prove { Y, s, Y } c c

(8)

whenever

Y Ec, c1 E : C

.

For this we consider as in [DS] (Chapter IX, $2) the (semisimple) subalgebra 9" of g generated by the root spaces ga and g-.. It has the Cartan decomposition 9" = I+ p" where P = g" n f, pa = p n 9"; the line a" = RA, is maximal abelian in pa so the corresponding symmetric space G"/K" has rank one. We have the orthogonal decomposition a = a" + nay (9) where n, is the hyperplane c1 = 0 in a. Let Y = r, + Y t be the corresponding decomposition of Y. Then r, = &(Y)A:,, and

{Y,S"Y}= { t r ,

(10)

+ Y i : -1

It

Il}.

Now Y = Q(u . H) for some u E K. Consider the mapping a:K" + a defined by a(k) = Q(ku . H). Then (11)

a(K") c c.

If = 0 then { Y, s, Y } is the single point Y so (8) is trivial. Hence we assume r, # 0. Then we have unique continuous functions aa: K" + R, a,': K" + n, such that

~ ( k =) a,(k)Y, + &k),

k E K".

We shall prove that the function a; is identically constant. For this it suffices to show that for each H , E nay( H , , o(k)) is constant in k E K". However, ( H O a(k)) ~ = (Ho, Q(ku * H ) ) = ( H o , k u . H ) = (k-'HO, u * H )

and k - ' H , = H o since [ A # , g"] = 0. Thus a;(k) = a,'(e) = Y,' so a(k) = aa(k)Y, Y,'. Select k, E: K" such that Ad(k,)Y, = - r,. Since

+

476

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

we have Ad(k,) = 1 on nu. Hence Ad(k,)a = a and a&,) = Q(k,u. H) = Ad(k,)(Q(u.H)) = - Y, + Y,'. This shows that o,(k,) = -1 so by the connectedness of K", a,(K") 3 [-1, 13. But then (10)and (11) show that { Y, s, Y } c a(K") c c,

and the theorem is proved. We have seen that the Iwasawa decomposition G = K A N , in particular the component H@) in g = k(g) exp H(g)n(g) plays an important role in the theory of the spherical transform. The Killing form B of g defines a norm I I on p and a distance function d on G / K . We know from [DS] [Chapter VI, Exercise B2(iv)] that if o = {K} then

(13) IH(g)l I d b 9 * 01, 9 E G, with equality holding only if g E KA. In particular, (14)

IH(ak)l 5 Ilogal,

a E A , kEK.

We shall now prove a result of Kostant [1973] determining the set {H(ak :k E K} quite explicitly. Again the result is closely related to one of Horn [1954b] and Thompson [1971] for G = SL(n, C). The relationship is explained in Kostant's paper. Theorem 10.5. Let a E A. Then the set {H(ak):k E K} equals C(log a), the conuex hull of the orbit W log a.

Let us interpret this geometrically in terms of the space X = G / K . Each point X E X can be written x = n, exp A ( x ) - o where n,E N , A(x) E a are unique. Since the orbit N - exp A ( x ) . o = exp A(x)N - o intersects A - o orthogonally at exp A ( x ) - 0 ([DS], Chapter VI, Exercise B.2) it is reasonable to call A ( x ) the N-projection of the point x . Note that for g E G,A(g) in $4. No. 2 equals the present A@K). Also, A ( g K ) = - H ( g - ') so A(kaK) = - H ( u - ~ ~ - ' )Theorem . 10.5 can therefore be stated Theorem 10.6. Let a € A. The N-projection of the orbit Kaao in X equals the conuex hull C(log a) of We log a.

Because of the orthogonality ( N a - o ) ~ I . ~(A - o ) ~ Theorem ~ , 10.2 can be viewed as an infinitesimal analog of Theorem 10.6. Lemma 10.7. Let X have rank one. Let b E A, n E N . Then for any a E exp C(log b ) there exist elements k, v E K , n' E N such that

kbnv = an'.

(15)

Proof. As used above

d(o, a - o) I d(o, b - o) I d(o, bn .0).

§lo1

477

10. Convexity Theorems

The distance d(o, an' 0) varies with n' E N from d(o, a . 0) to 00. Thus there exists an n' E N such that d(o, an' . 0) = d(o, bn .0). Hence kbn - o = an' o for some k E K so (14) follows.

Corollary 10.8. Theorem 10.5 holds for X of rank one. In fact if we take n = e in (15) we deduce { H ( b v ) : v E K } The converse is (14). We shall now prove Theorem 10.5. For b E A let 5

Lemma 10.9. The set

3

C(log b).

= { H ( b k ) :k E K } . 5

is W-invariant. I n fact, for each H E 5

and each a E X,

{H,s,H}

c 5.

Proof. Suppose first a is simple and consider g", F, p", a" = RA, as above. Corresponding to R + A , as a positive Weyl chamber in a" we have the Iwasawa decompositions g" = P + a" nQ, G" = K"A"N" compatible with g = f a n and G = KAN. As in $6 we put (a) = C+ n {a, 2a) and consider the subspace n' = gB. Then n = n" + n' and since a is simple, we have

+

+ +

IBEX+

[g", n'] c n'.

If y1 < y z < are the elements of C+ in increasing order and we write n(') = Cjzigr, then Lemma 6.8 applies to the subspaces n" and n' so we get the diffeomorphism decomposition (17)

N = N"N'

where N' = exp n'.

Let H E 5, a = exp H. Then there exist k, v E K , n E N such that

bv = kan.

(18)

+ Hi by (9), so by (10) = { t H , + Hi:- 1 It 5 1) = C,(H,) + Hi,

We decompose H = Ha

{ H , s,H} where C,(H,) is the convex hull of H, and - H a . We put a, = exp H,, a,' = exp H,' and write n = n,n' in accordance with (17). Then by (18) and (12) we have (19)

(20)

k-'bv = a,a,ln,n'

= a,n,ain'.

Let C,E exp C,(H,). By Lemma 10.7 there exist k,, v,EK", nx E N" such that

k,a,n,v, = c,n,O,

478

SPHERICAL FUNCTIONSAND SPHERICAL TRANSFORMS

[Ch. IV

whence by (20) and (12),

k,k-'bvv, = c,nzu,-'a~n'v, = c,a,lnzv; 'n'v,.

(21)

Using (16) we see that this last element belongs to c,a;N. Thus (21) implies

H(bvu,) = log(c,a,l)

(22)

= log

C,

+ Hi.

Since log c, was arbitrary in C,(H,) (22) and (19) show { H , s,H} c

(23)

B

if a is simple.

This proves that 5 is W-invariant since W is generated by the reflections s, for x simple. Now if a E Z o is arbitrary there exists a o~ W such that oa is simple. Then we have by (23), for each HE^, 5

= l

{ H , s,,(H)} = { H , os,o- ' H } = a{o-'H,sac- ' H } ,

and this relation implies the lemma now that we know that invariant. Lemmas 10.4 and 10.9 now imply that for each b E A

5

is W-

C(log b) c {H(bk): k E K } . (24) For the converse inclusion we may, since both sides are invariant under the maps log b + s(1og b) (s E W), assume that log b E C'. Then by Lemma 6.5, H ( b k ) c log b that is,

5 t

for each

SE

log b

+ -C

for all k E K ;

+ - C . But since 5 is W-invariant this implies 5 c s(log b + - C )

W, whence by Lemma 8.3

(25)

5

c C(log b).

This proves Theorem 10.5. Corollary 10.10.

G =KNK.

xasW

In fact if H Ea the sum H , = oH is fixed under each reflection s, (a E X) so H , = 0. Thus 0 E C ( H ) so H(exp H k ) = 0 for some k E K . Thus exp H k E K N , proving the corollary (since G = K A K ) . Theorem 10.11. (i) Assume bE A is such that the convex hull C(log b) satisjes

(26)

dim C(log b) = dim a.

$101

10. Convexity Theorems

479

Then the measure pb given by

(27)

/Kf'(H(bk)) dk =

/f

dpb(H)r

f'

Cc(a),

a

sat isjies

(28) (29)

suPP(pb) = C(l% b)? C(b

= Fb(H) dH,

where

FbE

L'(a).

(ii) Condition (26) holds whenever log b E a is regular and more generally if,for the decomposition X = Hi X i , a = ai, into irreducible factors Xi each component qf log b is #O.

1,

Proof. (i) It is clear that pb as defined by (26) is a measure (Chapter I, $1). Also, if supp(.f) n C(1og b ) = 0 we have l(b(f) = 0 by (25) so the support of pb is contained in the compact set C(1og b). Consider now the map gb: K + a given by gb(k) = H(bk). By Theorem 10.5, gb(K) = C(1og b ) whose interior, say C,(log b), is by (26) an open subset of a. Let K O = gbl(Co(log b)). By Sard's theorem (see, e.g., Guillemin and Pollak [1974]) for the map gb: K O+ C,(log b), almost every point H o in C,(log b ) is a regular value; that is, for each k E K O n gb-l(H0), gb is a submersion at k ([DS], Chapter I, $15). Since gb being a submersion at k amounts to the nonvanishing of certain determinants, we see that gb is a submersion on a set K - C , dense in K, where C is a certain closed null set for the Haar measure mK on K. Now, by (27) if S c a.

(30)

pb(S) = mKbb '(s))= mK(gb '(s)n ( K - c))*

Let k , E K - C. 'On suitable coordinate neighborhoods of the points k, E K - C, gb(k0) E a, the map gb is given by the projection (xl,. . . , x p ) -, ( x , , . . ., x,), ([DS], Chapter I, 415). Since dmK is there given by F d x , ... d x p ( F > 0), it follows from the Fubini theorem that if S c a is a null set then mK(gb'(S)n (K - C)) = 0. Thus by (30) pb(S) = 0 so pb is absolutely continuous and (29) is proved. From the above representation of gb as a projection near ko it is obvious that gb(k0) E supp(pb). Hence gb(K - C) c supp(pb), and since K - C is dense in K, c(lOg b, = gb(K)

suPP(&).

log bi according to the decomposition a = (ii) Suppose log b = 1, 1,a,. The Weyl group w for X i acts irreducibly on a, ([DS], Chapter X

$3, No. 3), so since log b, # 0 the set for each a,, hence also for a itself.

w..log bi spans a,. Thus (26) holds

480

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

Corollary 10.12.

With b E A as in Theorem 10.11 the function is the Fourier tranSform of e-PF,, E L'(a):

A -P

4,(b) = J;(i*p)(H(bkH

dk =

s.

,iL(H)(,-P(H)F

b(H))

dH.

In particular, by the Riemann-Lebesgue lemma, lim

4,(b) = 0.

A € O ' . ~ l ~ ~ r n

We shall finally use Theorem 10.11 to obtain a simple result about restrictions of the functions in C,'(G) to A. Let CY(A) denote the space of W-invariant functions in C,(A). Viewing A as a vector space, let C(a) denote the convex hull of the orbit W - a ,a being arbitrary in A. Proposition 10.13.

(i) The restriction from G to A induces a bijection of C:(G) onto CY(A).

(ii) Suppose X is irreducible and let b, c E A, b, c, bc all #e. Then there exists a unique W-invariant measure p b , , on A such that (31)

J-Kf(bkc) dk =

J f ( a )dpb.c(a),

f E c:(G).

A

Moreover, p b , , has support in C(b)C(c). Proof. The bijection (k, X ) + k exp X of K x p onto G identifies the functions in C:(G) with Ad(K)-invariant functions in C,(p) which under restriction to a are identified with CY(a) (cf. Proposition 5.18 in

Chapter I). This proves (i). This identification means that the functional

f

+

J f(Wdk K

(fE c m

can be regarded as a positive W-invariant linear functional on C r ( A ) . Thus the measure &,c exists. Also, has compact support; in fact let D c A be a compact set such that bKc c KDK. Then supp p b , c c D. Hence formula (31) holds for all f EC'(G). In particular, we can take f = bA.Using the functional equation for we then obtain

(32)

4db)4A(C)

=

J 4 l ( a )dpb.c(a). A

Our assumption bc # e implies that e b K c ; thus if ,f E C:(G) has support in a suitably small neighborhood of e in G we have ( f (bkc) dk = 0. In particular, pbsC({e}) = 0. Thus we can replace A by A - { e } in (32)

Exercises and Further Results

48 1

and use Corollary 10.12 on (a # e), 4A(b),41(c). Writing fa@) e-p(H)F,(H)if h = exp H , we deduce from (32)

fb * f , ( h ) =

(33)

=

f,(h) dpb.c(a), A - 14

for almost all ~ E A *, denoting convolution on A . We now have to prove that each a, 4 C(b)C(c),a, # e, has a spherical neighborhood A , such that pb,c(Ao)= 0. Superscript denoting interior, we select U O E C(a0)' -

C(b)C(c).

By continuity we can then find spherical neighborhoods U , of u,, A , of a,, such that (34)

U , c C(a)O - C ( b ) ~ ( c ) for

Let 4 E Cc(Uo),$ 2 0, 4 $ 0. Since outside C(b)C(c),(33) implies

UEA,.

fb * f , vanishes

almost everywhere

By the positivity we have, a fortiori,

(35) But the inner integral equals pa(e-Pc$), which by (27) depends continuously on a. Also, by (28)

U o c supp(p,)

if a E A , ,

so by general measure theory, p,(e-P4) > 0

for a E A , .

But then (35) implies & & l o ) = 0 as claimed. Thus supp(pb,c) c C(b)C(c) and the proposition is proved.

EXERCISES AND FURTHER RESULTS A. Representations

1. (i) Let n be a representation of a Lie group G on a complete locally convex space I/. If u E V show (cf. Corollary 1.4) that n(g(G))u is dense in the closed invariant subspace of V generated by u.

482

SPHERICAL FUNCTIONSAND SPHERICAL TRANSFORMS

[Ch. IV

(ii)* Assume V is a FrCchet space; let V" be the space of differentiable vectors in V and V, the set of linear combinations of vectors n ( f ) v (fE 9 ( G ) , v E V ) (the Gdrding subspace). Then V" = V, (Dixmier and Malliavin [19781). 2. Show that if two unitary representations are equivalent they are unitarily equivalent. 3. (i) Let I c R be an open interval and t + x, a mapping of I into a complex Banach space V, with dual V'. Assume that for each A E V' the mapping t + ( x t , A ) is of class C2.Then the mapping t + x, is C ' . (ii) Let n be a representation of a Lie group G on a Banach space V. Then a vector v E V is differentiable if and only if for each A E V' the function x + (n(x)u, A) on G is differentiable (Poulsen [1970]). 4*. Let G be a connected semisimple Lie group with finite center. Let K c G be a maximal compact subgroup. Let n be a quasisimple, K-finite representation of G on a Banach space V. Let V, denote the space of all K-finite vectors in V. Then if g is the Lie algebra of G, (i) V' c V" and dn(X)VK c V, (XE 9). (ii) Let dn, denote the representation of g on V, given by d n K ( X )= dn(X)IV,. Then dnK is irreducible (algebraically) if and only if n is irreducible (cf. Harish-Chandra [1953], 510). 5. Let G be a connected, noncompact simple Lie group, G/K the associated symmetric space, g and € the corresponding Lie algebras. We assume G imbedded in the simply connected Lie group GC with Lie algebra gc. Let w = € + ip and U the corresponding analytic subgroup of GC.

Let n be a finite-dimensional representation of U on a real vector space V, such that n(K) has a fixed vector v # 0 satisfying K

= {U E

U : n(u)v

=

u}.

Extending n to a representation of gc and GC on V n, prove the following:

=

V $ , also denoted

(i) The mapping I : g K + n(g)v is a bijection of G/K onto a (real) submanifold of V . (ii) The G-finite functions on G / K are the functions p o l where p is a (holomorphic) polynomial function on V. 6. Let K be a compact connected Lie group, 6 an irreducible representation of K with character xs. Let D ( K ) denote the set of left-invariant differential operators on K . Prove that D(K)Xa = Hd

Exercises and Further Results

483

the space spanned by the representation coefficients of 6 (cf. Helgason [1976], Theorem 4.1). B. Spherical Functions

1. Show through the indicated steps that the spherical function

$o(g)

=

/;-P(H(lik))

satisfies the inequalities

(*I

,-p(loga)

- &(a) I <

c(1

dk

+ [log

~I)~e-p("*~),

where a € A + , d = Card@,+), and c is a constant (cf. Harish-Chandra [1958al, 49). (a) For the left inequality use Lemma 6.5. (b)* Multiply the expansion for +A by the polynomial .(I) and observe that the function I --* c(sI)n(A) is smooth at I = 0. Viewing d(n) as a differential operator on a* we have $0

=4

WL~(4$J}A=O

where c is a constant. The right-hand side can be estimated by the expansion for $ A giving the right-hand inequality in (*). 2. With the notation as in Theorem 6.13 prove that for ,f E C,(a),

-

f(H(iil)

+ . + H(ii,))e-Zp'(H("'"dii, .. .dii,.

Here pi is the p-function for G f l , / K s i . 3. Let P E X ~and ps be the p-function for the rank-one symmetric space G , / K , in $6, No. 6. Then p(Ap) 2 Pp(As)

with equality if fl is simple. 4. Let d = Card@,+). Prove that if

E

> 0 then

For G of rank one this follows from Eq. (8) of $6. For G of arbitrary rank it then follows from B2 if we take B3 into account. The inequality is due to Harish-Chandra [1958a], who gave an instructive but more difficult proof. The inequality plays an important role in the theory of the spherical transform on the "Schwartz space" .Y2(G) (Problem C6).

484

[Ch. IV

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

5. Show that the spherical function is real-valued if and only if 2 and -1 are conjugate under the Weyl group. 6*. The spherical function + A is positive if and only if i 1 E a* (cf. Furstenberg [1965]; KarpeleviE [1965]). 7 . Using the convexity of the function x + ex on R show that if iA E a* then 4A(d 2 40(9), 9 G, (De George and Wallach [1978].) 8. Let G have real rank one. Then : X consists of one element a. The function t: 11- a(1ogh) can be taken as a coordinate on A whereby the differential equation (7) of 55 for 4Abecomes an ordinary second-order differential equation in t. Introducing the new variable z = -(sinh c ) ~ , this differential equation becomes the hypergeometricequation d24A ~ ( -z 1) - [(a dz2 where, with a, = a/(a, a),

+

+ b + l ) -~

C]

d4A

+ ab4, = 0, dz

a = %+ma

+ m2a + (i4 ao>),

b = Hima

+ mZa - (i4 a0)),

+

+

c = %md m2a 1). Thus 4Ais given by the hypergeometric function z = -sh2(a(log h)). 4,(h) = F(a, b; c; z), The behavior of $,(h) at 00, in particular the formula of Theorem 6.4 for c(A), can now be determined from a classical functional equation for F valid for a - b 4 Z: F(a, b ; c; z)/r(c) = T(b - a ) [ r ( b ) r ( c- a ) ] - ' ( - ~ ) - " F ( a , 1 - c + a ; 1 - b a;Z-') + T ( a - b ) [ r ( a ) r ( c- b)]-'(-z)-*F(b, 1 - c + b ; 1 - a b ; z - ' )

+

+

(cf. Harish-Chandra C1958a)). 9*. Because of Theorem 3.7 it is of interest to determine the set of A ~ a ffor which is positive definite. The complete answer does not seem to be known, but one has some partial results: (i) If A E a* then is positive definite. (ii) Assume G has real rank one and a is the single element in .:X Then (a) Suppose 2a 4 .'X Then + A is positive definite if and only if either A E a* or iA E a* and I(iA, a) 1 I( p , a).

Exercises and Further Results

485

(b) Suppose 2 a ~ Z ' . Then apart from the constant c$*~,,E 1, positive definite if and only if either 1 E a* or i l E a* and

I

a> 1 I

is

(ha + l ) ( p , a>

(cf. Kostant [19751; another proof is given in Flensted-Jensen and Koornwinder [1979a], and partial results were proved by Takahashi [1963] and Faraut and Harzallah [1972]). (iii) For G complex of rank 2, Duflo [1979b] gives an explicit (but complicated) description of the 1for which is positive definite. (iv) For a number of special cases, all the irreducible unitary representations of G have been determined, thus answering the question above. For a survey, see Knapp and Speh [1982]. 10. Let G = SU(2,l). Then the groups M and N can be described by M = exp m, N = exp n where

m

= RT,

n

= RX

T

+ RY + R Z ;

=[

3, (-8 ,)H x=

-ii

(i

Y = (0i 0i - i0). O

i

0

0 -i)

z=oo i

0

0 -i

([DS], Chapter IX, 53). Determine D ( M N / M ) and the spherical functions on M N / M NN N . Note that N is isomorphic to the Heisenberg group (Chapter 11, $4, No. 1). 11* . For 1 E a* the spherical function satisfies the following inequality (Harish-Chandra [1958b], p. 583): Let D E D ( G ) . Then there exists a k E 2' and a constant C such that IC 1(1 + I1l)-k.(n)(D~,)(a)eP''oB"'(

for a E A', 1 E a*. 12*. Deduce the boundedness criterion for c$l (Theorem 8.1) for complex G directly from the formula for c$l in Theorem 5.7. 13*. Let G be a unimodular connected Lie group, and K c G a compact subgroup. The following properties of G / K are equivalent: (i) The algebra D(G/K) of G-invariant differential operators on G / K is commutative. (ii) The convolution algebra C:(G) is commutative.

486

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

14*. Let U be a compact simply connected simple Lie group, and T c U a maximal torus. Deduce from 13 and Theorem 3.5, Chapter V that if dim T > 1, then the algebra D ( U / T ) of invariant differential operators is not commutative. 15. Using (4) in $4 prove the formula

A ( g * x , g * b =) A ( x , b )

+ A(g.0,g.b)

(gEG,xEX,bEB),

which generalizes (34) in Introduction $4. Using this formula and Proposition 2.4, show that for each p E a,* the function

.

ep.b*

+

er(4x.b))

is a joint eigenfunction of D(G/K). 16. With the notation of $3 consider the space X ( G ) = C,(G) with the inductive limit topology (from $1) and the sub-algebra C:(G) with the relative topology. In analogy with Theorem 3.3 show that: The continuous homomorphisms of the algebra C:(G) onto C are the mappings

where 4 is a spherical function on G. 17. Let S be a set and Ol,. .., Ow complex-valued functions on S. Show that if O1,. , . ,Ow are linearly independent over C then there exist xl,. . . ,x, E S such that the matrix (@i(Xj))l s i , j s w

is nonsingular. From this and (2) 56 deduce another proof of Lemma 6.1, generalizing the one for SL(2, R) in the Introduction, (51) $4 (Schlichtkrull and Flensted-Jensen). 18. Relate Lemma 5.21, Chapter 11, to the formula 4 - A ( g ) = 41(g- ') in (7) §4*

C. Spherical Transforms 1. As in $8 let Lh(G)denote the space of functions in L'(G) which are bi-invariant under K. (i) Suppose f E Lh(G) is such that

f(1) = 0

for all 1G a* (real dual of a)

Then f( 9 ) = 0

for almost all g E G .

Exercises and Further Results

487

(ii) If f e L P ( G ) ,show that F,(a) = ep(loga)

exists for almost all a E A and that the map f -+ F , is injective. These results are proved in Helgason [1970a], Chapter 11, $1; Part (i) sharpens Proposition 3.8. 2. Let bK(p)denote the space of K-invariant functions in b ( p ) with the relative topology. The restriction mapping in Chapter 11, Proposition 2.1 identifies the dual &(p) with the space of K-invariant distributions on p of compact support. Show that the spherical transform T + 3: where

3:w =

J$-md u a P

is a bijection of &(p) onto the space of W-invariant holomorphic functions @ on a: satisfying an inequality of the form l@(A)l I C(1 + IAI)NeRl'mAl,

AEa&

for some positive constants C, N , and R . 3. Suppose G has real rank one. Consider the integral

F,(a)

= ep(loga)

J

f ( a n ) dn = ep(loga) J f(iia) dii,

N

l v

where 8(dn) = dii. Here an explicit inversion off -+ F , can be given. (i) From [DS] (Theorem 3.8, Chapter IX) and its proof show that for U E A , ch'(a(A+(ia))) = [ch(a(log a)) + 4ceu(loga)I X 1']

+ ceZa('oga) I YI',

in the notation of the quoted theorem. (ii) Note that in the case G = SO,(n, 1) we can take either a or 2a as the element in Z' [compare Eq. ( 2 6 ) of Introduction 5443. (iii) Show that a(H)' = %ma + 4mZa)- (HI' for H E a. (iv) Renormalize the distance d on G/K and the norm 121' = - B(2,OZ) by

'

6

=

+

[2(mU 4mZa)]-'/'d,

11211' = &mu

+ 4mza)-' 121.'

Then (i) shows if a(H) = 1, a, = exp rH, ch2(6(o,riu,.o)) = (ch r

+ $erllXll')' + &"II

YII'.

488

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

Defining

[Ch. IV

4, $ on [l, co) by 4(ch2(S(o,g

0)))

= f(d,

$((ch

W2= ) F,(a,),

show that (*>

j--EJ - 2 M c h

W h W 2= )

+ llxl12)2+ 1I Y1I2>dX dY*

with a suitable normalization of the Euclidean measures on g-. and 9-211.

(v) Show that the integral transform f + F, or, equivalently, the integral transform 4 -+ t j in (*) can be expressed in terms of two Abel integral transforms of the type

H(u) = juWh(o)(u2 - u 2 ) P du. (vi) By inverting these two Abelian integral equations deduce the following principal content of Corollary 7.4 in the rank-one case: If for [[logall > R,

Ff(a) = 0 then

-

for d(o, g o) > R .

f(g) = 0

For G = SO& 1) this inversion of f + F , was given by Takahashi [1963]; generalizations to the rank-one case, related to our method above, have also been found by Faraut [1982], LohouC-Rychener [1982], and Rouviere [1982]. Earlier, Koornwinder [1975], using a different method, had represented f + F , as a similar composite of two fractional integrals. 4*. Suppose the measures d H and d l are regularly normalized and dg normalized by jGf(g) dg = l t f ( e x p H)

n

(ea(H) - e-a(H))mEd H ,

O€Z+

Then the Plancherel formula for the spherical transform c

l 3 ( 4 l 2 I c ( 4 r 2 ffl

fclf(e)l’ds = 0’

holds with c = I WI, the order of W.

f E @(G).

Exercises and Further Results

489

5. Suppose now the semisimple group G is complex. Put n(A) = (1E a:) and let the K-bi-invariant function A. E b(G) be defined by

flaET+(a, A)

Ao(a) =

C esp(loga),

a E A,

SEW

(cf. Chapter 11, Corollary 5.11). Using the formula in Theorem 5.7 for the spherical function prove that n(A)(A0f)-(1) =

1w f ( A + isp)n(A + isp),

f E @(G)

SE

(cf. Helgason [1970a], Chapter 11, 42). 6". For p 2 0 let SP(G) denote the set of all f E ~ ( G )satisfying the following two conditions: (a) f is bi-invariant under K . (b) For each D E D(G), q E Z', SUPl(1 + Isl)4~o(s)-2'p(Df)(9)l< esG

Here (91 = I log a1 if g = klak, (kl, k , E K , a E A). Because of (*) in B1, P ( G ) is contained in the space LP(G) of functions f on G for which If IP E L1(G). Theorem, The spherical tranSform f -3 is u bijection of 9 2 ( G ) onto .4Pw(a*). The substance of this theorem is proved in Harish-Chandra's papers [1958a, b]. See Helgason [1964a] for the additional remarks needed. A central step in Harish-Chandra's proof is the inequality in B11, which is proved by induction on dim G. By a refinement of this induction technique, Trombi and Varadarajan [1971] extended the theorem to a characterization of S P ( G ) - as the space of W-invariant rapidly decreasing holomorphic functions in the tube a* + i(2/p - l)Co(p) in a.: See also Helgason [1970a], Ch. 11, 42, for the rank-one case and p = 1. For G complex, p = 1, the identity in Problem C5 can be used to prove this by reducing the case f ' ( G ) to Harish-Chandra's case S 2 ( G ) (see Helgason [1970a], Chapter 11, 42 for the details). For G real, p arbitrary, ClerC [1980] has given a related reduction of the proof for S P ( G ) to the case S 2 ( G ) . D. A Reduction t o the Complex Case

Flensted-Jensen [19781 has discovered remarkable relations between the spherical functions on a real semisimple Lie group Go and the more

490

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

elementary spherical functions on the corresponding complex semisimple group G. These results are summarized in 1-4 below. Let go be a real noncompact semisimple Lie algebra, g its complexification, go = €, + p, a Cartan decomposition, and 8 the corresponding Cartan involution. Then u = f, + ip, is a compact real form of g and f = f, + ifo is a complex subalgebra of g. Let G be a Lie group with Lie algebra g (considered as a Lie algebra over R). Let G o , K O , K , and U be the analytic subgroups of G corresponding to go, f,, f, and u, respectively. Then K O c G o , U c G are maximal compact subgroups whereas K is noncompact. Let DR(K\G) denote the set of differential operators on the coset space K\G

={Kg:gEG)

invariant under all translations t(x): K g

--t

Kgx (9, x E G). Let

W \ G / U ) = {4 E :4(kgu) = 4(s)). 1*. There is a one-to-one correspondence 4 + 4q between the set of spherical functions 4 on Go/Ko and the set of functions II/ = @ on G satisfying

II/ E W \ G I U ) , $(el = 1 ; DII/ = &II/ for all D E DR(K\G), 1,

E C,

such that 4(xe(x)-

+ +

1)

= 4yx),

E G,.

2*. Let g = u a n be an Iwasawa decomposition of g. Then go = fo a, no with a, = a n go, no = n n 90 is an Iwasawa decomposition of go. By the Killing form identification of a, and a with their duals a t is imbedded in a*. Let 2p, and 2p denote the sums (with multiplicity) of the positive restricted roots for (go, f,) and (9,u), respectively. Similarly, let no and TC denote the products of the indivisible positive roots. Let 4A ( 1 ~ ( a & ) denote the spherical functions on G , / K , and O,, (A E a:) the spherical functions on G / U . If 1 E (a& define A E a: by

+ +

A

+ ip = 2(1 + ip,).

Then (1)

where du is the normalized Haar measure,on U .

Notes

49 1

This mapping 4A+ Q,, is injective whenever the mapping p : Z(Go)+ D ( G o / K o ) (Chapter 11, $5, No. 7) is surjective. If Go is simple, this fails only for certain real forms go of e6, e , , e , (cf. Chapter 11, Exercise D3). 3*. Suppose go is a normal real form of 9. Then if the Haar measure dk on K is suitably normalized, (2)

4A(xw-')

= lc(4121.0(412

/22A(w dk,

&Eat,

so, in particular, the Plancherel density for Go/Ko is given by (3)

For the case Go = SL(2, R) this formula becomes a,

sin I t dt = K tanh(y),

which enters into Godement's proof of the inversion formula for the spherical transform on SL(2, R) [see (27) in Introduction 4443. 4*. Formulas (1)-(3) have interesting applications to spherical transforms. For example, (1) reduces the determination of the bounded spherical functions (Theorem 8.1) to the case of complex G. For this case one can use the simple formula for Q2,, from Theorem 5.7; nevertheless, the proof of the boundedness criterion for this case does not seem all that simple (Exercise B12). Suppose now go is a normal real form of 9.Then formulas (1)-(3) reduce the Paley-Wiener theorem, the inversion formula, and the Plancherel formula for the spherical transform on Go/& (Theorems 7.1 and 7.5) to the corresponding theorems for G / U (these are elementary because of Theorem 5.7). For this and further applications we refer to the paper of Flensted-Jensen quoted above.

NOTES 81. For further information about representations of Lie groups on locally convex spaces see Bruhat [1956], Harish-Chandra [1966], Bore1 [1972], and Warner [1972]. In finite-dimensional spaces there is a simple correspondence between representations of a Lie group G and of its Lie algebra 9. For infinite-dimensional representations this correspondence requires modification. Thus Girding [19471 replaced the representation of G on a Banach space V by the representation of g on a certain subspace of the space V" of differentiable vectors. (In this context see Exercise Al.) This was further improved by

492

SPHERICAL FUNCTlONS AND SPHERICAL TRANSFORMS

[Ch. Iv

Harish-Chandra’s space of well-behaoed or analytic vectors [1953]; the proof of Lemma 1.2 is from this paper. For a general locally compact group where Haar measure is the principal analytic concept, the Hilbert space L2(G) and the irreducible unitary representations become the central objects in analysis on G. However, when G is a Lie group we have [in addition to Lp(G)] spaces like O(G), 6(G), O‘(G), and C ( G ) forming the core of analysis on G. It has accordingly turned out. at least for semisimple G, that the quasisimple representations (introduced by Harish-Chandra [19533) form a more natural class than the unitary ones (cf. Langlands [1973], Zelobenko [1974], Duflo [ 19753, Borel and Wallach [1980], Vogan [1981], Knapp and Zuckerman [1982]). Theorem 1.5 goes back to Gelfand and Raikov [1943]; see also Godement [1948]. For references on the orthogonal decompositions (9) and (16) see notes to Chapter V. The functional equation in Theorem 1.6 is from Weyl [1931, Appendix 21 and Weil [1940, $243. The decomposition in Lemma 1.9 is from Harish-Chandra [1966]; see also Borel C1972.831. 12-3. Spherical functions on compact symmetric spaces U / K were introduced by Cartan [1929] as coefficients (x(u)r, r ) of representations L of U for which L ( K ) has a fixed vector e. Gelfand [195Oa] showed the commutativity of the algebra C:(G) (Theorem 3.1; the unimodularity of G was remarked by Berg [1973]) and introduced the spherical functions as its homomorphisms into C (Lemma 3.1). He also obtained the functional equation (1) of $2 and the characterization by differential equations which we have used as a definition. Some of these results were also obtained by M. G. Krein. The correspondence in Theorem 3.4 is essentially from Gelfand and Naimark [I9521 and Godement [1957a]; the first paper gives an explicit formula for the spherical function for G = SL(n, C). The mean value theorem in Proposition 2.5 is from Godement [1952a]. For harmonic spaces (in particular for two-point homogeneous spaces) the result is due to Willmore [1950]. The generalization to arbitrary eigenfunctions (Proposition 2.4) and the semisimplicity (Proposition 3.8) are from Helgason [1962a]. $4. The integral formulas in Theorems 4.2 and 4.3 and Eq. (7) are due to HarishChandra [1958a]. Lemma 4.4 and the correspondence theorem (Theorem 4.5) were proved by Helgason ([1970a], p. 116 and [1972a], Chapter 11). Kostant ([1975a], Theorem 1.3.5) proved an algebraic analog of Theorem 4.5. Proposition 4.8 was observed by Gindikin [I9671 and Korlnyi [1979], Theorem 4.7 and Proposition 4.10 by Helgason [1972a], Chapter 11. $ 4 - 7 . A Plancherel-type formula for the Fourier transform on a unimodular locally compact group G was given by Segal [1950] and Mautner [1950]. For the spherical transform corresponding to a “topological” symmetric space G / K a simpler Plancherel formula was given by Mautner [1951]. A more satisfactory version (with the set of positive definite spherical functions playing the role of the dual space) was given by Godement [195l, 1957al; see also Harish-Chandra [1954c]. For the case of a noncompact semisimple Lie group with its rich structure and explicit parametrization of the (cf. Theorem 4.3) it became a natural problem to set of spherical functions by a:/” determine the Plancherel measure explicitly in terms of this parametrization. HarishChandra embarked on this project in his papers [1958a, b]. The first of these papers gives the expansion (Theorem 5.5) for $ A , introduces the c-function, and relates it to the behavior of at co. Some improvements in the proof (namely Lemma 5.3 and the proof of Proposition 5.4) come from Helgason [1966b] and [1972a] (Chapter 11). On the basis of the spectral theory of ordinary differential equations, Harish-Chandra expected the Plancherel measure to be given by le(l)l-’ dl. The proof of this result is in

Notes

493

his paper [1958b] reduced to two conjectures; the first one is about the c-function and the second concerns the transform f + F,. While Harish-Chandra [1958a1 had determined the c-function for the case G complex and the case when G has real rank one, Bhanu-Murthy determined c(A) for all (except one) classical G which are normal real forms (see [1960a, b]). Guided by his formulas, Gindikin and KarpeleviE [I9621 proved the general product formula (Theorem 6.13), which when combined with Harish-Chandra’s rank-one formula gives a general explicit formula for the c-function. This formula implies easily Harish-Chandra’s first conjecture mentioned above (see Helgason [1964a], #2-4). In C1966, $211, Harish-Chandra proved the second conjecture, thereby completing the proof of the Plancherel formula for the spherical transform. In [I9551 Ehrenpreis and Mautner proved a Paley-Wiener theorem for the spherical transform on G = SL(2, R); this was extended by Takahashi [I9631 to G = SO(n, 1). For general G the Paley-Wiener theorem for the spherical transform is given by Theorem 7.1 and Corollary 7.4. These were proved in Helgason [1966b] except for full justification of the term-by-term integration (15). This justification (ix., Lemma 5.6) was given by Gangolli [1971]. (Actually Gangolli proved a stronger estimate than that of Lemma 5.6; the proof of Lemma 5.6 in the text is a simplification from the author’s paper [197Oa]). While the proof of the Paley-Wiener theorem indicated above used Harish-Chandra’s inversion formula for the spherical transform, Rosenberg in [I9771 discovered that this is not really necessary; on the contrary, he showed that the proof of Theorem 7.1 together with some additional arguments ($7, No. 3, in the text) gave the inversion formula as well as the Plancherel formula. However, the surjectivity in Harish-Chandra’s result on the Schwartz space .!J2(G) (see Exercise C6) does not seem to be accessible by this method. We now give some references for the remainder of @5-7. The formula of Theorem 5.7 is the counterpart to the character formula on a compact group (cf. Theorem 1.7, Chapter V). As indicated earlier, it was proved by Gelfand and Naimark [1952] for the case G = SL(n, C); the general case was given by Harish-Chandra [1954b, 1958a1 and Berezin [1957]. Lemma 6.2 and Proposition 6.3 are also from Harish-Chandra [1958a] and so is the rank-one formula (Theorem 6.4) for c(A). His computation of c(A) using the hypergeometric equation is outlined in Exercise 88. The method of the text is from Schiffmann [1971] and Helgason [1970a]. Lemma 6.8 is from Harish-Chandra [1957b], and the proof in the text is an elaboration of that of Bore1 and Harish-Chandra [1962]. Lemma 6.9 and Proposition 6.10 are from Iwahori [1964]. If G has real rank one, the inversion formula and the Plancherel formula can be viewed as parts of the spectral theory of the singular ordinary differential operator A(L,) in Eq. (8) of $5 (also in Exercise B8). Flensted-Jensen [I9721 generalized these rankone results as well as the Paley-Wiener theorem, allowing the multiplicities m,, m2. to be arbitrary real positive numbers. Further generalizations have been given, for example, by Chebli [I9741 and Koornwinder [1975]. 48. The determination of the bounded spherical function is from Helgason and Johnson [1969]. In Lemma 8.3 we combine convexity arguments from this paper and from Kostant [19731. Lemma 8.8 has certain relevance for the c-function. Gindikin and KarpeleviE [1962] attached a c-function in a natural way to an arbitrary half-space R c a with 0 on its boundary. Lemma 8.8 shows that this is not more general than their functions r,(l) (s E W )in Theorem 6.13. Morera’s theorem in connection with Cor. 8.2 was suggested by R. Kunze. $9. The main results of this section are from the author’s paper [1980]. Some simplifications by N. Bopp have been taken into account. Lemma 9.4 is essentially from Kostant [1963,gj3]. For G complex, (17) is in Flensted-Jensen [ 1978, $33.

494

SPHERICAL FUNCTIONS AND SPHERICAL TRANSFORMS

[Ch. IV

$10. Theorems 10.1 and 10.5 are from Kostant’s paper [1973]. Special cases had been proved by Horn [1954a, 1954bI and Thompson [1971]. The proof in the text follows that of Kostant except for a variation in the geometric Lemma 10.7 which makes the reduction to the normal form unnecessary. A different approach, leading to some natural generalizations, is given by Heckman [1980]. Theorem 10.11 and Corollary 10.12 are from FlenstedJensen and Ragozin [1973]. They had been proved by Koornwinder (cf. [1975]) in the rankone case. For rank one a stronger version of Proposition 10.13 occurs in Flensted-Jensen and Koornwinder [1973].

CHAPTER V

ANALYSIS ON COMPACT SYMMETRIC SPACES In this last chapter we turn to analysis on compact homogeneous spaces with emphasis on the symmetric ones. For the study of functions on such spaces the representations of compact groups play a fundamental role, analogous to that of exponentials for Fourier series. Through the theory of maximal tori for compact Lie groups these representations can be determined very explicitly. In $1 we discuss this description of the representations through weights and characters. In 42 we extend to compact nonabelian groups a certain sample of well-known theorems about Fourier series on the circle. While these classical theorems seem at first rather unrelated, the group-theoretic generalizations provide certain links between them and thereby throw a little light on some classical results for Fourier series. In $3 we discuss Fourier decompositions of vector-valued functions with applications to this I is brought into more specific form for comcompact homogeneous spaces. In & pact symmetric spaces and the relevant representations are described more explicitly.

$1. Representations of Compact Lie Groups

Let U be a connected, simply connected, compact, semisimple Lie group. Let LI denote its Lie algebra and g the complexification of u. Let V be a finite-dimensional vector space over C. A representation of the real Lie algebra u on the complex vector space V is of course an R-linear mapping of u into the complex Lie algebra gI(V) preserving brackets. There is then an obvious one-to-one correspondance between the representation of U , u, and g on V. In this section we develop the standard theory of weights and characters and present some applications. While the weight and character theory is customarily developed purely algebraically for the complex Lie algebra g we shall for the sake of brevity of proofs take advantage of the compactness of U and will use some analytic tools already developed. A function j’ (or a distribution) on U is said to be central if it is invariant under conjugation, i.e., f’(uxu- ’) = .f(x) (x, 11 E U).

495

496

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

1. The Weights

Let T be a maximal torus of U , to its Lie algebra. Then the subalgebra t of g generated by to is a Cartan subalgebra. Let t* denote the dual space of t. Definition. Let n be a representation of g on a vector space V over

C.An element p E t* is called a weight of n if there exists a vector u # 0 in V such that

H E t. ~(H)= u p(H)u, In particular, the weights of the adjoint representation of g are the roots of g with respect to t. Now assume dim V < co. As mentioned in Chapter IV, $1. No. 2, there exists a positive definite Hermitian form on V x V such that K ( U ) consists of unitary operators. Hence n(T), and therefore also n(t), form a commutative family of semisimple endomorphisms of V and we have a direct decomposition P

where p runs over the weights of

II

and for each 1 E t* we put

(2) V, = { u E V : x(H)u = I(H)u for H E t}. The spaces V, # 0 are called weight subspaces and m, = dim(b) is called the multiplicity of the weight p. Since n(t) is a unitary diagonal matrix for each t E T, we see that each weight is real-valued on the space tR = it,, (cf. [DS], Chapter 11, @55, 6). As usual, let t, denote the unit lattice for U , t, = { H E to : exp H = e } . (3) We know from [DS](Chapter VII, Corollary 7.8) that

(4)

t,

=

{integral linear combinations of (4ni/(c1, a))H, : c1 E A}.

Here A is the set A(g, t) of roots #O and ifp E t*, H, E t is determined by (H,H,,)= p(H), (HE t) where ( , ) denotes the bilinear form on t and t* induced by the Killing form of 9. Put 1.1 = (-,- ) l I 2 . Theorem 1.1. Let t, denote the unit lattice and A the set of all weights of all finite-dimensional representations of U . Then

(i) A = {A E t* : 1(t,) c 2niZ), (ii) t, = {HE t : A ( H ) c 2niZ). Proof. Let A E A and let n be a representation of g on a finitedimensional vector space V such that A(H)u = 1(H)u for some u # 0 in

v1

1. Representations of Compact Lie Groups

497

V and all H E t. Then n(exp H)u = el(% ( H E t) so I(t,) c 27riZ. This proves the inclusions c in the equalities (i) and (ii). For the converse inclusion in (i) suppose I E t* such that I(t,) c 2niZ. There exists a homomorphism x : T C such that x(exp H ) = e'('') ( H E to). We claim that there exists a finite-dimensional representation 7r of U on a space V such that 7r(t)u = x(r)u for some u # 0 in V. For this consider the function space

C,( U ) = { f E C( U ) : f(ut

- l)

= f(ulx(t)).

The mapping (6)

(dt is the Haar measure) then maps C ( U ) onto C,(U); this latter space is nonzero as we see by choosing ~ E C ( U by ) the Tietz extension theorem such that the restriction f 1 T equals Re(X). Considering the left regular representation of U on C,(V) we know from Lemma 1.9 of Chapter IV that there exists an invariant finite-dimensional subspace V c C,(U). Pick g E V such that g(e) # 0. Then the function gYu) =

belongs to V, g'(e) = g(e)

J g(su)x(s)ds T

ST l x ( t ) I 2 dt # 0, and (gy = x(t)s )

The representation n on V so constructed satisfies 7r(exp H)u = el(% ( H E to) where u = 9'. It follows that I E A. This proves (i). Finally, suppose H E t satisfies I ( H ) E 2niZ for each I E A . Then H E to and (1) implies that for each finite-dimensional representation 7r on, say, V, n(exp H)u = u for each D E V. Then the Peter-Weyl theorem implies f(exp H ) = f ( e ) for each f~ C(V), so H E t,. This concludes the proof of the theorem. If p E A we define the function er on T by eP(t) = ep(")

if t = exp H , H

E to.

Because of (i) above this is a valid definition. Taking (4) into account the proof above-gives the following result. Corollary 1.2. The mapping p group of T ; moreover,

--*

e' identijies A with the character

498

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

This last integrality condition shows again that A c A (cf. [DS], Chapter 111, Theorem 4.3). Note also that A is the group T(A) considered in [DS](Chapter X,$3. No. 6). As there we put G = 2a/(a, a ) for a E A. Now we select a Weyl chamber t+ c tR and let A + denote the corresponding set of positive roots. Let a l , . . .,a, denote the corresponding simple roots and let t i be ordered lexicographically with respect to the basis a l , . . . ,a,. Let ol,. . . ,o,be the "dual basis" given by ( o j ,Gi) = dij. The weights wl,. . . ,o,are called the ,fundamental weights and we have A= Zoj (see loc. cit. Proposition 3.31 and its proof). The vectors H,, form the edges of the Weyl chamber t+. For each a E A + we select nonzero vectors X,, in the root spaces g" and g-", respectively. Let p =f a. We have of course the root-space decomposition

caEA+

g =t

+

Cg". UEA

Since the elements A E A are characterized by (A, 15)E Z (a E A) they are often called integral functions. If A E A is such that (A, G) E Z + for all a E A+ then A is called a dominant integral function. Let A( +) denote the set of all dominant integral functions. In particular, p E A(+). Also A(+) =

xi Z'W,.

We shall now study the set A(n) of weights for a single finitedimensional representation n of g. Clearly A(n) is invariant under the Weyl group W = W(u) = W(g,t). As usual (cf. [DS],Chapter 11, Proposition l.l), n extends to a representation of the universal enveloping algebra U(g). Because of the intervention of the compact group U , n can be decomposed into irreducible representations (cf. [DS], Chapter 111, Exercise B3). Thus we can assume irreducibility. Theorem 1-3. Let n be a finite-dimensional irreducible representation of g on V. Let A denote its highest weight, V, the corresponding weight space. Then

(i) dim V, = 1 and A E A( +). (ii) If v # 0 in V, and A + = {B1,. . .,fir} then V is spanned by n( Y;l:- - - Y;l;)v, ni E Z + . (iii) Each weight p E A(n) has the form I

p =A -

C miai,

mi E Z + .

1

Moreover,

(7) (8)

Jp1 I111 with equality only ifp

Ip

+ PI < I A + PI

E

ifp #

W

A.

911

1. Representations of Compact Lie Groups

(iv) If p E A( + ) and

499

if 1

p =I

- C m i a i (mi E Z') 1

then p E A(n). (v) Each p ~ A ( n )belongs to the conuex hull C ( I ) of the orbit W.I. Remark. Note that (iv) is not an exact converse of (iii) since p in (iii) need not be dominant integral. However, (iii) and (iv) show that 0 E A(n) if and only if I = miai with all mi E 2 ' .

El

Proof. Since [H,Xu]= a ( H ) X , ( H E t) we have n(X,)u n(X,)u = 0 if u E V,, a E A'. Hence, using the irreducibility,

= V,+" so

V = n(U(g))u = span of n( Y,"; . . . Y,";)u, ni E Z + .

However,

n( Y;;* * * Yj;)u E

v,-n,p,... -n,Pr,

xi

so (ii), the equality dim V, = 1, and the formula p = I - miai follow. Next select s in the Weyl group W(u) such that H , = H , , E ~ Then . v E A(n) so v = I - P, where P = Ci kiai (ki E Z'). Hence, if /3 # 0,

( A I > = (v, v> + 2(v, P> + (P, P> > (v, v >

= (P, P).

This proves (7) and H, E p.For (8) we have

I I + PI2 - IP + PI2 =

1w - IpIZ + 2 0 , P> - 2(P, P> 1

=

1112 -

+ 2Cmi(ai,p) > O 1

since H , E t'. For statement (iv) we first prove the following lemma (cf. [DS], Chapter 111, Theorem 4.3). Lemma 1.4. Let v

(9) is an interval p In

E A(n)

and let a E A. The set

S = { n E Z : v + naEA(n)}

s q and - ( v , i) = p

Proof.

+ q.

Select nonzero vectors X E g", Y Eg-", H [ H , X] = 2X,

[H,Y] = -2Y,

E

t such that

[X,Y] = H.

500

[Ch. V

ANALYSIS ON COMPACT SYMMETRIC SPACES

Then II gives by restriction a representation of CX + CY + CH on V. Suppose p’ = min S , q‘ = max S. If uo # 0 in K+q,athen by Lemma 1.2 of the Appendix, n(Y)’(v0)# 0 for 0 Ii I(v

+ q’a)(H) = ( v , a ) + 2q’.

Thus v + n a E A ( n ) for - ( v , di) - q‘ I n I q’, so p’ I - ( v , G) - q’; i.e., p‘ + q‘ I - ( v , ji). Now replace a by -a. Then p‘ is replaced by -q’, q’ by -p’, so -4’ - p‘ I-(v, -di). Thus p’ + q’ = - ( v , d i ) and the lemma is proved. For (iv) suppose p = I - miai E A( +). Suppose v E A(K) satisfies v -p = nlaj, with nj > 0. Then since ( v - p, v - p ) > 0 we have ( a j , v - p ) > 0 for some j E 1. But then

cjE,

ci

(aj, V ) = (aj, v

-P)

+ ( a j , P ) > 0,

so by the lemma v - ajEA(n). Using this inductively [which we can since (v - aj) - p = x k P k a k (Pk > O)] starting with I, we conclude that c1 E M

I.

Finally, for (v) let p ~ A ( n be ) arbitrary. We choose sp is dominant. Using (iii) we have

SE

W such that

in the notation of Lemma 8.3 of Chapter IV, which now implies

HspE C(H,). Thus p E C ( I ) and Theorem 1.3 is proved. The importance of the highest weight concept is further affirmed by the following fundamental theorem. Theorem 1.5. (i) Two irreducible finite-dimensional representations of (1 with the same highest weight are equivalent. (ii) Giuen a 1 E t* which is dominant integral, there exists an irreducible finite-dimensional representation of g with highest weight I .

This theorem will follow easily from analytic results proved in the next subsection. The theorem shows that the set 0 of irreducible finitedimensidnal representations of U is parametrized by A(+) = { I E A : H A E I T }= A f W = TIW,

where is the character group of T (cf. Corollary 1.2). The representations ni which correspond to the fundamental weights wl,. . . ,olare called the ,fundamental representations.

013

1. Representations of Compact Lie Groups

50 1

2. The Characters

Let n be a representation of U on V, dim V < 00. Then n is a differentiable function on U with values in Hom(V, V) (cf. [DS], Chapter 11, Theorem 2.6). If X E u let 2 as usual denote the left invariant vector field = X. Then we have (Ad(u)X)- = XR("-') [Chapter on U such that 11, $4, Eq. (9)]. On the other hand,

xe

and more generally, if D E D(U), (11)

( D ~ ) ( u=) n(u)n(D), u E U.

But if u E U , n(WR("))= n(Ad(u-')X) = n(u-')n(X)n(u), so by (1 1) (12)

(DR("'n)(~) = II(U)II(U- ' ) ~ ( D ) x ( u ) .

Lemma 1.6. Let n be an irreducible representation of U and let D be a bi-invariant diffetential operator on U . Then

(i) n(D) is a scalar multiple of the identity n(D) = c D I ; (ii) n is an eigenfunction of D, Dn = c D ~ . In fact, DR(")= D implies by (12) that n(D) commutes with n(u), so (i) follows from Schur's lemma; then (1 1) implies (ii). Theorem 1.7. Let n be an irreducible representation of the group U (simply connected, semisimple, and compact). Then the character x of n is given by

if A is the highest weight of n, the summations extending ouer the Weyl group.

Proof. We use the radial part of the Laplacian L , on U as given by Chapter 11, Proposition 3.12. Because of (I),

(13)

C

X(exp H ) =

m,,e"("),

H Eto.

CEA(X)

On the other hand, by Lemma 1.6, x is an eigenfunction of Lu, so, using the quoted formula for the radial part, the function (14)

(61/2x)(expH)

(det s)rn,e("P+c)'H)

= s,

c

502

ANALYSIS ON COMPACT SYMMETRIC SPACES

+

[Ch. V

+

is an eigenfunction of L,. But by (8), Ip spl < (sI spl if p # sl, so the identity LTeSP+P = - Isp + ,u12esp+P (15) together with (14) imply (d'/'x)(exp H ) =

c (det s)msAe(SP+s')(H). S

But by Theorem 1.3, msL = mA = 1, so the theorem is proved. We also see from (15) (with p = sI) that (16) Lux = P> - ( I + P, I + P > k . Proof of Theorem 1.5. Part (i) is immediate from the character formula since we have already observed (Chapter IV, 51. No. 2) that two representations with the same character are equivalent. Next suppose J o e t* is dominant integral such that the statement of Part (ii) fails. According to Chapter I, Lemma 5.14 and Proposition 5.15, the function (det S ) ~ F ( ' ~ + ~ ) h= c s ( d e t s)rsp (

0

7

Is

is a continuous function on T and is invariant under the Weyl group. Since U-conjugate elements in T are W-conjugate ([DS], Chapter VII, Lemma 7.10), h extends uniquely to a central function h on U . Clearly h is bounded and by [DS] (Chapter VII, Lemma 6.3), h" is continuous on the regular set U , whose complement U - U , has measure 0. If x is as in Theorem 1.7 we have, using Chapter I, Corollary 5.16,

because I # I , by assumption, so the elements H',,,,

which lie in

t cannot be W-conjugate. Using the Peter-Weyl theorem we conclude +

that h

= 0, which is a contradiction.

Theorem 1.8. Let rc be an irreducible representation of U with highest weight A. Then

dimn =

n (2 +

aeA+

PI a> '

Proof. We must calculate x(e) on the basis of Theorem 1.7. As in Chapter I, Lemma 5.14, we consider the algebra F over R generated by em, w E A. Let x be an indeterminate. For each v E A consider the homomorphism h, of F into the ring R [ [ x ] ] of formal power series in x

013

1. Representations of Compact Lie Groups

503

defined by h,(e") = e ( v * O ) x(w E A). By (13), dim R equals the constant term in the power series h&). With the linear mapping A : F + F given by A(e") = (det s)esw, we have

Is

so, using Chapter I, Proposition 5.15,

(18)

h,(A(e")) = h,(A(eP))

n

-

=

n

e(w*P)x

(1 - e-(',w)x)

aeA+ (e(1/2)
- e-"'2'(".">x

asA+

The lowest-degree term in this series (in x) is the equation (19) implies

1.

nueA+

(a, w)x. But then

h,(A(e"P)) = h,(x)h,(A(eP))

n

(a, 2

asA+

+ P>

= dim(4

n

(a, P >

aeA+

as desired. Let U ( g ) denote the universal enveloping algebra of g and Z(g) its center. Viewing the members of U ( g ) as the left invariant differential operators on U , we identify the elements of Z(g) with the bi-invariant differential operators on U (Chapter 11, Corollary 4.5). By considering the radial parts for the action of U on itself by conjugacy we shall set up an isomorphism between the algebra Z(g) and the algebra Z(t) of Weyl group invariants in the symmetric algebra S(t). Viewing this algebra as the universal enveloping algebra of t its elements can be regarded as translation-invariant differential operators on T. As in Chapter 11, Proposition 3.12, we use the submanifold exp Po c T as a transversal manifold for the action of U on itself by the automorphism Z(uo) : u --+ uouu, ( u o , u E U ) . The density function 6 on T has a well-defined square root 6''' given by (cf., Chapter I, Proposition 5.15),

for H E to. Theorem 1.9. Let D E Z(g) with A(D) its radial part (on expP,). Then there exists a unique y(D) E Z(t) such that

A(D) = 6-'"y(D) Moreover, the mapping D

+

o

6'12

on exp P o .

y(D) is an isomorphism of Z(g) onto Z(t).

504

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

Proof. (i) We begin with a simple remark on divisibility in &(T). A function f~ b ( T ) is said to be skew if f(exp(sH)) = (det s)f(exp H ) for s E W, H E to. The function S l i 2 given by ( 2 0 ) is skew. We claim that each skew f~ &(T) is divisible by d'12 in &T). For n E Z , a E A' the reflection ua,nin the hyperplane a = 2nin in t, is given by

(.21)

Ca,n(H)

= sa(W

+ n4niHa/(a, a),

so (f~exp)(a,,~(H)) = -(foexp)(H) (cf. [DS), Chapter VII, 57). Hence f 0 exp vanishes on the hyperplanes ua, ( n E 2)and is therefore divisible by the function H + 1 - e-a(H).Thus by (20), f o exp = (S1l2 0 exp)g where g E &(to). Also, g depends continuously on f 0 exp, and g(H H e ) = g(H) for H E to, H e E t,. Hence ,fS- ' I 2 belongs to b ( T ) as claimed and depends continuously on ,f. (ii) Next we prove that if P E Z(t) and we consider P as an invariant differential operator on T then there exists a E E Z(g) such that

+

A(E) = d-'I2P

For F

E b(U)

o

d1I2.

put F = FI T ,

and consider the functional (22)

S:F

+

[S- '/2P(d'12(F')-)]e,

F

E b(U).

Since (F9)- is W-invariant the remarks above show that S is a distribution. Moreover, S is central and has support contained in {el. Hence the is the image of S under the map convolution E : F + F * (where u + u-' of U ) is a bi-invariant differential operator on U . To compute the radial part A(E) we note that if F is central then (A(E)F)(t) = (EF)(t) =

F(tu) dS(u) = S(F"'-")

If F is the character x of an irreducible representation with highest weight I then by Chapter IV, Theorem 1.6, the above integral equals x(t)x(s)/x(e). Thus by Theorem 1.7 the entire expression equals p(n

+ P)F(t).

§]I

1. Representations of Compact Lie Groups

505

This, however, is also equal to (6-1'2P(6'/2F))(t). Thus A(E)F = 6- 1/2P(81/2F) if F = x, so by the decomposition of central F E b ( U ) into characters (Corollary 3.6 of this chapter) it follows that A(E) = 6- ' l 2 po b1l2

as desired. (iii) For the last step we prove that if D E Z(g) then there exists a Q E Z(t) such that (DF)(e) = (6- 1/2Q(8'/2F))(e) for F E & ( U ) central, the bar denoting restriction to T. For this we construct a W-invariant distribution S on T with support at e such that S ( F ) = (DF)(e) for each central F E b ( U ) . To define S consider an open ball V around 0 in u on which exp: u U is a diffeomorphism and exp(V n to) = (exp V ) n T. (cf. [DS], Chapter VII, Lemma 6.4). Suppose ~ E B ( T is) W-invariant and has compact support in (exp V ) n T. Passing to 4 exp and using Theorem 5.8 of Chapter I1 we can extend 4 to a function (D on U which is smooth near e and invariant under conjugation. We put S(4) = (D(D)(e) and if f E b ( T ) has compact support inside (exp V) n T we put S ( f ) = I W 1- 'S(E,f"). We then extend S to a W-invariant distribution on b ( T ) with support { e } . Then there exists a Q' E Z(t) such that --$

0

(DF)(e) = S ( F ) = Q'(F)(e) for all central F E d(U). The differential operators Q' and 6- 'l2Q' 0 dl/' have the same leading terms so we see inductively that there exists a Q E Z(t) such that (DF)(e) = (6- 1/2Q(61/2F))(e) for each central F E b(U). Since (DF)(e) = (DF'Xe), it follows that (DF)(e) = (6- '/2Q(6'/2(Fh)-))(e), F E d(U). The proof of (ii) then shows that A(D) = 6-"2Q o 61/2.

Defining y: Z ( g ) -,I(t) by y(D) = Q parts (ii) and (iii) above imply that y is a surjective homomorphism. Also y is one-to-one because if Q = 0 we have DF = 0 for all central F E b ( U ) . But then (DF)(e)= (DF')(e) = 0 for all F E &(U) so D = 0. It is of interest to combine Weyl's character formula with HarishChandra's integral formula in Chapter 11, Theorem 5.35, also taking into

506

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

account the formula for the order of the Weyl group (Corollary 5.36). The function

is holomorphic and W-invariant so by Chapter IV, Lemma 9.5 it extends to a holomorphic function j on uc invariant under Ad(U). Theorem 1.10. Let x be the character of an irreducible representatation of U with highest weight 1. Then

(24)

j(X)x(exp X) = x(e)

du,

X E u,

and Ij(X)12 = det, (1

;&?*),

x E u.

In fact, the roots are purely imaginary on to, so if X =u*H

(UE

U,H E to)

we have

by [DS], Chapter VII, $4. The formula for x now follows by putting H' = HA.+, in the cited Theorem 5.35 (Chapter 11), taking Theorems 1.7 and 1.8 into account. Formula (24) has an interesting interpretation in terms of Fourier transforms. The point iHA+pbelongs to to and we denote by p A the Uinvariant measure on the orbit OA= U . ( i l l A + & c u normalized by pl(OA)= 1. Then pi can be viewed as a distribution on u of compact support; as observed in Chapter I, 52. No. 8, its distributional Fourier transform is the function P A ( X ) = e - i ( y . x )&( Y ) . Formula (24) can therefore be rewritten as follows.

su

Corollary 1.11. The function j ( x exp) on u is a constant multiple of the Fourier transjorm PA, i.e., 0

(25)

j(X)x(exp X) = x(e)PA(X),

X E w.

1321

2. Fourier Expansions on Compact Groups

507

Let K be the fixed point group of an involutive automorphism of U ; K is actually connected (cf. [DS], Chapter VII. Theorem 8.2). For analysis on symmetric spaces we are interested in the representations n of U which are spherical (with respect to K); that is, n ( K ) has a nonzero fixed vector. In $4, No. 1 of this chapter we shall give an explicit characterization of the weights of these representations. $2. Fourier Expansions on Compact Groups 1. Introduction. L 1( K ) versus L 2 ( K )

Let K be an arbitrary compact group with Haar measure dk normalized by jK dk = 1. Let denote the set of all equivalence classes of irreducible (finite-dimensional) unitary representations of K . If 1E I? let U , be a member of the class 1acting on the d,-dimensional Hilbert space X,. If for some orthonormal basis of Xa, U , has the matrix form U, = (u;) the Schur orthogonality relations together with the Peter-Weyl theorem express the following result (already used in Chapter IV, 41). Theorem 2.1. The ,functions d:" uh (1E a complete orthonormal system in L2(K). +

K, i,

j = 1,. . . , d,) ,form

To each f~ L'(K) we associate the Fourier series (Tr denotes the trace)

where the Fourier coeficient is the endomorphism of X, given by A, =

1

, f ( k ) U , ( k - ' ) dk.

K

I f f is a K-finite function on K (under the left regular representation on K ) or, equivalently, i f f is a finite linear combination of functions u; then the orthonormality in Theorem 2.1 implies that the sign in (1) can be replaced by an equality sign. If g E L1(K) has Fourier coefficients {B,} then

-

Also, if A* denotes the Hilbert space adjoint the completeness in Theorem 2.1 implies that

(3)

p, W A a AX)

I€K

508

[Ch. V

ANALYSIS ON COMPACT SYMMETRIC SPACES

for f c L 2 ( K ) . This relation and (2') imply that EL' is uniquely determined by its Fourier series. Then (3) even holds in the sense that if for a sequence {A,},,m the right-hand side of (3) is finite then (1) holds for some ,f E L2(K). Iff is a central function then (1) takes the form (with x, = Tr(U,))

f(k)

-

a, = S x / ( k ) x , ( k - l ) dk

Ca,x,(k), dEK

and the characters 2, form a complete orthonormal system in the space of central functions in L ~ ( K ) . Since f in (1) is uniquely determined by the series (1) it is of interest to study properties of the family { A , } n E k Let . 11 11 denote the operator norm and Put Ilf l, = (JK If(k)IP dk)"P.

Let f~ L'(K). Then the function I llA,ll vanishes on the discrete space R. In other words, for each E > 0 the set

Proposition 2.2.

at

00

-.f

{JER:II&Il

>El

is finite. Proof. If f € L 2 ( K ) , this is obvious from (3) since 11AI12 5 Tr(AA*). II .f - 911 and the In general (2) implies llA,ll IIlflll, so IlA, - B,I proposition now follows from the fact that L 2 ( K ) is dense in L'(K). In this section we shall discuss some properties of the expansion (1)

which emphasize the difference between L ' ( K ) and L2(K). For motivation we first discuss briefly the case of the circle group.

2. The Circle Group Let T denote the circle group {e'":O < x I2 ~ ) We . recall that the irreducible representations of T are given by the characters xn : eix+einx, n varying over Z . Identifying functions f on the circle with periodic functions F: x + f ( e i x ) on R, the Fourier series (1) becomes the classical Fourier series (4)

F(x)

-C

aneinX,

ncZ

where 1

r2n

F E L'(T),

621

2. Fourier Expansions on Compact Groups

509

Here Proposition 2.2 becomes the Riemann-Lebesgue lemma: a, -,0 as In1 --* co. Moreover, (3) generalizes the Riesz-Fisher theorem: The convergence

is a necessary and sufficient condition for the Fourier series (4) to represent a function F E L2(T);in this case we have the Parseval formula,

(7) Condition (6) has two interesting features: It only involves the absolute value of a, and is independent of the order. Thus if (6) holds and if 6 :Z + Z is a permutation the series a,~,,ei"x is the Fourier series of an L2 function. Similarly, if I y, I = 1 for all n E 2 the series

EneZ

C a, yneinx noZ

is the Fourier series of an L2 function. We are therefore led to the following questions where for a subspace E c L' the notation n

means that

a,einx is the Fourier series for a function in E.

Ena,ei""

I. Let

E L'(T)

and assume

(9) for every sequence (y,) such that Iy,, I = 1for all n E Z. Is then T h e answer is yes (Littlewood [1924]). 11. Let

C,, a,,einXEL'(T)

EnIa, l2 <

and assume that for every permiitation

the integers we have (10)

C a,(,,e'"x

E

L'(T).

n

Is then C 1u,,l2 < co? T h e answer is yes (Helgason [19583).

We quote three more results with a similar conclusion. 111. (Sidon [1932]). Let (a,) be a sequence such that for each

E b,einx E C(T) n

we have

1,la,,bnl < a.Then C,,1~1.1~

c co.

00

6

?

of

510

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

1V. (Zygmund [1930]). Lef

1 akeinkxE L ' ( T ) , k

where nk+Jnk> 2 ,for all k. Then m

Fourier series of the form (1 1) where nk+l/nk > q > 1 are called lacunary Fourier series. V (Kolmogoroff [1928]). Consider rhe compact group T" = '& where & = T for each k. Assume f E L'(T") has Fourier series of the,form

np=l

f ( x 1 , x 2 , . . .)

-c m

ake'Xk.

1

Then m

1 laA2 < 00. 1

We shall now generalize these results to compact groups K (abelian or not). We shall define lacunary Fourier series for such K ; then IV and V above will be special cases of the same result and I1 (generalized to abelian groups) will be a corollary. Finally, the generalization of Littlewood's result I to the group K will at the same time give the extension of Sidon's result 111.

3. Spectrally Continuous Operators Let p be a complex-valued measure on the compact group K. Let L, and R, denote the bounded operators on L P ( K )( p 2 1) defined by

(R,F)(k) = ( F * p)(k) =

F(kx-') dp(x). K

If p = f ( k ) dk where f~ L ' ( K ) , we write L, for L,, R , for R,. Let f * denote the function k + ( f ( k - l ) ) - (the bar denoting complex conjugate) and p* the measure f -+ (p(.f*))-. Let J denote the involution .f + f*. Then a routine computation gives

L,,

= JR,J.

$21

2. Fourier Expansions on Compact Groups

Also, L,. acting on Lq(K) (p-' L P ( K ;) thus

(13)

+ q-'

L,. = (L,)*,

=

51 1

I ) is the adjoint of L, acting on

R,, = (R,)*.

In the case p = q = 2 we see from (12) and (13) that L, and R , have the same norm. Definition. Let , f e L ' ( K ) and consider L, and R , as operators on L 2 ( K ) .The common value of the operator norms llLIII and llRrll will be called the spectral norm of .f. We denote it by IIfllsp. Then llfllsp

=

SUP gcL2

Since 1I.f* s l l 2 5

II/II

1 Ilsll2

I l f * 9112 llsll2

*

for all 9 E L2W)we have

Ilflll,

fELZ(K). In the case K = T, Parseval's formula (7) shows quickly that

(14)

(15)

llfllsp

llfllsp

5

= max (a,l

iff

-C

a,,e'"".

n

n

This will be generalized in Lemma 2.5. Because of (14) the spectral norm topology on L ' ( K ) is weaker than the usual topology. A linear transformation T on L ' ( K ) is said to be spectrally continuous if it is continuous from the spectral norm topology to the L 1 topology. Equivalently, T is spectrally continuous if SUP (IITfll,) < a.

1 1 1II.P<

1

Of course (14) shows that a spectrally continuous operator is continuous. Theorem 2.3. Let K be a compact group. The spectrally continuous operators S on L 1 ( K ) commuting with all right translations on K are precisely the lefi convolutions S = L , with f E L 2 ( K ) .Furthermore,

Example. Consider the case when K is the circle group T. Then Theorem 2.3 implies Littlewood's result I in the preceding subsection. In fact, let C,(Z) denote the Banach space of sequences y = (y,,) such that yn -,0 for In1 + m, the norm being llyll = sup,, I ynl. Each y E C,(Z) can be written y = tl + /i' where tl and p are sequences (a,,) and (P,,) where

512

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

la,l and [@,,Iare independent of n. Thus assumption (9) implies that for each y E C,(Z) there exists a function f , E L'(T) such that fy

-C

anyneinx.

n

The mapping S : y f, of C,(Z) into L'(T) is linear and has a closed graph { ( y , f y ) : y E C o ( Z ) }in the product space C,(Z) x L'(T). By the closed-graph theorem of Banach, S is continuous. But by (15) this means that S is a spectrally continuous operator on L'(T). Since it clearly commutes with all translations on T,Theorem 2.3 implies that 1 lanlZ< co as claimed. See Exercise B2 for a nonabelian analog. For the proof of Theorem 2.3 and related results we start with a simple lemma (cf. Wendel [1952], where, however, the continuity of S is assumed). Lemma 2.4. Let S be a linear mapping of L'(K) into itself: The forlowing properties are equivalent:

(i) S commutes with all lejl translation L(k), k E K. (ii) S commutes with all left convolutions L,, g E L'(K). (iii) S has the form Sf = f * P ,

where p is a complex-valued measure on K .

I n particular, (i) implies that S is continuous. Proof. Let 1E R and, as in Chapter IV, 81, let H, denote the span of the matrix elements u&) of UA(k). If xA is the character of 1 we know from Chapter IV, Corollary 1.8, that H A= 1, * L'(K). Thus (ii) implies SHAc HA.Since H, consists of the K-finite elements of L'(K) of a given type it is also clear that (i) implies SHAc HA.Assuming either (i) or (ii) we can therefore define the matrix S, by (SA)ij= (Suij)(e). We shall now prove that each of the properties (i) or (ii) is equivalent to

(iv)

Sf

-

dATr(SAAA UA(k)) for

f

-2

dATr(AACIA@)).

1

A

Suppose S satisfies (ii). Then by the above SH, c HA.The orthogonality relations imply U t l * untj

= (1/d&mIUij,

so by (ii) UII

* SUmj = (l/dJ&tlSUij*

923

2. Fourier Expansions on Compact Groups

513

It follows that Suij is a linear combination of the uiq (1 I q IdJ, so,

Then, with f as in (iv) we have (XA

* Sf)(k)= S(XA * f ) ( k ) = S(Tr(AA UA))(k) = Tr(SzAzU,(k)).

This shows that (ii)

(iv). Also, (iv) =$ (i). Next assume (i). Since

ui,xuk) =

c ~i1(u)u,i(k),

u, k

E

K,

1

we deduce (17), which as we saw implies (iv), which in turn implies (ii) because of (2'). Thus (i) o ( i i ) o(iv). Now assume (iv). Then if fn + f and Sfn + g in L'(K) we see from the inequality llAzll Illflll that g = Sf. Thus S has a closed graph in L'(K) x L'(K), so by the closed-graph theorem, S is continuous. The complex adjoint operator S* : L"(K) + L"(K) will also commute with left translations. Since (S*f)Uk' - S*f = S * ( f Q k )

- f),

the boundedness of S* shows that S*(C(K)) c C(K). The functional v :f + (S*f)(e)is therefore a well-defined measure on K and by the translation invariance, S* = Ri, where i, is the image of v under the map k + k-' of K. Hence by (13), S = R, where p = (t)*. Since (iii) obviously implies (i) the lemma is proved. Lemma 2.4'. Lemma 2.4 also holds with L'(K) replaced by C(K).

The proof is the same except that the last step (passing to S*) is now unnecessary. Lemma 2.5.

Let f~ L'(K). Then

I l f l l s p = ma? IlAzll AEK

in terms of the expansion (1). Proof. Let g E L 2 ( K )and consider the expansion (2'). Then

514

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

The last two expressions are majorized by

respectively. We see this, say for the first, by diagonalizing A,A: and using \ \ A j . A ; \ \ = IJA,)/*. This proves (18)

II.f*glI2 I

max IIAiII IIgII2,

1I.f*gII2 5 max IIBlII I I . f I I 2 ,

1

1

so ll,fl[sp I max, IIA,I1. On the other hand, suppose IIA1ll reaches its maximum for ;1 = p. Then for each endomorphism B, of R,

lll'll:p

2 T a pA,*~,*~,)/Tr(B,*B,).

Now choose a basis in 2, diagonalizing A,A;I* and then choose the matrix B, having all entries 0 except at the place in the diagonal where the largest eigenvalue of A,A,* occurs. Then IIjllsp 2 IIAJ, whence the lcmma. If , / ' E L 2 ( K ) and if g E L ' ( K ) has Fourier coefficients B , we have by (18) (19)

IIJ'*911, 5 II1'*91l2 5 max IIBi.II 111'112, 1

which shows that L, is spectrally continuous and the right-hand inequality of Theorem 2.3 follows. To prove the converse (and the nontrivial) part of Theorem 2.3 we assume that S is a spectrally continuous operator on L ' ( K ) commuting with the right translations. By Lemma 2.4, we have Sg = p

* g,

g

E L'(K),

where p is a complex-valued measure. To p we assign the FourierStieltjes series

where A j.

=

SKU#

- 1)

tlp(k).

Again Theorem 2.1 implies that p is uniquely determined by the family A,

(A E R).

Let { A ,,..., AN} be an arbitrary finite subset of Z?, U , , . . ., U N corresponding representations of K , and dl, . . . ,dN their degrees. Then by

921

2. Fourier Expansions on Compact Groups

515

Lemma 2.5 we have if d V , is the normalized Haar measure on the unitary group U(rl,) and dV = dVl . ‘ . dVN on U ( d , ) x . . x U(tlN), l l ~ l l n p <1

the last identity the result of interchanging the integrations and using the right invariance of dl.: Putting N

V = (V,, . . . , V,),

d, Tr(A, V,),

F( V ) = I

we have proved

In order to cstimate this last integral further from below we use the inequality

which is obtained from Holder’s inequality j I jg1 I (j 1 f Ip)>”P(<s Ig(q)llq by putting .f‘ = lF12’3, g = IF14’’, p = $, q = 3. The inequality (22) is chosen because the integrals j lF12 and J IFI4 are more manageable than j IFJ. For the first we have

J

I$

N

d, Tr(A, V,) dV = C d, Tr(A,AX)

(23) U ( d t )x ... x U(d.v) I j I because the function F on U(tlI) x ... x U(d,) has the Fourier series

- zd,Tr(A,V,/,). N

F(V1,..., V N )

1

For

IF l4 we prove the following result.

Lemma 2.6.

Let A he an arbitrary n x n complex matrix ( n 12).

Then ITr(AV)I4 rlV =

(

1

[Tr(AA*)I2 - - Tr(AA*AA*) n2 - 1 n

~

516

[Ch. V

ANALYSIS ON COMPACT SYMMETRIC SPACES

Proof. We begin by writing A

= PO1,where

P is positive definite,

O1 unitary, and then B = O,PO;', where B is a diagonal matrix and

O2 unitary. Clearly Tr(AA*) = Tr(BB*), so by the invariance of dV it suffices to prove the lemma when A is a diagonal matrix. Let e l , .. . ,en be the corresponding orthonormal basis for the Hilbert space on which A and V act. Then ITr(AV)I4 dV =

1 aiqakal

uicjukz,

dV,

U(n)

i.i.k.1

where ai = (Ae,, ei), uj = ( Vej, ej) (inner products). Then we have uitjukcl

(24)

dV = 0

unless (i, k ) = ( j , 1) or (1,j).

S,(n)

In fact, let V , be a diagonal matrix with diagonal elements a, of modulus one. Writing the integral in (24) as

S,(n)

(Vei, ei)(ej, Vej)(Vek, ek)(el,ve,) dv,

we find, by replacing V by V V , and using the invariance, that this equals

a.E.a~2 I

J

(Ve,, e,)(ej, Vej)(Vek,ek)(el,Ve,) dV.

k 1 S,,n)

Now (24) follows, the a, being arbitrary. Note that the equation uiujdV = 0,

(25)

1 Ii, j In,

S,(n)

follows in the same way. Using (24) we obtain

Thus it suffices to compute the integrals

§21

2. Fourier Expansions on Compact Groups

517

Let

U , ( n - 1) = { T EU ( n ) :Tel = el}, U1.2(n- 2) = { S E Ul(n

Cn = U(n)/Ut(n

-

L - 1

119

I ) : Se2 = e,},

-

U,(n -

=

I)/CJ1,2(n

-

21,

and let dP and d T denote the unique normalized invariant measures on the unit spheres Z,, c C", c C n - l , respectively. Then

Writing

Ve, = wlel +

1 . .

+ w,e,,

Te, = Oe,

V-'e, = blel

+ z2e2+ + z,e,, -

a

.

+ + b,,e,,,

518

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

On the other hand,

=1-

[Em

I w 1 14 d

~ ,

and if dR is the invariant measure on S Z n - lnormalized by J dR = 1 then

~ z n l w l 1 4 d Y=

+ x:)'

(x:

dR

=2

s2n- 1

But by Chapter I, Lemma 2.30,

s + s

x;' dR =

s2n- I

and since (x:

+ ...

3 4n(n

+ I)'

dR = 1, x:x: dR =

s2n- I

1

4n(n

+ 1)'

Substituting, we get the desired formula

Second,

so

Substituting into (26) we obtain the lemma. Corollary 2.7. The ,function N

F(V1,. . . , VN) =

1d, Tr(A, V,) 1

satisfies

(29)

JIF(V)14 dV I 2(slF(V)12 d V ) 2 .

2. Fourier Expansions on Compact Groups

921

519

(Xi

Using Schwarz's inequality (Ciai)' I aZ)n for the eigenvalue ai of AA* we obtain a simple estimate for the expression in Lemma 2.6, (30)

2

(CTr(AA*)]' - -1 Tr(AA*AA*)

n

Using the orthogonality relation as we11 as (23), (25), and (30) we obtain n

N

r

1

+2 i#j

IdiTr(Aiv)IZd v

U(dj)

N

I 2

di[Tr(A,A,*)]2

1

IdjTr(AjVj)I2dVj

W j )

+ 2 1 d i d j Tr(AiAT)Tr(AjAf), i fj

1

and now (29) follows. Combining now (21), (22), and (29) we have

I N

1

d, Tr(A,A,*) I IIS1l2. 2 1 The coefficients A . . ,AN were chosen arbitrarily from the FourierStieltjes series (20). But then (31) shows that S = L, for a function , f L~ 2 ( K ) and (16) holds. This proves Theorem 2.3. (31)

-

4. Absolute Convergence A Fourier series (1) on K is said to be absolutely convergent if (32)

1 d, T r ( l 4 I ) <

leR

00.

Here lAll denotes the unique positive self-adjoint operator such that ] A l l 2 = A l i l f . [If A = PU is a polar decomposition of a matrix A where P is positive definite and U unitary we have P = I A J ; also A*A = U - * ( A A * ) U ,so use of A f A , instead of A,Af in defining lAll would have led to the same trace.] If K is abelian then each d l = 1 so definition (32) amounts to the usual one. Definition. Let A ( K ) denote the set of f convergent Fourier series.

E

L ' ( K ) having absolutely

Lemma 2.8. (i) A ( K ) is invariant under lefi and right translations of K . (ii) A ( K ) c C ( K ) . (iii) A ( K ) = { g * h : g , h E L 2 ( K ) } .

520

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

Proof. Let M be an n-dimensional Hilbert space with inner product ( , ) A, B two operators on A = PU a polar decomposition. Let e l , . . . ,e, be orthonormal vectors diagonalizing P , i.e., Pe, = ciei where ci 2 0 (1 I i I n). Then

ITr(AB)I = ITr(PUB)I =

ci(UBei,ei) I i

Ci CiIIBeiII IIeiII, so

(33) Since lABl (34)

ITr(AB)I 5 Tr(IAI)IIBII* = ABV where V is unitary this implies (as I(BVI1 I 11B11)

Tr(lABI) ITr(lAI)IIBIl,

TrOABI) I Tr(IB0lIAII.

Second, the mapping (A, B) -+ Tr(AB*) is a positive definite inner product on the space of complex n x n matrices. Thus

(35) Tr(IAB1) = Tr(ABV) I Tr(AA*)’/’ Tr(BB*)’/’. Now part (i) follows directly from (34). Also, (33) shows that the Fourier series (36)

f ( k)

ly

c d, TNA, U,(k)) 1

for f~ A( K) converges uniformly on K, so J E C( K) . For (iii) consider g, h E L’(K) with Fourier coefficients {B,} and {C,}, respectively. Since

(37)

[d:l2

Tr(B,BX)’12 d:‘’ Tr(C,CX)’/’]

I11g11211h112,

1

(35) implies g * h E A(K). On the other hand, if f~ A ( K ) and we write A , = lA,l V,, where V, is unitary, then by (32) the series

c d, Tr( I A , I l / z U , ( m c d, Tr( I A , I

112 V, U,(k))

A

1

4, rl/ E L’(K) for which f = I+, * 4. Remark. Writing IA I = A U (U unitary) we have, using (33),

represent

Tr(IAl) 2 sup (Tr(AB)I 2 ITr(AU)I = Tr(lAl) IlBll

so Tr(lA1) = sup ITr(AB)I. lPll s 1

§21

2. Fourier Expansions on Compact Groups

+

Thus Tr(I.4, A , I ) I Tr((A,I) subspace of C ( K ) .

+ Tr(lA,I)

52 1

and hence A ( K ) is a linear

Theorem 2.9. Let S : C ( K ) + A ( K ) be a linear mapping commuting with all left translations L(k), k E K . Then there exists a function F E L 2 ( K ) such that

Sf=f*F,

fEC(K).

Example. Consider the case when K is the circle group T. Suppose (a,) is a sequence such that for each

(38)

f-Cb,einx,

fEC(T),

a,b,ei"" the map f + f , of lanbnl< 00. Putting &(x) = we have C ( T ) into A ( T ) then satisfies the assumptions of Theorem 2.9, so we oblanJ2< co. (Cf. I11 in subsection 2 above.) tain Sidon's conclusion

For the proof of Theorem 2.9 we shall construct a suitable spectrally continuous operator and then invoke Theorem 2.3. Because of Lemma 2.4, S has the form Sf = f * P where p is a measure on K. We consider the Fourier-Stieltjes series

(39)

where (41) Then

(43)

for all f~ C ( K ) . Consider now a family

(44)

B

=

( B A } A e t such that

JJBJJ = sup JJBJ< co, A

522

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

B , being an endomorphism of X1.Then by (34) and (43),

(45) and the mapping which maps the function f to the function in A ( K ) given by

commutes with all left translations L(k) ( k E K ) . Hence by Lemma 2.4'

1 d1 Tr(BASAA1U1(k)) 1

j ' * pldk).

where pB is a measure. Consider now the mapping M: B + p B from the Banach space of families (44)t o the Banach space of measures on K . Since p B has Fourier-Stieltjes series d , Tr(B,S1 U,(k)) it is seen as before that M has closed graph, hence is continuous. If the family B = {B,} forms the set of Fourier coefficients of a function 4 E L ' ( K ) then pB = p * 4 E L ' ( K ) . Since M is continuous, the map A?: 4 -+ p * 4 of L ' ( K ) into itself is spectrally continuous. Moreover, fi commutes with right translations on K . From Theorem 2.3 we now conclude that p is given by an L2 function and this proves Theorem 2.9.

5. Lacunary Fourier Series Let n' and n" be two finite-dimensional representations of K on V' and V", respectively. Let n 0 n" denote their tensor product as defined in Chapter IV, $1. Let (el) be a basis of V', (eJ a basis of V". Then the elements e: @I e; form a basis of V' @ V". Let d' = dim V', d = dim V", and let E' and E" be endomorphisms of V' and V", respectively. Defining the matrices A = (ski), B = ( b l j ) by d'

d"

E'e: =

a&.,

E"e; = Cbfje;', I= 1

k= 1

we have

(E' 0 E")(ef @ e;)

(47)

=

C akibfje;0 er. k,I

Putting

fi+d'(j-l)

= ef @ ey

the matrix expression of E' @I E is given by

s= 1

§21

523

2. Fourier Expansions on Compact Groups

where

(49) C i + d ' ( j - l ) , k + d ' ( l - 1 ) = aikbjlThus C = (cs,) is the d'd" x d ' d matrix obtained by taking the matrix B and replacing each entry blj in it by the entire matrix Ab,. We also consider the direct sum representation n' + n" of K on V' x V" as defined in Chapter IV, 51.

Lemma 2.10. Denoting the character of a representation n b y have Xn'+n" = Xn,

+ &*,

&Bn" = Xn, *

xr

we

Xn.'.

In fact, the first relation is obvious and the second follows from (49). Definition. A (discrete) set S is called a hypergroup if there is given + of S x S into the set of measures on S . A suba mapping (a, ,!l) set T of the hypergroup S is called a subhypergroup if all the measures pa, (a, p E T ) have support contained in T. The group K being compact, the set R is a hypergroup in the following sense. For each pair a, p E R the tensor product a @ /3 has a direct decomposition into irreducible unitary components. This decomposition defines a measure pa,s on the discrete set R with values in 2 ' . Two finite-dimensional representations of K are said to be disjoint if no irreducible component of one is equivalent to an irreducible component of the other. It will be convenient to denote by E the identity transformation of an arbitrary vector space and to denote by e the trivial representation k --* 1 of K . For M c K put

M'

=

{ a E I? : U,(k) = E for all k E M }

and for X c I? let

A?'

= { k E K : U,(k) = E for all a E A?}.

It will also be convenient to call a subhypergroup X of R a normal subhypergroup if e e X and if c r ~ Ximplies E E X (the bar denotes complex conjugation). Proposition 2.11.

(i) I f M c K , M' is a normal subhypergroup of R and (Ml)' is the smallest closed normal subgroup of K Containing M . (ii) I f X c R, A?' is a closed normal subgroup of K and (A?')' is the smallest normal subhypergroup of R containing X . (iii) If N is a closed normal subgroup of K , (KIN)* = N'.

524

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

Proof. We start with (ii). Let &' c R. Then it is obvious that &'' is a closed normal subgroup and that (&")' is a normal subhypergroup. The matrix elements from &' can be regarded as a family of continuous functions on the factor group K / S ' separating points. Let &'* denote the normal subhypergroup generated by &' and let W denote the set of linear combinations of matrix elements from members of &'*. Then W is an algebra, it is closed under complex conjugation, it separates points on K/&", and it contains the constants. By the Stone-Weierstrass theorem, W is uniformly dense in C(K/&). Hence, if a€(&")' - &* each matrix element u(k) from Ua(k) can be uniformly approximated by members of 9, but on the other hand, u(k) is orthogonal to 9 by the orthogonality relations. This shows that (&")' = &'*. (i) Let a, PE M'. Corresponding to the decomposition of a @ p into irreducible components ai we have for the respective characters

(50)

Xa(k)Xp(k) = X a , ( k )

+

**.

+ Xa,(k),

and the corresponding dimensions satisfy (51)

dad, = d l

+ + d,.

If k E M we have ~ , ( k )= d,, xa(k) = d,, and since maxk IX(k)l = d , we conclude from (50) and (51) that ~ , , ( k )= di for 1 Ii < n. But, U,, being unitary, this implies that U,,(k) = E, so a, E M '. Thus M' is a subhypergroup which is normal. It is obvious that (M')' is a closed normal subgroup containing M. If N is an arbitrary closed normal subgroup containing M , then (M')' c N because if IE(M')' - N we can (by going over to the factor group K I N ) find a representation D A (A E R) such that D,(k) = E for k E N but DA(l)# E. Thus A E N' whereas A # [(M')']'. This last set equals M', as we see by applying (ii) to &' = M '. Thus we have A E N' - M', which contradicts M c N. (iii) This is implicit in the proof of (ii) if we let &' = N' = &'*, because then &" = (N')' = N by (i). Definition. A subset S = R is said to be lacunary if the two following conditions are satisfied.

L,. Whenever a, 8, y, ~ E such S that the sets {a, P}, {y, 6 ) are different the representations U , @ U , andU, 8 Ud are disjoint. L,, . The measures pa, (a, fl E S) are uniformly bounded.

,

This last condition amounts to that the number n,,, of irreducible components of a @ 8, counted with multiplicity, has a bound: na,


(a, P E S).

021

2. Fourier Expansions on Compact Groups

525

xIeSd A Tr(A, U,(k)) is called lacun-

A Fourier series on K of the form ary if the subset S c I? is lacunary.

Remarks. If K is abelian, condition L,, is automatic and a Fourier series a,x(k) on K with nonvanishing coefficients is lacunary if and only if there is no nontrivial relation

xXES

(52) x 1 x 2 = x3x4 between the elements of S. In particular, the series (11) on the circle group is a lacunary Fourier series in the sense of the general definition above. But it is also clear that the Fourier series

-C OD

f ( x , , x2, . . .)

akeiXk

k= 1

on T" (cf. V, No. 2) is a lacunary series on T". We shall now consider a more general situation. Let I be a set and to each i E I attached an integer di > 0. We consider the product group

u=

(53)

n U(di). ic I

The projection Di of U onto U(di) is a unitary representation which clearly is irreducible; it follows that I can be regarded as a subset of 0. We consider Fourier series on U of the form di Tr(AiDi(u)).

(54) iel

Theorem 2.12. Suppose .f E L'(U) has Fourier series of the ,form (54). Then f E L 2 ( U ) and 2-1'211fl12 II l f l l l IIlfl12. Moreover, the Fourier series (54) is a lacunary Fourier series on U. Proof. Let J c I be any finite subset and consider the sum

dj Tr(BjDj(u))

s(u) =

(55)

jeJ

for arbitrary Bj. It is then clear that (56)

1"

IS(^) Y du =

S,, I x j

1

m

dj Tr(BjDj)

~ D J

where du is the normalized Haar measure on U and dDJ is the normalU(dj). But the integral on ized Haar measure on the product V, = the right-hand side of (56) is dealt with in (22) and (29), so we conclude

nEJ

(57)

526

[Ch. V

ANALYSIS ON COMPACT SYMMETRIC SPACES

By the general Peter-Weyl theorem f can be approximated in the L' norm by functions sI(u), s2(u), . . . , with a finite Fourier series of the form (55). By (57) (s,) is a Cauchy sequence in Lz(U), so we conclude f E L 2 ( U ) and (57) also holds for f. Finally, the lacunary property follows from the following statements. (a) Di @ D, is irreducible if i # j . (b) Di @ Di is irreducible if di = 1. (c) Di @ D iis disjoint from D,@ Dj if i # j . (d) If di 2 2 then Di @ Di decomposes into two irreducible parts [of dimensions g d : + d i ) and g d : - di)]. Here (a) and (b) are obvious. For (c) we remark that the number of irreducible components common to Di @ Di and Dj@ D, is equal to J(xo,)2(&j)2

du =

J

(Tr Di)2(Tr Dj)2 dDi dD,

Wr)x W

j )

[dD, is the Haar measure on U(d,)], and this last integral vanishes by (25). Finally, Lemma 2.6 shows that

5,

IXDi@Di(U)12

du =

1

ITr(Di)14

dDi

= 2,

Wi)

so Di @ Di decomposes into two irreducible parts. Since the symmetric and skew-symmetric second-order tensors do form invariant subspaces they give the decomposition of Di @ Di. This proves Theorem 2.12; note that it is the lacunary property of the series (54) which lies behind the estimate (29) of IIFI14, which in turn was the main ingredient in the proof of Theorem 2.3. Next, we consider the case of central functions on K ; that is, functions invariant under inner automorphisms. If K is abelian this restriction is of course vacuous. Theorem 2.13. Let f be a central function in L ' ( K ) and suppose f has a lacunary Fourier series. Then f E L2(K).

-

Proof. The Fourier expansion o f f has the form f ( k ) x A E S a A ~ A ( k ) where aAE C and S c I? is a lacunary subset. We consider a finite partial sum s(k) = a,xn(k). Then

s2 =

c P

a;x;

+2

c

P'4

apaqxpxq.

PI

2. Fourier Expansions on Compact Groups

527

If we now expand x: and xpxq into a sum of characters xi of irreducible representations [cf. (50)] at most one pair ( p , q ) gives rise to the same xi because of condition L,. Using now L,, as well we have N

n

C being a constant. Again by (22) this leads to an estimate

so we conclude f

E L 2 ( K )as

before.

Remark. Taking K as the circle group T we obtain results IV and V in No. 2. We can also use Theorem 2.13 to derive the following simple result (cf. I1 in No. 2).

Corollary 2.14. Let K be a compact abelian group. Suppose that for each permutation cr of R the series

(58)

is the Fourier series for some L' function on K . Then C-lax12 .c co. XEK

Proof. We may assume R infinite. Then R contains a countably infinite lacunary subset s. In fact, if R has an element xo of infinite order we can take S = { x o , x i , x : , ...}. If on the other hand all members of R have finite order we pick x1 E R - { e } arbitrary, xz

# (XI>,

x3

# (Xl,

x 2 h

etc., ( L ) denoting the subgroup of R generated by the subset L . Then S = {xl, x 2 , . . .} has the desired property. Now we can write

1x

=

c a,x(k) + c a,x(k), P

Q

where P and Q are countably infinite subsets of R and Cplaxl < co. It follows that the series x x E Q a x ( a . x ) ( kis) an L' Fourier series for each

528

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

permutation D of I?. Choosing this permutation in such a way that this series is a lacunary series Theorem 2.13 gives the desired conclusion. Example. As the proof above showed, for each infinite compact abelian group K there are infinite lacunary subsets of I?. We shall now look into the possibility of an analog for the simplest nonabelian case K = SU(2). For the torus

the single positive root

o!

is given by

The set A( +) of dominant integral functions consists of the multiples = *la ( I E Z’). Let U ndenote a representation corresponding to 1 and xn = xl its character. By Theorem 1.7 we have

A

(59)

The Haar integral on K of a central function f is given by

Proposition 2.15.

Let K = SU(2). Then

(i) I? has no infinite lacunary subset. (ii) The series

c m

XI

1=0

is the Fourier series for a central function belonging to LP(K),forp < 3. Remark. The example in (ii) shows that the “abelian” condition cannot be dropped in Corollary 2.14; part (i) explains why.

Proof. For (i) we observe that

(61)

~ ~ x . x . ( k ) 1 2dk = 1 + min(n, m),

which is unbounded as n and m run through any infinite subset of Z’. Since this number equals the number of irreducible components of the

931

529

3. Fourier Decomposition of a Representation

tensor product U, 0 U , (if v = jna, p = $my), it follows that infinite lacunary subset. For (ii) consider the function

k

has no

Then in the notation of (59) f ( t ) = (1 - cos O)-‘ so by (60), f E L p ( K ) for p < +. Since jKf(k)xl(k)dk

=

s’

1 n

(1 - cos 6)-’xl(t)sin’ 8 d8

-n

= L J( 2 + eie + e-”)xl(t)d8 2n

-n

because of the formula Xr(t)= eile + e W - 2 ) e + . . . + e-i10

=2

I

(ii) is verified.

$3. Fourier Decomposition of a Representation 1 . Generalities

-

a,,einecan be viewed The usual Fourier series decomposition f as a “decomposition” of the regular representation of the circle group T into irreducible representations. We shall now generalize this to an arbitrary representation n of a compact, connected Lie group K on a complete, locally convex topological vector space V. Let Y denote the set of all continuous seminorms on V. Let { u , } , ~ ~ be an indexed family of members of 9. The series aeA

is said to be absolutely convergent if

for each v E Y.For each finite subset F c A put sF = CUE0,. The series (1) is said to be convergent if for each neighborhood V, of 0 in V there exists a finite subset F , c A such that sF, - s F 2 € V, for any two finite subsets F,, F , of A containing F , . By completeness of V the sums

530

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

sF will then have a limit s E V ; s is called the sum of the series (1) and we write s = LEA v,. It is clear that absolute convergence implies convergence. As in Chapter IV, $1 let R denote the set of equivalence classes of finite-dimensional unitary irreducible representations of K. For 6 E k? let ad = d(6) conj(x,) where d(6) and xa denote, respectively, the dimension and character of 6. Then we have seen (Chapter IV, Lemma 1.7) that n(ad) is a continuous projection of V onto the closed subspace V ( 6 ) of K-finite vectors of type 6. For the cited case of the circle group T if 6 is the representation eie + eineand n the regular representation then n(ad) is the map

where 1 a,, = 2n ~02'f(eie)ei"e do.

We recall (Chapter IV, $1) that a vector v E V is said to be differentiable if the mapping z,: k + n(k)u of K into V is differentiable. We shall now prove a sharpened version of Lemma 1.9, Chapter IV. Theorem 3.1. Let v E V be a differentiable vector. Then the "Fourier series "

p w v

dEK

converges absolutely to v. Proof. The Lie algebra € of K is the direct sum € = 3 + [€,€1where is the center of € and [€, €1is semisimple. Let Q be a positive definite quadratic form on € which on [€,€1 coincides with the negative of the Killing form of [€,€3 and under which 3 and [€,€3 are orthogonal. Let XI,. . . , X, be a basis of €,orthonormal with respect to Q, and put j

R

= 1-

2: - ... - 2; E D(K).

Since Q is Ad(K)-invariant, R is a bi-invariant differential operator on K . It is clear from Lemma 1.6 (which does not require semisimplicity of K ) and Eq.(16) of $1 that (3)

Rad = cdad,

where cd 2 1. On the other hand [Chapter IV, $1, Eq. ( S ) ] , we have for U€K

n(a,)n(u)u = n(u)n(ad)u= n(a$(u))u= n(a;("))u

931

3. Fourier Decomposition of a Representation

53 1

so

(4)

n(a,)n(R)u= n(Rad)u = c,n(a,)u.

Lemma 3.2. Let v E Y . Then there exists a vo E Y such that

v(n(a,)u) 5 c;"'

d(6)' vo(n(R"')u)

for all 6 E l? and m E Z + . Proof. By the compactness of K we can choose V , E Ysuch that v(n(u)w)5 vo(w) for all w E V , u E K . If .f : K -+ V is a continuous map we have

so, since sup la, I I d(S)2, we have (5)

v(n(a,)w) Id(6)' vo(w).

On the other hand, iterating (4)we have n(a& = c;"'n(ad)n(R"')u,

m EZ+.

Using ( 5 ) the lemma follows. In order to use the lemma we observe that if m is large enough

If K is semisimple this is clear from Eq. (16) of $1 and Theorem 1.8. If K is not semisimple let Z and K' denote the analytic subgroups of K corresponding to 3 and [€,€1, respectively. If 6 E R then the restriction 6' = 6 I K' belongs to (KO-, the elements 6(z) (z E 2)being scalars. Thus (7)

d(6) = d(6'),

~,(zu') = Tr(d(z)d(u')) = 6(z)~,,(u'), (8) for Z E Z ,u ' E K ' . At the identity element we have LK = Lz + LK, for the respective Laplacians, and this combined with (7) and (8) shows that c, r: c6'. This proves (6) in general. Now Lemma 3.2 shows that the series in Theorem 3.1 converges absolutely. Let uo denote its sum and put w = u - uo. If 6, E R is arbitrary,

n(ado)vo= xan(ad,)n(ad)o= n(a6& dell

by the orthogonality relations. Hence n(aa)w = 0 for each 6 E R. Then if u E U , n(a,)n(u)w = n(u)n(a,)w = 0.

532

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

If I is arbitrary in the dual space V' of V this equation implies (9)

X* * A,

6 E R,

= 0,

where * denotes convolution on K and I , is the function u + I(K(u)w) (cf. proof of Lemma 1.9, Chapter IV). But (9) implies that I , = 0, so, since I is arbitrary, w = 0. This proves Theorem 3.1. Let X be a manifold with a countable base and suppose K is a compact, connected Lie transformation group of X. The spaces 9(X) = C:(X) and b ( X ) = Ca(X) with their usual topologies (Chapter 11, $2) are locally convex and complete. The mapping K which to k E K assigns the transformation f ( x ) + f ( k - x) of b(X) is a representation of K on &'(X). Lemma3.3. Let f E&(X). Then f is a diflerentiable vector for the representation n.

Let k E K , T E f. Then the lemma amounts to the existence of the limit lim -1 (fr(kexptT) - fr(k)) 1-0 t in the topology of b(X). This, however, is clear from the definition. Remark. The analogous result holds also for the natural representation of K on 9(X). Defining as in Chapter IV, $1

(10)

(4 * . f ) ( x ) =

J

4 ( W ( k - ' x ) dk

K

for 4 E C(K), f E C(X), we have ( n ( + ) f ) ( x ) = 4 * f , so Theorem 3.1 implies the following result. Corollary 3.4. Let f E b(X). Then

the series being absolutely convergent. 2. Applications to Compact Homogeneous Spaces

Let K be a compact, connected Lie group, M c K a closed subgroup, and X the manifold KIM. We can apply the results above to the natural representation K of K on the spaces C(K/M), 9 ( K / M ) , L 2 ( K / M ) , etc., and combine these with the formalism of Chapter IV, $1.

031

3. Fourier Decomposition of a Representation

533

For each 6 E R let V, be a vector space (with inner product ( , )) on which a representation of class 6 is realized and let this representation also be denoted by 6. Let R , denote the set of elements ~ E for which the subspace V," = {uEV,:d(m)u = u for m E M }

is nonzero. If ~ E Ithen ? ~ by Eq. (12) of Chapter IV, 01, the contragredient representation 6 also belongs to R,. Let l(6) = dim V,", d(6) = dim V,, and let ul,. . .,ud(8) be a basis of V, such that u l , . . . ,ut(8) span V,". If C E Hom( V,, V,") then 6(m)C = C (m E M ) so we can define the function F , on KIM by F,(kM) = Tr(G(k)C),

(11)

k E K.

For 6 ~ let k C , ( K / M ) denote the subspace of C ( K / M ) consisting of the K-finite vectors (under n) of type 6. Theorem 3.5. Let 6 E R. Then

(i)

Cd(K/M) # 0 if and only if 8 E R,. In this case,

C $ ( K / M )= {Fc : C E Hom(V,, V,")} and the functions kM -, (6(k)uj, ui) = 6,,
1 Ii Id(6), 1 I j I1(6),

constitute a basis of C i ( K / M ) . (ii) W e have the orthogonal Hilbert space decomposition L 2 ( K / M ) = 0 C,(K/M),

(12)

,€kM

(13)

f=

C 461% * f,

f E L2(K/M),

~ G K M

* being as in (10). (iii) If ,f E &(KIM), the series (13) converges absolutely and uniformly. (iv) Let L : ( K ) denote the set off E L 2 ( K ) that are bi-invariant under M. Let C:(K) = C Cdij, ~ E R M . 1s i ,j s W )

Then we have the orthogonal direct sum

R

534

[Ch. V

ANALYSIS ON COMPACT SYMMETRIC SPACES

The convolution algebra L:(K) is commutative i f and only

l(6) = 1

if

for each 6 E R,.

Proof. If the space C;(K/M) is # {0}, it contains a function F such that F ( { M } ) # 0. By averaging over M we may take F M-invariant so 6 E Z?,. The rest of (i) and (ii) is already proved in Chapter IV, $1. Part (iii) follows from Theorem 3.1 when used on the Banach space V = C ( K / M ) with the uniform norm. It remains to prove (iv). The space C ; ( K / M ) is by (i) spanned by the first l(6) columns of the matrix 6(k). Thus (13’) follows from (12) and the analogous decomposition of L 2 ( K / M ) . The last statement now follows from Chapter IV, Theorem 1.6 (iii), which shows that C:(K) is a convolution algebra with center C(6, + . - . + 6,(6)l(6)).

Corollary 3.6. Let f E b ( K ) be invariant under Conjugation, i.e., f ( k 1 k - l ) = f(l). Then

and the series converges absolutely and uniformly.

In fact, take M = e in (13) above and observe that xa * f is a function in C s ( K ) that is invariant under conjugation. Writing xa * f in the form (x6 * , f ) ( k ) = Tr(G(k)C), the condition Tr(G(k1)C) = Tr(G(1k)C) implies Tr(d(k)6(I)C) = Tr(6(k)C6(1)), which again implies 6(l)C = Cd(1) so C is a scalar operator. Thus xa * f = cxa (cEC) and now (13) implies (14).

$4. The Case of a Compact Symmetric Space

We shall now put Theorem 3.5 into a more explicit form in the case of a symmetric space of the compact type (cf. [DS], Chapter V, 51). First we need some results about representations. 1. Finite-Dimensional Spherical Representations

Let G be a connected noncompact semisimple Lie group, G = K A N an Iwasawa decomposition, and M the centralizer of A in K . Let g, €, a, n, and rn denote the corresponding Lie algebras. As in [DS] (Chapter VI), we extend a to a maximal abelian subalgebra l~ of g. Then t, = IJ n € + a, t, is a Cartan subalgebra of g and, superscript C denoting complexification, lf is a Cartan subalgebra of gc.

§41

4. The Case of a Compact Symmetric Space

535

Each representation 6 of G on a finite-dimensional vector space V over C induces a representation, also denoted 6, of g (and gc) on V. If G is simply connected this process can be reversed. In any case we can use the notion of a weight from $1 for representations of G. The weights (like the roots of gc with respect to 6') are real valued on the real subspace bR = i(t) n €) a. We put bt = b n €. The choice of n corresponds to an ordering of the dual space a*. Let C+ denote the corresponding set of positive restricted roots. Choosing a compatible ordering of the dual of fjR we have an ordering of the set of all weights. The Killing form ( , ) of gc induces an inner product on a, bR, and their duals. We can now characterize the spherical representations.

+

Theorem 4.1. Let 6 be an irreducible representation of G on a finitedimensional vector space V over C .

(i) 6 ( K ) has a nonzero fixed vector if and only if 6 ( M ) leaves the highest-weight vector of 6 fixed. (ii) Let A be a linear form on bR. Then 1 is the highest weight of an irreducible jinire-dimensional spherical representation of G if and only if

Let GC be the simply connected Lie group with Lie algebra a, and n and let U be the (compact simply connected) analytic subgroup corresponding to u = f + ip. Here p denotes as usual the orthogonal complement of € in g. We can then identify finite-dimensional complex representations of g, gc, u, G, GC,and U . For the proof of Theorem 4.1 we first assume G = G. Suppose first 6 ( K ) has a fixed vector v, # 0. Let A denote the highest weight of 6 and vA # 0 a highest-weight vector. Since G = K A N , V is spanned by the vectors 6(K)v,. Put P = j K 6(k) dk. Then Pd(k)v, = Pun, so PI/ = C P v , . Since v, = Pv, it follows that Proof.

gc and

(2)

G,R, x,fl the analytic subgroups corresponding to g, €,

v, = cPv,

for some CEC.

Suppose now H E fj n f. Then

I

v, = 6(exp H ) v , = cPe"")v, = el(%,,

so el(") = 1 for all H E t, n f, whence (3)

A(i(lj n f)) = 0.

536

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

Next we recall ([DS],Chapter VII, Theorem 8.5) that since U is now simply connected the set aK = {HEia:expHEKJ

(4)

consists of the integral linear combinations of the vectors (5)

where A, E a is determined by (Z-Z, A , ) = a ( H ) for H E a. Thus by (2) uK = 6(exp e,)uK = cP6(exp e,)u, = exp(2ni(A, a)/(a, a))uK

so (1) is proved. Now we recall ([DS], Chapter VI, Lemma 3.6 and Chapter IX, Exercise A3) that M has the expression

M = Mo(exp(ia) n K ) , (6) where M o is the identity component of M and

mc = (6 n QC

+

c

(!3c)B + (gc1-P.

PEP-

Here P - is the set of positive roots /? of gc with respect to 6' which vanish identically on a. Since A + /? is not a weight of 6 we have S((gC)@)u,

= 0.

Also, S((gc)-p)u, = 0; in fact, since (A, /?)= 0 [by (3)], the linear form 1 - /? is the image of A + /? under the Weyl reflection sP and thus is not a weight of 6. Now (3) and the formula for mc show that u, is fixed under 6(Mo), and (1) and (4) show it fixed under G(exp(a,)). Hence by (6), 6 ( M ) leaves ua fixed. This proves half of (i) and (ii) for the case G =

G.

On the other hand, suppose 6 ( M ) leaves u, fixed. Then condition ( I ) follows readily from (6). The vector r

is of course 6(K)-fixed. To see that uo # 0 we use Theorem 5.20 of Chapter I, by which

,.

HI

4. The Case of a Compact Symmetric Space

537

Consider now an inner product ( , ) on V for which S(U) are unitary operators. Then, * denoting adjointness, we have 6(T)* = -6(T)

( T E t),

6(X)* = 6 ( X ) (XE p),

so 6(Z)* = -6(OZ) for Z E ~ whence , 6(g)* = 6(8g-') for g EG. Thus (6(ii)uA,ui) = (uA,S(Oi-')v,) = (ui, uA) # 0, so, taking the inner product above, we get (7)

In particular, uo # 0 as claimed. It remains to prove the "if" part of (ii) and then to remove the assumption G = G. Let /? be a positive root of (g', If) and the restriction Pla. Then by [DS] (Chapter VII, Lemma 8.4),

(B, B>

= m(B,

P>,

m = 1,2,4.

Moreover, if rn = 4 then 2p E Ct Assuming (1) and writing

we see that the left-hand side belongs to Z'. Thus by Theorem 1.5 there exists an irreducible representation 6 of GC with highest weight 1. As we saw earlier in the proof, condition (1) implies that the highest-weight vector ui is fixed under S(M). This proves the theorem for the case G =

G. Consider now the case of a general G. The representation 6 induces, via g and g', a representation 8 of G with the same highest weight 1. By Schur's lemma the center 2 of G is mapped into scalars by 8. This center is contained in I@, the centralizer of A" in I?, so if &I?) has a fixed vector or if b(@) fixes uA, 8 equals the identity on the center. Similar remarks apply for 6 relative to the center Z of G. By connectivity of K and I? we have Ad,(K) = Adg(I?), whence KIZ = I?/Z. Also, if k E K such that Ad(k) Ad@) = Ad(a) Ad(k) (a E A) we have kak- = az, (z, E 2). Since z, E K the uniqueness in the Cartan decomposition G = K exp p implies z, = e, so k E M. This shows that Ad,(M) = Adz(@), so M I 2 = 1$?/2. Using the above remarks about the centers Z and 2 we conclude for UE

v,

u is G(K)-fixed o u is 8(R)-fixed; ua is G(M)-fixedo ui is 8(A)-fixed.

538

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

This proves the theorem for G. If the vectors uK and uA are taken as unit vectors we have from (2) since P* = P = P2 1 = ( V K , OK) = (CPUA, CPUA) = c(uA, OK),

and then (7) takes the form (with uo = PuJ vA)12 = c ( - U

(8)

+ P)).

The proof of Theorem 4.1 has the following consequence. Corollary 4.2. Let U be a compact simply connected semisimple Lie group, K the fixed point group of an inuolutiue automorphism of U . Then condition (1) characterizes the highest weights of the irreducible spherical representations of U .

2. The Eigenfunctions and the Eigenspace Representations

With U and K as in Corollary 4.2 we shall now write out Theorem 3.5 in a more explicit form. Let OK denote the set of equivalence classes of spherical representations of U . For each 6 E 0, let V, be a representation space for 6. We know from Chapter IV, Lemma 3.6, that now V f is spanned by a single unit vector e. Theorem 4.3. For the symmetric space U / K we haue the Hilbert space decomposition (9)

and i f f E ~ ( U / K )this expansion converges absolutely and uniformly. Moreouer, (i) Cd(U/K) consists of the functions

uK

+ (u,

6(u)e),

U E V,.

The natural representation of U on the function space C,(U/K) is of class 6 ; in particular, it is irreducible. (ii) Each C 6 ( U / K )contains a unique spherical function, namely 4&K) = (e, 6(u)e). If the homomorphism cd: D ( U / K )+ C is determined by

D4a = C d ( D W d ,

D E D(U/K),

91

4. The Case of a Compact Symmetric Space

539

then we have for the joint eigenspace

(1 1) { f E Cm(U/K):Df = c,(D) f

for all D

E

D ( U / K ) } = C,(U/K).

(iii) The class OK coincides with the class of eigenspace representations for U / K . In particular, each eigenspace representation is irreducible and finite-dimensional. Proof. Part (i) is clear from Theorem 3.5 because if f ( u K ) = (u, 6(u)e) then f ( u ; ' u K ) = (6(u,)u, 6(u)e). For Part (ii) we first note from Chapter IV, Theorem 3.4 that the function 4,(uK) = (e, 6(u)e) is

a spherical function in C,(U/K). Because of Lemma 3.6 in Chapter IV, the irreducibility in (i) implies the uniqueness of 46as well as the fact that its translates span C , ( U / K ) . This again implies the inclusion =I in (1 1). For the converse inclusion as well as part (iii) let f be a joint eigenfunction of D(U/K), i.e., Df = v ( D ) f for all D E D ( U / K ) , v being a homomorphism of D ( U / K ) into C. Applying D to (10) we conclude that

However, by Corollary 2.3 in Chapter IV, all the c6 are different, so (12) contains just one term, whence f E C,(U/K). This concludes the proof. While Theorem 4.3 gives for U / K an answer to Problems A, B, and C in Introduction, $1, we can still look for a more specific description of the joint eigenfunctions than that provided by (i) above. We shall now obtain such a description by going over to the complexification Uc = GC.We adopt the notation of $4, No. 1 and choose for G the group G. As in Chapter IV, $4 we let, for g E G, the vector A(g) E a be determined by g E N exp A(g)K. We consider uc = gc with the usual Hilbert space inner product (X,Y ) = -(X,r Y ) , r being the conjugation of uc with respect to u. Because of the vector space direct sum of the complexifications

uc = nc + ac + rC (13) the map ( X , H , T ) + exp X exp H exp T is a holomorphic diffeomorphism of a neighborhood of (0, 0,O)onto a neighborhood UE of e in Uc. The map (14)

exp X exp H exp T

+H

is therefore a well-defined holomorphic mapping of Ug into ac extending the map A . We denote this extension also by A . We can take UE as the diffeomorphic image (under exp) of an open B(0) c uc with center 0.

540

ANALYSIS ON COMPACT SYMMETRIC SPACES

Then Ug is invariant under the conjugations u k E K and so is the set U o = Ug n U .

[Ch. V

kuk-' by elements

Theorem 4.4. Each joint eigenfunction of all D E D ( U / K ) has the

form (15)

f ( u K ) = jK,~e-,("'*lYk"F(kM)dkw,

u E Uo,

where p E a* and F satisfy

Conversely, if p and F satisfy (16) then the function f defined b y (1 5 ) extends uniquely to an analytic function on UIK and this function is a joint eigenfunction of all D E D(U / K ) . Proof. Let f be a joint eigenfunction. By Theorem 4.3 there exists a 6 E OK,u E V,, such that f ( u K ) = ( u , 6(u)e). The corresponding spherical function &uK) = (e, 6(u)e) can be extended to the function (17)

&u) = (e, d(u)e>,

uE

uc,

because 6 automatically extends to a representation of Uc. Let p denote the restriction to a of the highest weight of 6 and let u, be a corresponding highest-weight vector. Then by (2) since P is self-adjoint, (18)

(e, 6(u)e> = I c 12
Let k E K , g E G. Then gk Hence

= k'

exp( - A(k- lg- ' ) ) n where k' E K , n E N .

PG(gk)o, = exp( -p(A(k- Ig- ')))PG(k')u, = exp(-p(A(k-'g-')))Pu,,

so (19)

PG(g)Po, = / ; - p ( A ( k a - l ) )

dk Pu,.

But P 2 = P so Ic12(u,, Pu,) = Ic12(Pu,, Pu,) = 1, whence by (17)-(19) (20)

&g) =

je - p ( A ( k e - ldk,) ) K

g E G.

WI

4. The Case of a Compact Symmetric Space

54 1

We write p = il - p (AE a;, p half the sum of the positive restricted roots) and recall from Chapter IV, $4, that 4A(g) = 4-A(g-') and q5sA = for all s in the Weyl group W . Thus by (20) (21)

&g) = 4A(g) =

dk,

gEG,

j;(s*fl)(A(kg))

where s* is the Weyl group element mapping C+ to -X+. Using now the extension A : U: + a we can extend by the formula

4A(u)= J K e W + p l ( r l l k -

W )dk,

u E U:.

Then the integral formula (Chapter IV, Lemma 4.4) n

holds by holomorphic continuation for g, h, g - ' h E U:. The vector u above is a linear combination

(23)

u =

1ai6(ui)e,

a, E C,ui E U ,

i

and here we may assume the elements u i contained in an arbitrary neighborhood U , of e in U . In fact, if a linear form /? on V vanishes on 6(U,)e then the analytic function u + /?(G(u)e) vanishes on U,, hence on U . Since 6(U)e spans V we conclude /? = 0. Now by (23) ~ ( u K=) C ai+(u; 'uK). i

If u is in a sufficiently small neighborhood of e we can write 4(u,: 'uK)= can substitute from (22). This gives formula (15) with p replaced by -(is*A + p) = -s*p. Since -s*p satisfies the integrality condition (16) formula (15) is proved. On the other hand, let ,u E a* satisfy (16) and let by Theorem 4.1 6 be the irreducible finite-dimensional spherical representation of G on a vector space V with highest weight having restriction to a given by p. Let V* be the dual of V, uo # 0 a vector fixed under 6 ( K ) , and u* a vector in V* such that u*(uo) = 1. Let ug E I/* be defined by

4s*A(ut:1u) and

u E V,

u$(u) = /;*(6(k)u) dk,

and the function 9 on G/K by 4(gK)

= uo*(&7-

%o).

542

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

Comparing with Eq. (12) of Chapter IV, $4 and the ensuing discussion we see that 4 is a spherical function on G / K and that 6 is equivalent to the natural representation 6, of G on the vector space V, spanned by the translates of 4. Let $ be a highest-weight vector for 6,. Then, writing o = { K } , we have $(na . o) = (6,(a- 'n- ')$)(o) = e-"'"g") $@I, so, taking $(o) = 1, we have $ ( g K ) = e-fi('A(f7)).

But 6, extends to Uc, so $, being a representation coefficient, extends to a holomorphic function on Uc. By analytic continuation this extension $ satisfies (24)

$(uK) = e-fl(A(")), u E U o .

Being a joint eigenfunction of D(G/K), $ satisfies J$(gk

- x ) dk = $(g . o)

JK

$ ( k x . o) dk, g , x E G

(cf. Chapter IV, Proposition 2.4). By holomorphic continuation, $ satisfies this functional equation on U / K ; hence it is a joint eigenfunction of D(U/K). But then (24) gives the desired conclusion about the integral (15). This concludes the proof. Note that the formula

is valid on all of U / K .

3. The Rank-One Case

Suppose now the symmetric space U / K above has rank one. We shall then give more explicit formulas for the spherical function 4 = 46 corresponding to the spherical representation 6. As before, let p denote the restriction to a of the highest weight of 6 and put p = il - p. By (16) we have

(i) p = na, (ii) p = 2na,

n E Z + ifX+ = { a } ; n E Z+ if C+ = {a, 2a).

Exercises and Further Results

543

According to Exercise B8, Chapter IV, the spherical function q5A in (21) is given by the hypergeometric function

+ n), -+n, &ma + l ) , -sh2t) = F ( h , + rn2. + n, - n, $(ma + m2u + l), - sh2t)

4,(h) = F(&rn,

4A@)

in the two respective cases; here t = a(log h). Using the Gauss-Kummer transformation formula (Erdelyi et al., Vol. I , p. 65) F(a, b ; a

+ b + i;4z(1 - z)) = F(2a, 2b; a + b + 3;z),

the first formula above can be written

Letting now /3 denote the larger element in Z ' both case (i) (ii) can be combined in the formula

(25) 4A(expH ) = F(imsl,

+ mS + n, -n,

&ms12

+ ms + l), -sh2 ("'2"))).

which by the expansion for F reduces to a polynomial. Both sides are holomorphic on all of a' so in particular we can take H ~ i aand have then proved the following result. Theorem 4.5. Let UIK be a simply connected compact symmetric space of rank one and p the larger element in X+.Let 6 be a spherical representation of U and let p denote the restriction to a of its highest weight. Then p = n/3 where n E Z+ and the spherical function q& is given by the (Jacobi) polynomial

Each n E Z+ appears for a suitable 6.

EXERCISES AND FURTHER RESULTS A. Representations

1. If g is a simple Lie algebra over C,show, using Eq. (16) of 01 that

(6 + P I 6 + P > - ( P , P ) = 1, where 6 is the highest root, and p half the sum of the positive roots.

544

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

2. (i) For s E W = W(g,t) let n(s) denote the number of positive roots which s maps into negative roots. Then

det s = (- l)n@). (ii) Show that the orbit {sp:s€ W }

equals the sphere { A E A : ( ~=I I p I } . 3. Let R be as in Theorem 1.7, A(n) the set of weights p of n, m, the multiplicity of p. Let 1 denote the highest weight of n. Writing Eq. (19) of $1, in the form

fl sinh(f(a, 1+ p>x) = 1 m p e ( p i P )nx sinh(%a, p>x), aeA+

PEA\(=)

asA+

deduce

1 m,(p9 P Y = &dim

n)((A +

P 9

A + P>

-

( P , P>).

P E A(=)

(cf. Freudenthal and de Vries [1969], $47). 4. Specializing A3 to the adjoint representation and using A1 deduce dim, g = 24(p, P> for each simple Lie algebra g over C. 5. Let n and M, ( p E A(K))be as in A3 and for v E A (cf. Theorem 1.1) let 9 ( v ) be the number of unordered partitions of v into a sum of positive roots. Because of the analogy between xn and $k (for G complex) as expressed in Theorem 5.7, Chapter IV and Theorem 1.7, Chapter V, the multiplicity m, corresponds to the coefficient T,. Thus prove the following analogs of (25) and (12) in Chapter IV, $5, (9

m, =

1w (det s ) m ( A + P ) - (P + P)),

SE

(ii) ((1

+ P, A + P> - ( P + P , p + p>)m, = 2k > l .Ca c A t

mp+ka(a,p

+ ka)

(cf. Kostant [1959b] and Freudenthal [19541, respectively). Deduce from (i) the recursion formula 9(p) = -

1

(det W ( p - ( P

-

v)),

P E A.

saW,s#e

6. Let U be as in Theorem 1.7 and let A( +) denote the set of all 1E A which are dominant integral. For each A E A(+) let nk denote the corresponding irreducible representation of U and xA its character. Let 1, P

E

A(+).

545

Exercises and Further Results (i) Show that XlX, =

1

m a , P)XW

YEA(+)

where my@, p ) =

1w det(ss')P(s(A + p ) + s'(p + p ) - v - 2p)

s. s' E

with 9 having the same meaning as in A5 (cf. Steinberg [1961]; for an earlier formula see Brauer [1937]). In view of Lemma 2.10 this result gives the direct decomposition of nl @ n, into irreducibles. (ii) Show that

%+,(A

p) = 1 ;

in other words, n d + , appears exactly once in the decomposition of nl 0 n, into irreducibles. This representation nl+, is sometimes called the Cartan product of nl and n,. (iii) Let V = V, be the weight space decomposition of V. Let p E A(nl) and a E A + such that p + a E A(nl). Using Lemma 1.2 in the Appendix show that

x,,EA(zA)

nl(X&,

f 0.

7 . Let U = SU(2). For the torus

T = {exp(ie0 - i dO ) :0 E R}, the single positive root a is given by

and A(+) consists of the multiples 1= +la (I E 2'). Show that sin(/ + l)6 Let p then (ii)

E

A( +), p

mv(A

= ima.

dim nl = I

+ 1.

Deduce from Exercise A6 that if v = $no! (n E 2')

= 1,

0,

n=I else,

+ m , l + rn - 2, . . . , I

l - ml

546

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

(Clebsch-Gordan formula). Another proof comes by writing

(iii) Let V, denote the set of linear combinations Cp+q=lap,qzPwq. Show that the natural representation of U on V, has character 2,. 8. Deduce from Lemma 2.6 that if J denotes the canonical representation u -, u of U(n) (for n > 1) then J @I J decomposes into two irreducible components (the spaces of second-order skew-symmetric and symmetric tensors, respectively). 9. For the simply connected group SU(n + 1) let J denote the canonical representation u + u of SU(n + 1) on the vector space V = Cn'l. The action of SU(n + 1) on V induces for each r a representation of SU(n 1) on R V , given by

KJ

+

(RJ)(U)(Ul A

* * *

A 0,) = U

*

U1 A

a

*

A U . U,,

(cf. [DS], Chapter I, $2). Show that the representations J (1 S r S n ) are irreducible and constitute the fundamental representations of SU(n + 1). Geometrically, amounts to the action of SU(n + 1) on the family of the r-dimensional subspaces of V.

KJ

10. Let n be a finite-dimensional representation of u and 5 its contragredient. Show that

(i) A(5) = -A(.). (ii) If n has highest weight Iz then ii has highest weight -sl, s being the Weyl group element which interchanges t+ and -t+. In particular, if w is simple and fa, (1 > l), b2k+ e6 then every n is self-contragredient ([DS], Chapter X, Exercise B6). (iii) Suppose w is simple and n irreducible. The diagram of n is obtained from the Dynkin diagram of g = wc by attaching to each simple root a in the diagram the integer 2(1, a)/(a, a). In the exceptional cases under (ii) n is self-contragredient if and only if its diagram is invariant under the automorphism --s of the Dynkin diagram. (iv) If n is fundamental so is 5. 11. Let g = sI(2, R) and consider the Cartan decomposition g =

f

+ (a -I-q) where f, a, and q, respectively, are spanned by the vectors XI

=

(O -1

I),

0

xz

=( ), 1 0 0 -1

x3

=c ;)%

547

Exercises and Further Results The Killing form of g is given by B(X, X) = 8( -x:

+ x: + xi)

3

xiXi

if X = 1

and we put a+ = (R+ - O)X,, n = R(X, + X3). The space ac = CX, is a Cartan subalgebra of gc and the highest weights of the finite-dimensional representations of gc are I = j n a where n E Z+ and A+($, aC) = { a } . The spherical representations of G are (by Theorem 4.1) given by 1 = na (n E 2'). In particular, the adjoint representation AdG of G = SL(2, R ) is sphershow I that ical. With K , A, N , and M as in & (i) The mapping gK + AdG(g)X1 is a bijection of G / K onto the quadric

> 0)Ad,(g)(gX, + X , ) ) is a bijection of

B(X, X) = WXi, X i )

(ii) The mapping g M N G / M N onto the null cone

(*I

--.)

B(2,Z)

=0

(XI

(xi

> 0).

(iii) The horocycles in G / K are given by the plane sections B(X, 2) = Z being a fixed element of the null cone (*).

- 1,

B. Fourier Series

1. Let K be a compact group. To each ~ E L ' ( K )we associate the Fourier series

as in $2. No. 1. A hyperfunction on I? is a mapping r which assigns to each I E k a linear transformation r, of XL(the representation space of UJ.A hyperfunction r is said to be unitary if each T r is unitary. Let 6 and B be function spaces on K . A hyperfunction r on I? is called an (6,9 ) m u l t i p l i e r if for each f~ d with Fourier series (1) the series

represents a function f r E .E Show that (i) If r is an ( L ' ( K ) , L'(K))-multiplier [or a (C(K),C(K))-multiplier] then fr = f * pr

where p,- is a measure on K .

for all ,f,

548

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

(ii) If r is a (C(K), A(K))-multiplier then for a certain function F E L2(K) f r =f * F for j c C ( K ) . 2. Suppose f € L ' ( K ) such that for each unitary hyperfunction r the series (2) represents a function 1;.E L'(K). Then f~ L2(K). (Use Theorem 2.3.) Compare with Littlewood's theorem (52, No. 2). 3*. (Failure of Theorem 2.3 for noncompact groups.) Let G be a nonpact, connected, separable, unimodular locally compact group. Then every spectrally continuous operator T on L'(G) commuting with the right translations on G is 0, (cf. Helgason [1957]; for a further reduction of the assumptions on G see Sakai [1964]). 4. Let U be a compact, simply connected (semisimple) Lie group. Identifying A(+) with 0 (Theorem 1.5), we call a hyperfunction on A( +) rapidly decreasing if for each k E Z+,

lim IllkTr(rArX)l/Z= 0. I1l-m

Then the mapping j + {A1}ne,,(+)defined by (1) is a bijection of b ( U ) onto the space of rapidly decreasing hyperfunctions on 0 (cf. Sugiura [1971] even for U just assumed to be compact and connected Lie group). 5. Let K be an arbitrary compact group. A subset S c 0 is said to be distinguished if for each continuous function f(k) the subseries

-

c

p

1

W A A U,(k))

AEK

d1

Tr(A1 UA(k))

1ES

also represents a continuous function f s . Show that (i) If S is distinguished then

.fs = .f * Ps where ps is a central measure on K which is idempotent in the sense that ps * ps = ps. Conversely, each central idempotent measure arises in this way from a distinguished set S. (ii) The distinguished sets which preserve positivity in the sense that .fs 2 0 whenever .f' 2 0 are precisely the normal subhypergroups of R.

NOTES 81. The basic weight theory for finite-dimensional representations of simple Lie algebras over C was founded by Cartan [I9131 (later he considered real Lie algebras as well [1914]; see also Iwahori [1959]). Here appears Theorem 1.5 (on the highest weight) and

Notes

549

a description of the fundamental representations. While Theorem 1.3 is a well-known standard result I believe inequality (8) and the remark following the theorem are due to Freudenthal [1954, 19561, and I learned part (iv) from Kostant in 1965. Using integration methods, Schur [19241 developed the representation theory for the orthogonal group O(n) including the character formula (Theorem 1.7); an algebraic treatment of SO(n) is due to Brauer [1926]. Weyl in [1925, 19261 adapted Schur’s method to arbitrary compact semisimple Lie groups and proved the character formula (Theorem 1.7) and the dimension formula (Theorem 1.8). In Weyl’s derivation of Theorems 1.5 and 1.8 from Theorem 1.7 we have followed Cartier’s exposition in Sminaire S. Lie [1955]. In [1954], Freudenthal gave an algebraic proof of the character formula; this proof can be found in Jacobson [1962]. Cartier’s paper quoted gives it also along with Weyl’s original proof. The proof in the text is an analytic modification of Freudenthal’s proof; it goes back to Sugiura [1960], although I was unaware of this fact in [1972a]. The isomorphism Z(g) % I ( t ) from Theorem 1.9 was first obtained by Harish-Chandra [1951a] (Part 111). who gave an algebraic definition and proof. The analytic proof in the text is from Helgason [1972a] (Chapter I, $2). Theorem 1.10, which is a corollary of Harish-Chandra’s integral formula (Theorem 5.35 in Chapter 11). represents the character in terms of the Fourier transform of the characteristic function of an orbit. Kirillov [I9621 proved a formula of this type for nilpotent groups and in [1968, 19693 he posed the problem of generalization to arbitrary Lie groups. For semisimple groups positive results on this question have been proved by Gutkin [1970], Duflo [1970], Rossmann [1978b, 19801, and Vergne [1979]; for solvable groups by Duflo in Bernat er a/. [1972]; and for general groups by Khalgui [1982]. 52. Schur’s orthogonality relations (Schur [ 111) and the Peter-Weyl theorem (Peter and Weyl [1927]) are proved in Weil [1940] and in many more recent books, including Sugiura [1975], Lang [1975], Schempp and Dreseler [1980], and Wallach [1973]. The results in this section (in particular Lemma 2.6) which deal with spectrally continuous operators, absolute convergence, lacunary Fourier series on groups, and the hypergroup structure on f? are from the author’s papers [1957a, 19581 in which the group-theoretic lacunary condition was introduced. Further work on lacunary series was done by Hewitt and Zuckerman, Figa-Talamanca, Rider, Edwards, Ross, and others. See Hewitt and Ross [I9701 for an extensive treatment. However, as Prop. 2.15(i) indicates (Helgason [1958]) and as is shown more generally by Ceccini [I9721 and others, the notion, and certain natural variations of it, is primarily of interest for abelian compact groups and for infinite-dimensional compact groups. The example in Proposition 2.15(ii) occurs in Coifman and Weiss [1971, Chapter IV]; it was pointed out to me by Travaglini. According to Theorem 2.3, the constant

I l f * 911I

inf sup ~, f

g

11/11211~11rr

which only depends on the group K, lies between 2-l” and 1. The exact value is not known even for K abelian; see Edwards and Ross [I9731 for work in this direction. An extensive exposition of Fourier series on compact groups is given in Hewitt and Ross [1970]; see also Edwards [1972], Dunk1 and Ramirez [1971], and Rudin [1962]. Further integrals of the type in Lemma 2.6 (Helgason [1957]) can be found in Hewitt and Ross [I9701 ($29) and in James [1961]. #3-4. Theorem 3.1 and its corollaries 3.4, 3.6, and Theorem 3.5(iii) are due to HarishChandra [1954b, 19663. Theorem 4.3 on Fourier expansions on compact symmetric spaces goes back to Cartan 119291 with some amplifications by Sugiura [1960]. The expansion in Theorcm 3.5 goes back to Weyl [1934]; see also Weil [1940] ($23).

550

ANALYSIS ON COMPACT SYMMETRIC SPACES

[Ch. V

Theorem 4.1 on the spherical representations is stated and proved in Helgason [1965b] and [1970a] (Chapter 111, 53). An informative extension was given by Schlichtkrull [1984a]. A result of the type of Corollary 4.2 is indicated in Cartan [1929] (gVI); it is stated more clearly but without proof in Sugiura [1962]; the first proof seems to be in Helgason [1970a], Chapter 111. A complexified version is in Kostant [ 1975aI (Chapter 11, #6). Formulas (7) and (8) occur in Helgason [1970a] (Chapter 111, 83) and Lasalle [IY78] (55), respectively. The eigenfunction integral representation in Theorem 4.4 was proved by the author in [1977]. The mapping (14) was used by Clerc [I9761 and Stanton [I9761 in a different context, and it was observed by Sherman [I9771 that Lemma 4.4, Chapter IV extends to the present case [as formula (22) of $41. Also, formulas (18)-(20) are due to HarishChandra [1958a] (Lemma 5). Theorem 4.5 is given in Cartan [1929], #Vll-Vlll for mg,2 = 0 or I ; it is stated in Gangolli [1967], p. 179, that a similar proof works in general. For an analytic generalization see Koornwinder [1973, 19743 and for further results on the spherical functions for compact spaces of higher rank see Vretare [1976], Hoogenboom [1983].

SOLUTIONS TO EXERCISES I NTR 0D UCTI0N A. The Spaces R"and S".

A . l . (i) The space spanned by the translates off can be decomposed into subspaces on which O(n) acts irreducibly. Each of these will contain a function & for some k, so by the irreducibility of the spaces E , these subspaces are among the E k . (ii) The group O(1, 1) is generated by the transformations

ti)

cht sht x1 -x2 (sht cht)(x2)' Consider the function f on x: - x: = - 1 defined by

(zi)

--*

"(*)

f(x,, x2) = sinh-'(x,). Then f a t ( x , , x2) = ,f(x,, x2) - t, so f is O(1, 1) finite, yet is not the restriction of a polynomial. A.2. Consider for 1 # 0 the transform F + f given by

Generalizing Lemma 2.7, we see that the map F + f is one-to-one. Using Theorem 2.7 of Chapter 11, we let S A be a sphere in R" with center 0 such that each f ~ 8 ~ ( Ris" determined ) by its restriction f l S , to S A . Because of Theorem 3.1, the space Ek = is exactly the space of functions , f C(S"~ ') which are O(n)-finite and for which the representation of O(n) on the space of translates is equivalent to 6 (in Prop. 3.2). The space Ek(SA) is similarly characterized. The maps F

+

.f ISA,

F

+f,

F E L2(Sn-'),

are, by the above, one-to-one and commute with the action of O(n). By the characterization of Ek indicated the first map sends Ek(Sn-') injectively into Ek(SA), hence surjectively. Hence each O(n)-finite f~bA(R") 55 I

552

SOLUTIONS TO EXERCISES

has the form stated. The proof of the irreducibility criterion for TAnow proceeds as for Theorem 2.6. A.3. There is a natural bijection of x k I x k - 1 onto Ek commuting with the O(n) action. Thus we see that already O(n) acts irreducibly on %klxk

- 1.

A.4. The Lie algebra sI(2, C)of SL(2, C)consists of the complex 2 x 2 matrices of trace 0 viewed as a real Lie algebra. Given X E sI(2, C), u E &‘(R2)we put

By the quasi-invariance of L, X*u is harmonic if u is harmonic. The statement to be proved is that if A : So(R2)+ &‘o(R2)is a continuous linear mapping such that A X * = X * A for all X E SW(2, C ) then A is a scalar. Taking X as

(00

0l )

Or

6 A)

we find that A commutes with a/ax and a/ay, and therefore maps the subspaces 2 and 2 (of holomorphic and antiholomorphic functions, respectively) into themselves. Taking X as

we find that A commutes with the operator xa/ax + ya/dy, which on ! A coincides with za/az. This implies easily that A is a scalar on .X and similarly on hence on &‘o(R2). A.5. (i) Let Dk(n) =dim pk and dk(n) = dim H k . Writing a polynomial in (xl,. ... x,) as a polynomial in x, with coefficients in (xl,. . .,x,- 1) we see that

so by induction on k Dk(n)

Since by Eq. (4) part (i) follows.

= Of

+ n we have n(n + l)..-(n + k - 1) k!

53 dk(n) = &(n)

=(n + )‘ k k

- 1

- D k _ z ( n ) = Dk(n - 1) + Dk- l ( n - I),

Introduction

553

Part (ii) follows from Eq. (3) of 43. For (iii) we have by the mean-value theorem for harmonic functions

i f f is any rapidly decreasing radial function. Here we can take f ( x ) = e-(1i2)1x12 and since h is a polynomial we can replace y by any complex z E C". This gives

Using Cauchy's theorem on the holomorphic function +

h(z - ~y)e-"/2"Z.z'

we can shift the integral on the left to R" + iy. Then (iii) follows. (The result is given in Bochner [1955] with a proof involving Bessel functions.) (iv) Let cj E E, and let h, denote its unique extension to an element of ffk. Note that d(h,)(Ix12-") is a harmonic function of homogeneity 2 - n - k. Writing it as I ~ l ~ - " - ~ $ ( x / l xwe l ) see using Eq. (3) of 43 that $ is an eigenfunction of L with eigenvalue -(2

-

n - k)(2 - n - k

+ n - 2),

so $ E Ek. The mapping d, + d(h&IX12-")Isn-1

=$

thus maps Ek into itself. Since it clearly commutes with the rotations, Schur's lemma and Theorem 3.l(ii) show that the mapping is a scalar multiple of the identity. This proves (iv) except for the value of the constant Ck.", which can easily be calculated by using h k + , from part (ii). A computational proof (using induction on k ) can easily be given for h of the form (a, x), a E C" isotropic. B. The Hyperbolic Plane

B.1. In fact, if g E G the point g . o can be written nu . o (n E N , a E A ) so q - ' n a E K . 8.2. Part (i) follows from [DS] (p. 548). which also shows that SU(I, 1 ) and the conjugation I? + 5 generate the group of isometries of D. Since the conjugation 2 + Z of D corresponds to the map w + l / W of the upper half planc part (ii) follows.

554

SOLUTIONS TO EXERCISES

(iii) Since c maps Euclidean circles into circles it is clear that as a set S,(i) is a Euclidean circle. It passes through the points (0, eizr) so the

statement is clear. B.3. This follows from (24) and (32) combined with the classical Paley-Wiener theorem for R. Another proof can be given by means of Abel's integral equation $(u) = J/(u

+ y2) tly,

appearing in the proof of Theorem 4.6. 8.4. (i) Clearly 4 - A ( z ) is real for all z if and only if , f ( A ) is real for all real,/'E9b(D),which by B3 is equivalent to

1m

_e-'"+(r) dr

being real for every real 4 E BP(R).This in turn is equivalent to cos(Ir) = cos(Xr) and (i) follows. (ii) Let I = 5 + ir] where (, r] E R. Assume first 1 ~ I1 1. Then if . / ' E L ' ( D )and is radial

1

I~

( Z M- i.(z)

J

I dz I I j , ( z )I4 i q ( z ) dz rm

r

roo

because we have

and FISIis even. Thus 4Ahas a finite integral against any L'-function and so is bounded. bounded. If r] = 0 there is nothing to On the other hand, suppose prove and since 4A= we can assume r] < 0. Then Theorem 4.5 applies and e(-iA+' I r 0) has a nonzero limit as r --+ + 00. But the boundedness of 4Athen implies Re( - iI + 1) 2 0 and so r] 2 - 1 as desired. B.5. The calculation is the same as that of Lemma 4.9 (cf. Helgason [1976], p. 203). B.6. This follows by approximating f by functions j ; ~ g ' ( D )and using (32).

Introduction

555

B.7. As in text expand the function . f ~ 9 ( D ) , f(z) =

C .fm(z),

me2

where ,fm(eiez)= eimefm(z). Then

fm

and

fm

have the form

fm@, eie) = eim8f,(A, I),

fm(eiB,t ) = eimejm( 1, t),

and

Also, by (58),

and since p m ( i l ) and p,(

- in)

are relatively prime this implies

3mr-4

1) = pm( - iJ)h(l),

where h ( l ) even and of exponential type. Now the result follows from the Fourier inversion fm(

'

1, t)e' = ( 2 ~ ) -

i

fm(l, l)eiA'd l .

8.8. These are the circular arcs connecting - 1 and 1 inside D. They are the curves x , : r --* x ( r , t ) in Exercise B9 and consist of points of fixed non-Euclidean distance from the geodesic A 0. B.9. We have x(r, t ) = a,. x(0, t ) = a, (tanh t i ) where a, E SU(1, 1) is as in $4. This gives the formula for x ( r , t ) . Next substitute ( = X(Y,t ) into the formula a

B.10. We have by (24)-(27)

f(tanh s) = F((chs)2) if F ( x )

= x - ( ' ' ~ ) ( '). ~ ~It+follows

that

556

SOLUTIONS TO EXERCISES

C. Fourier Analysis on the Sphere

(i) Immediate from $3. No. 1. (ii) The first relation and the second for (s, a ) > 0 are clear from the uniqueness of 4 k in Prop. 3.2. For (s, a) < 0 the second relation holds at least up to a constant factor which however is 1 because by the first relation C;( - 1) = ((iii) Denote the integral by F(s, s'). Since F( -s, -s') = F(s, s') we may assume (s', a) > 0. Next observe F(us, us') if u is a rotation of B. Let ug be a rotation of angle 4 in the x,x,+~ plane. If we can prove

F ( p g s , ugs') = F(s, s') for all 4 with (u4s', a) > 0 then (iii) is reduced to the case s' = a and then (ii) applies. Thus it suffices to prove aF(ugs, uos')/a4 = 0. Each b E B can be written

b = (cl cos 8, c2 cos 8,. . . ,c,-

cos 8, sin 8, o),

c E 9-2.

Putting g(c,

0 9 4 9

s) n- 1

=i

cos 8 C cjsj

+ i sin e(s,

cos 4

- s,+

sin 4)

+ s, sin 4 + s,+

cos 4

j= 1

we have eb,k(u&S) .fa,k(U,$s')

= b(c, 6, 4, s)Iki = b ( c , 6,

$9

s')]-k-n+l.

For each m E 2 we have the differential equation

and the volume element decomposition

db = (const)cos"-28 dB dc

(- ~ / I 2 8 I4 2 ) .

Using these facts we get after simple computation

aF(u,s, u,s')

a4 =

-i

a

~esm-2J-n,2 e,4, n12

- MC,

de

sikg(C, e, 4, s ' r k - n + l cosn- 1 el de = 0.

(The result is from Sherman [1975], the proof from J.-G. Yang [1983].)

Chapter I (iv) Let f#&)

=

557

C;((s, a)) ( k E 2’)so by Lemma 3.5 m

and the result follows using the expansion above. (With a suitable definition of f the result holds for all , f 8~( S n ) ;cf. Sherman [1975].)

CHAPTER I A. Invariant Measures

A . l . (i) If H is compact, Idet(Ad,(H))l and Idet(Ad,(H))( are compact subgroups of the multiplicative groups of the positive reals, hence identically 1. (ii) G/H has an invariant measure so ldet Ad,(h)I = (detAd,(h)I, which by unimodularity of G equals 1. (iii) Let Go = {g E G : ldet Ad&)[ = l}. Then Go is a normal subgroup of G containing H . Since p ( G / H ) < 00, Prop. 1.13 shows that the group G/Go has finite Haar measure, and hence is compact. Thus the image ldet Ad,(G)I is a compact subgroup of the group of positive reals, and hence consists of 1 alone. A.2. The element H = (-: A) spans the Lie algebra o(2) and exp Ad(g)tH = g exp tHg-’ = exp( - t H ) . A.3. We have det Ad(exp X) = det(eadX)= eTr(adX), so (i) follows. For (ii) we know that G / H has an invariant measure if and only if

exp(Tr(ad, T ) ) = exp(Tr(ad,, T)), TElj. Put T = t X i (Y < i In), t E R, and differentiate with respect to t . Then the desired relations follow.

558

SOLUTIONS TO EXERCISES

A.4. To each g E M ( n ) we associate the translation T, by the vector x = g - o and the rotation k given by g = Txk. Then kT,k-' = K.x,so

glgZ = q 1 ~ 1 T , =~ T~~ 2l + k l . ~ ~ k l k 2 * Since gkl,xl * g k z , x z = g k l k z . x l + k l . x Z this shows that the mapping is an isomorphism. Also sf@ko.

xogk, x) dk dx = jf(gkok,x o + k o . =

since dx is invariant under x

+ x,,

x)

+

gk.x

dk d x

1.f

(gk.x) dk d x

+ k, .x.

A.5. By [DS], Chapter 11, $7, the entries oij in the matrix 0 = X - ' d X constitute a basis of the Maurer-Cartan forms (the left invariant 1-forms) on GL(n, R). Writing d X = X R we obtain from [DS] (Chapter I, $2, No. 3) for the exterior products

fl dxij = (det X)n n wij,

ni,

i. j

i, j

so ldet XI-'' dxij is indeed a left invariant measure. The same result would be obtained from the right invariant matrix ( d X ) X - ' so the unimodularity follows. A.6. Let the subset G' c G be determined by the condition det X , , # 0 and define a measure d p on G' by d p = ldet X l l 1 - '

n

dxij.

( i . j ) # ( l , 1)

If dg is a bi-invariant Haar measure on G we have (since G - G' is a null set) r

r

r

where J is a function on G'. Let T be a diagonal matrix with det T = 1 and tl, . . ., t, its diagonal entries. Under the map X + T X the product dxij is multiplied by t;-'t; ...t: and ldet X 1 , I is multiplied by t 2 t 3 . . - t n .Since det T = 1, these factors are equal, so the set G' and the measure p are preserved by the map X + T X . If A is a supertriangular matrix with diagonal 1, the mapping X + AX is supertriangular with diagonal 1 if the elements x i j are ordered lexicographically. Thus ,) dxij is unchanged and a simple inspection shows det((AX), ,) = det(X, l).It follows that G' and dp are invariant under

n(i,,3+cl,l,

n(i.j,+cl.

Chapter I

559

each map X -,U X where U is a supertriangular matrix in G. By transposition, G' and dp are invariant under the map X + XV where V is a lower triangular matrix in G. The integral formulas above therefore show that J ( U X V ) = J ( X ) . Since the products U V form a dense subset of G ([DS], Chapter IX, Exercise A2) p is a constant multiple of dg. For another, more down-to-earth proof see Gelfand and Naimark [1957] (Chapter I, #4). A.7. A simple computation shows that the measures are invariant under multiplication by diagonal matrices as well as by unipotent matrices; hence they are invariant under T(n, R ) ; cf. Gelfand and Naimark [1957] (Chapter I). A.8. G leaves invariant the metric dy2) on H ([DS], Chapter X, Exercise G1) and the Riemannian measure is y - 2 d x dy, so D has finite area. Since G = Tn-'(D) where n: G -,H is the natural projection g + g .i, p(G/T) < m.

+

Remarks. (i) D is actually a fundamental domain for r in the sense that each r-orbit in H meets D and if two distinct points of D are in the same r-orbit they lie on the boundary of D. Also, T and S generate r (see, e.g., Serre [1970]). (ii) The finiteness ,u(G/T) < m for G = SL(n,R), r = SL(n, Z) is classical (Hermite, Siegel; for an elementary proof, see Bourbaki [19633, IntPgration, Chapter 7, Exercise 7, $3). For an extension to semisimple groups see Bore1 and Harish-Chandra [19621.

A.9. Part (i) is straightforward. For Part (ii) we just indicate the proof in the case of G2,4. The invariant measure is of the form

D(a, b, c, d ) da db dc dd where the density D is to be determined. Requiring invariance under rotations in the (j1, ,f2)-plane and in the (el, e,)-plane gives, respectively, (1) D(a cos 4 =

- b sin

4, a sin 4 + b cos 4, c cos 4

-d

sin 4, c sin 4

D(a, b, c, 4,

(2) D(u cos 4

-

c sin 4, b cos 4 - d sin 4, a sin C#J

= D(a, b, C,

+ c cos 4, b sin C#J + d cos 4)

d).

Invariance under rotations in the (el, f,)-plane gives (3)

+ d cos 4)

D(a', b', c', d') = (a sin 4

-

cos 4)"D(a, b, c, d )

560

SOLUTIONS TO EXERCISES

where

cos $c+ sin 4

(Q

d cos 4 From (1) we obtain with r = (a2

+ b2)”2,

D(r, 0, ~ - ‘ ( u c+ bd), r-’(ad

(4)

b + (bc - ad) sin 4

-

bc)) = D(u, b, C, d),

and from (3) we derive

D(0, 0, -(a2

(5)

+ l)-’c, d ) = (a2 + l)’D(a, 0, C, d).

Combining (4) and ( 5 ) we get (6) D(a, b, C, d ) = (r2 + l)-’D(O, 0, -(rZ

+ l)-1’2r-1(a~+ bd), r-’(ad - bc)).

Also, by (1) and (5), D(O,O,

C, d ) = D(O,O,

0, (c2

+ d2)’l2,

D(u, O,O,d ) = (a2

+ l)-’D(O,

D(u, O,O,0) = (a2

+ 1)-2D(0, O,O,

O,O, d),

whence 0).

But (1) and (2) imply D(u,0, 0,O) = D(0, a, 0,O) = D(O,O, a, 0) = D(O,O, 0, a).

The formula for D now follows easily. A.lO. (i) This comes from the usual formula for surface area in R”. (ii) The invariant measure can be written f(x1,

* * * 3

x p + q + 1)

dx1 * * * dx,+,

letting x l , ...,xp+qserve as coordinates. Since O(p,q) acting on xl,. xp+qleaves the measure invariant f is a function of + . . . + x; - x;+l - .. . - xi+,

. .,

and x , , + ~ +only, ~ hence of x , + ~ +only. ~ Now use the invariance under the group ( ~ 1 ,~2

9 .

. . ,x p + q + 1)

+

(XI

ch t + xp+q+1sh t , x2

---

This shows that f(Xp+q+l)lXp+q+ll

= const.

9

sh t + x p + q + 1 ch t).

56 1

Chapter I

A.11. (i)-(ii) follow by routine computation. For (iv) it is sufficient just to verify that given a, /l E C,ap # 0, I a l2 + I 1' = 1 there exist unique t, 8, 4 such that

o s t < 2,

o Ie < 22,

-22 I 4

22,

and such that (1) holds. For further information concerning this exercise see, e.g., Vilenkin [1968]. B. Radon Transforms

B . l . Let

4 E g H ( P n and ) put

@(A,

o)= /4(o,p)e-iPAdp. R

Then @ is holomorphic in 1. Let a E C" be isotropic and consider the expression

($s,"-w,o ) ( a ,

d o ) A=O 9

where I, k E Z + , I < k. Using 4 e g H ( P nwe ) see this expression vanish. Now apply Theorem 2.10 of the Introduction to 0. B.2. (i, ii) As in Introduction, 53, let S k m [I I rn Id(k)] be an orthonormal basis of the eigenspace Ek of L = L,.-,. Then we have the following result. Theorem. The range 9(R")- consists of the functions $ E 9 ( P " ) which when expanded,

$(o,P ) =

1

$km@)Skm(m), kLO. 1s m s d ( k )

have the property that, for each k E Z + , 1 Im Id(k), $km(P)

dk dP

= 7$ k m ( P )

where &,,E 9 ( R ) is even.

By Theorem 2.10, Q(R")- is characterized by $km(P)pldp = 0

for 1 < k,

which (by Fourier transform) is equivalent to the description above. [The evenness of 4 k m comes from the property $( -0,- p ) = $(a,p).]

562

SOLUTIONS TO EXERCISES

= 0. Then T(f)= 0 for each f E 6.3.Let T Eb'(P") such that 9(R"). With notation from B2 define qm E &(R) by Tkm(4)

=

dTb,

J4b)skm(w)

Since the expansion for $ (in B2) converges in &(P)(Introduction, Theorem 3.4 or Chapter V, Corollary 3.4),

T($) =

Tkm($km)*

k. m

Thus by B2, dk/dpk(Gm)annihilates all even functions in 9 ( R ) . On the other hand, it annihilates all odd functions because J4(-p) dGmb) = ( - 1 ~J4(p) d q m b ) .

Thus dk/dpk(&,) = 0, so, since

G,,,has compact support, Gm= 0.

8.4. Because of Eq. (46) of 82 we have

in other words, N equals the annihilator of 9(R")- in &(P"). But then (since d t = dw d p ) Theorem 2.10 implies that JV contains each space E, @ p' if k - I > 0 and even. Denoting now

(4, $)

=

J 4(t)$(t)4, P"

we shall prove for 4 ~ 9 ( P " ) (*)

4 E 9(R")- o ( 4 , Ek @I p')

It suffices to verify the implication

The condition

(4, E, @ p ' ) J+@)p'

for k - 1 > 0, even.

=0

=0

-=. For

this we expand

then implies

d p = 0,

k - 1 > 0, even.

But since 4km( - p ) = (- l)k&,,@) this integral vanishes for k - 1 odd. Hence + E ~ ( R " as ) ~ stated (cf. B2). Now suppose $ E JV and decompose

Chapter I

563

1

The decomposition $ = $k is the one given by Corollary 3.' of Chapter V, so the various $k transform under inequivalent representations of O(n).Thus the condition = 0 implies Ij/k = 0 for each k . In analogy with (**) we decompose each f E Q(R")

4

f = k1 .fk 2O [Corollary 3.4, Chapter V or Introduction, Eq. (19), 53.1 Then by (*) we have for each k and each ~ E L @ ( P " ) ~

9 E (Q(R")k)*

0

(9, E k @ p ' )

= 0,

k - I > 0, even.

For a fixed k the spaces Ek @ p' ( I < k, k - 1 even) span a finite-dimensional space E k whose annihilator (i?k)' in Q ( p n ) k is (Q(R")k)-. But t j k = 0 implies that $k lies in the double annihilator ((&)l)', which by the finite dimensionality equals &. Since (**) converges in the topology of 8(P")(Corollary 3.4, Chapter V), $ lies in the closed subspace generated by the spaces Ek @ p'. 8.5. (Sketch) It is easy to prove by approximation that each C in the image (8')has the property stated. Conversely, suppose Z E JW" has the stated property. Because of the Remark following Theorem 2.17 we define SEQ'(R") by S ( / ) = Z(Af).

Let dg be a Haar measure on the isometry group G = M(n). If g E G let S9 denote the image of S under g [i.e., Sg(,f) = .f(g x) dS(x)] and define Cg similarly. If F E L@(G)the distribution .f-

i

&wY,f)dg,

.f E %R"),

is given by a function s on R". Also, the distribution

9

+

J', F(g)C9(9) dg,

9 E Q'(P"),

is given by a compactly supported function CT on P". Since Z vanishes on the kernel .N'from B4 we have a(.N)= 0 so by Theorem 2.10 CT = h for . Remark following Theorem 2.17 now shows some h ~ 9 ( R " ) The s(x) = c h(x) so s E g(R"). Hence, by Exercise E2, Chapter 11, S E b'(R"). Since s^ = c CJ a suitable limit argument shows = c C as desired. A related result has been proved by Ambrose (unpublished) and a generalization by Hertle [ 19833.

s

564

SOLUTIONS TO EXERCISES

C. Spaces of Constant Curvature

C . l . Let I : Q - -+ R"+' denote the identity mapping. We shall prove I*@*(ds2) = dx?

l * Y * ( d ~= ~ dx: )

+ + dxi - dxi+ + . . . + d x i - dXi+ * * *

1, 1,

which is the substance of C1. But by [DS] (Chapter I, 83. No. 3)

+ l)-'dxi

@*(dyJ = ( x, + I

- x~(x,+]

+

dx,+l,

so @*(ds2) =

n

n

i= 1

1

C dxjZ + (x,+ + 1)-2 C X? d x i +

But on Q - we have n

so

=

f dx? - d x i + 1

Also Y * ( d z i ) = (x,+] - x,)-'dxi -xi(x,+] - x,)-2(dx,+l - dx,) Y*(dz,)

=

-(x,+1

(l
- x,)-2(dx,+1 - dx,).

But since I * [ i x i dxi

+ X , dx, - x , + ~d

~ , + ~

1

a computation similar to that made before gives I*(Y*(~o~ =)d)x l

+ .. . + dxi - dxi+

C.2. We may use as the hyperbolic plane the unit disk IzI < 1 with the metric ds2 = 4

+

dx2 dy2 (1 - x 2 - y y

Chapter I

565

which gives curvature -1 and the distance

[Introduction, 91, Eq. ( 6 ) ] .This means that

6

=

tanh(d/2)

It suffices to verify (2) for a triangle ABC = 0zlz2with z1 on the x axis. Then A = o, B = tanh(c/2), C = tanh(b/2)eiA, so a I tanh(b/2)eiA - tanh(c/2)( tanh - = 6(B, C ) = 11 - tanh(b/2)eiAtanh(c/2)1 * 2 Now (2) follows by a simple computation using the formulas cosha =

1 + tanh(a/2) 1 - tanh2(a/2)’

sinh a =

2 tan h(a/2) 1 - tanh2(a/2)’

Formula (1) can be derived directly from (2). For curvature -c2 we use the model D,: ds2 = 4

dx2 + dy2

+ y2))2

IZI

= tanh[+cd(o,

z)].

(1 - &2(X2

< &-l.

Then we find

E(ZI

We consider again a triangle ABC = ozlztwith z1 on the x axis. Then E

&

The mapping

z

+ (z - Zl)/(l

- &22,Z)

maps the disk IzI c c - l isometrically onto itself so a = d(zl, z2) = d

1 - &2Z1Z2

and c ( ( z 2- z1 I)/[ 1

- c22,z2(= tanh(ea/2).

This equation can be written tanh(7)

=

I tanh(cb/2)eiA - tanh(&c/2)1 11 - tanh(&b/2)eiAtanh(cc/B

566

SOLUTIONS TO EXERCISES

Thus (1) and (2) hold for D, if we replace a, b, and c by &a, E b , and EC. Letting E -,0, D, formally tends to the flat plane and formulas (1) and (2) converge to the standard trigonometric formulas a

b sin B

- - _-=-

sin A a' = b2

C

sin C'

+ cz - 2bc cos A.

D. Some Results in Analysis

D.l. As in Introduction 53 consider the bilinear form

on the space Pk of homogeneous polynomials of degree k. It is nondegenerate and the annihilator of the set of polynomials x -,( x , w ) as ~ o runs through P-'is 0. D.3. It is obvious that lim r SomF(re',re-')e-' dt = 0 r=O

and if r > 0 rm

r

J

r"

rm

F(re', re-')e' dt = J F(r, r't-') d t

0

t++m

J

F(r, 0) dr. 0

0

lim sup po(t) +

4

f pj(t)ek" j= 1

I a.

Chapter I1

lim

t-m

po(t)

I-+m

+ sCj s rp,(t)ekjr

567

= 0,

1

lim r++m

CO

+ s1j s rcjekjr = 0, I

CHAPTER II A. The Laplace-Beltrami Operator

A.1. The differential equations for geodesics {[DS], Chapter I, 45, Eq. (3)) show that T f j ( p )= 0. Now use Prop. 2.6. A.2. The results are local so we may assume M has an orthonormal basis X I , .. . , X, of the vector fields. Let wl,. . . ,w" be the dual basis of 1-forms. Then wk = wxk and w1 A A w" is the volume element. The Lie derivative O(X,) satisfies

8(xk)oi =

1

(rjk

- rLj)w'

j

(use [DS], pp. 540, 44-45), whence by Lemma 2.2,

(rik - rLi).

diV(Xk) = i

568

SOLUTIONS TO EXERCISES

On the other hand, * w l = w2 A ... A u”,etc., and 6 u k can be calculated by means of the first structural equation ([DS], Chapter I, Theorem 8.1) in terms of the rfik. One finds 6wk = -div Xk and now 6(ux)= -div X follows by using Eq. (14) of 82. Finally, if , f b~( M ) we have

1 (xif)xi, 4f = 1( x i , f > m i * div(grad j’) = 1 X:f + c ( X i f )div X i , i gradf =

i

i

i

Af=

-sq=

(F

-6 Xi.f)*

=

*d[

=

Ci ( X i , f )div X i + * c ( X i X j f ) d A * ui i. j

= div

grad ,f.

A.3. The space G / H is in each case symmetric, hence reductive, so D ( G / H ) is determined by the algebra /(m) of Ad,(H)-invariants on m (cf. Theorem 6.1 in Chapter I). The space G / H being isotropic, each q E I(m) is constant on each “sphere” {X # 0 in m : g,(X, X ) = c } in m ; hence q ( X ) = F(g,(X, X ) ) where F is a function. But then F is a polynomial; now use Corollary 4.10. A.4. (i) We have g’j = gj’ so by Theorem 4.3 ( ( g i j X i X j+ g j i X , i X i ) , f ) ( e )

For the pseudo-Riemannian structure given by B the one-parameter subgroups are the geodesics through e ; the Christoffel symbols therefore vanish at e for the canonical coordinates. Thus by Proposition 2.6, Q coincides with the Laplace-Beltrami operator LG at e, hence everywhere by their invariance. Now R E Z(G) by Corollary 4.5. (ii) Choosing the basis ( X i ) compatible with the Cartan decomposition g = f + p the argument above shows that p(Q) and LG,Kcoincide at the origin {K},hence everywhere by the invariance.

Chapter I1

569

xi

A.5. Using notation from A2 we have g = mi 0 mi so (i) follows from the proof of A2. Formula (1) follows from LexprX= h(exp t X ) L by differentiation. (iii) Let M have pseudo-Riemannian structure g and put h = z'g. Then hij = r2gij, hi' = z-'g'j, h = ~ ' ~ and g , by (22) 42 the new Christoffel symbols are (1)

r *.k .--rfJ .. + @dJ{log T) + 6idi(log T) - g i j 1g'kal(logT) IJ

i

Thus by Proposition 2.6 the new Laplacian L* is given by L*f = r-'[&f

+ ( n - 2)(grad ,f)(log T)],

which by (17) 42 can also be written (2)

L*f = T-"/2-"L(T"'2-'f)

-

L(@-l).f].

Now we need the identity

(3)

-

L(.f""-') = cn(K*Tn/2+1- Ktd2-1 ),

where K * is the scalar curvature for the Riemannian structure h (see Yamabe [1960], where the scalar curvature has the same sign). The desired result now follows from (2) and (3) because, L and K being invariant under isometries, we can take T as the identity mapping. This quasi-invariance of L occurs several places in the literatures; see, e.g., Friedlander [1975], Qrsted [1981b], and also Kosman [1975]. To a large extent, it goes back to Cotton [1900]. A.7. Let U be an arbitrary coordinate neighborhood on M and let p be given on U by

570

SOLUTIONS TO EXERCISES

A.8. (Harish-Chandra [1960].) During the proof of Proposition 5.23 we saw

Y, = (coth c)Z, - (sinh c)-'Z;-',

c = j(H),

so

X, = (sinh c ) - ' { e C Z p- Z j - ' } . Hence -X,X-,

= (sinh

c ) - ~ { Z , Z - ,+ (Z,Z-,)"-' - ecZ,Z'Yi

-

e-'Z;-'Z-,}.

Using

z,z$

= z"& =

Zt-iZ,

+ [ Z , , ZY-;]

+ (cosh c ) [ Z , , Z - , ] + (sinh c ) [ Z , , Y-,]

we obtain -X,X-,

=(sinh C ) - ~ { Z , Z - + , (Z,Z-,)p-' - e'Z'liZ, - e - ' Z ; - ' Z _ , -+(e2' I ) [ z , , Z - , ] - +(e2' - I)[z,, Y-,]}

+

Here we apply 8 and note that O(Z;-') = Z;, etc. This gives a formula for 8(X,X-,) in which we replace j by - j and a - 1 by a. Adding, we obtain -(X,X-, + W - p X , ) ) = (sinh c ) - ' { Z , Z - , + Z - , Z , + ( Z , Z - , + Z-,Z,)"-' - 2e-'~;-'~-,} -2e'z"_-,'z, + e'(sinh c ) - ' { [ Z - , , Y,] - [ Z , , Y-pl}.

+

Put R + = C,,p+(X,X-, X-,X,). Then 8R+ = R, so that R, = #2+ OR+). Also, [ Z , , Y-,] [Yo, Z - , ] = A , so that the formula above gives

+

R,

=

+

1 Coth~A, - 1 (sinh c ) - 2 { Z p Z - , + Z - , Z , + ( Z , Z - , + Z-,Z,)"-'

BEP+

PEP+

-

2 e ' ~ " _ ' ~-, 2 e - ' ~ f ; - ' ~ - , }

Since the mapping /?-+ -Be = j*is a permutation of P , we have X,,= cp8X-,, X-,* = c-,BX, where c, E C. Since 8 preserves B and since B(X,, X-,) = 1 , we have c,c-, = 1 . Hence

z,*z-,* = c a c - s Z - p Z , = z-,z,. This gives the desired formula. See also Anderson [1979].

Chapter I1

57 1

6. The Radial Part

6.1. Let uo E V be such that the orbit H - uo has maximal dimension. Then dim(H . u) = dim(H. uo) for u in a neighborhood N o of uo in V. According to Chevalley [1946] (Chapter 111, $7, Theorem 1) there exists a coordinate system {xl,.. . ,x,} near uo in V such that

xi(uo) = 0 (1 I i I n )

and such that each slice x,+ = a,, l , . . . ,x, = a, is a part of an orbit H . u. Then the submanifold x, = 0,. . . ,x, = 0 can serve as W (remark by R. Palais). 6.2. This is clear from (33') because here 6(r) = A@).

B.3. By (i) the orbit H . w is locally compact and thus homeomorphic to HIH". By (ii), H" is compact for each u E I/* so no change is required in the proof of Theorem 3.7. 6.4. The right-hand side of the formula is independent of the choice of basis ( H i ) . But with ( H i ) orthonormal the computation is the same as in the proof of Proposition 3.9. C. Invariant Differential Operators

C.l. The group G = GL(n, R) acts on the space P, of n x n symmetric positive definite matrices by

p

+

gp'g,

if . p E P,, g E G, 'g = transpose of g,

and the isotropy group of I is K = O(n). For the Lie algebras we have g = f + p and p is the space of all n x n real symmetric matrices X. If k E K , then Ad,(k)Ip is given by X + kXk-'. The algebra I(p) of invariant polynomial functions is generated by Tr(X), Tr(XX),

. . . , Tr(X").

This boils down to the fact that the symmetric polynomial in n variables

A1,. . . , A , (the eigenvalues of X) is a polynomial in A1 + *..+ A,, ...,A! + - - . + A ; .

By Corollary 4.10 the algebra D ( G / K ) is generated by the corresponding operators Di (1 Ii In). With the inner product Tr(XY) on p the matrices e, = E i i , 1 I i I n, eij = 2- l/'(Eij + Eji) ( i > j ) form an orthonormal basis. Writing the matrix X = ( x i j ) in the form X = Siiei + t i j e i j we have ti, = xii, tij= 2 ' / ' x i j . If Tr(Xk) = Pk(rpq)

572

SOLUTIONS TO EXERCISES

we have to determine Pk(a/atpq).But for a = 25111

X=

*.

1/2

., a5n1

( i

i )

aL1, . . .,

5nn

and we have

a ax

--

a

...)

+ Sij) a/axij. Since Tr(Xk) = Pk(Spq)

the symmetric matrix with entries +(I we have

Thus the operators

generate D ( G / K ) . A different description of D ( G / K ) is given Maass [1955] and Selberg [1956).

C.3. (i) Since Ad,(m) fixes H each P E I(a

+ I + q) can be written

P ( H , T, X) = C HkPk(T, X). k

But each Pk(T, X) is invariant under the action (T, X) + (mT, m X ) ,

m E M = SO(n - 1).

By a standard result about invariants (cf. Weyl [1939], p. 31) Pk is a polynomial in I TI2,T X, and I X 12. This proves (i). 7;] = (ii) We have the commutation relations Xi = [ T , HI, [Xi, dijH, [Xi, H] = T {[DS], Eq. (17) p. 407). The symmetrization A satisfies +

573

Chapter I1 Using the commutation relations this can be written

and also as 1

n- 1

n-1

C H q T + -2 i C ( x i T + q x i ) +n --61 i= I =]

H.

By subtraction, [D,,,Dl~12]= -2DT..; the other formulas from (ii) follow similarly. C.4. In the notation of Theorem 5.16 we have D4v = Y(D)(iV)$V

D E &(GI.

Let s E W , let m, E M' be a representative and put N s = m, Nm;'. Then k E K , AS(g)E a, n

g = n exp A"(g)k,

E

N'.

Let 7': DK(G) + &(A) be the homomorphism defined by using the Iwasawa decomposition G = N'AAK in place of G = N A K and put

4 ~ y (=~J)

,p+sp)bWks))

dk.

K

Then by Eq. (38') of $5,

D&

=

y'(D)(i~)&.

Comparing the two Iwasawa decompositions we see that As(g) = sA(m,-'g).

As a result &, = dV,so ys = y. The result is given in Schiffmann [1979] with a different proof. D. Restriction Theorems

D.4. For R > 0 select C, such that

- &H')I 5 C R I H - H" for all H, H' in the ball I H I < R in a. The K-invariant extension

(D satisfies such an inequality on each maximal abelian subspace b of p. Given X , Y E p the distance from X to the orbit Ad(K)Y is minimized at a point Ad(k,)Y for which X and Ad(k,)Y commute (Prop. 5.18, Chapter I) and so lie in the same b. Thus I(D(X) - (D(Y)I IC,IX - YI if both 1x1, I Y I c R. (cf. Helgason [1980a], Proposition 2.4).

574

SOLUTIONS TO EXERCISES

E. Distributions

E . l . By definition

f(F * Ds) =

=

JJF(xy) d(Ds)(x) d t ( y ) = (Ds* t)(F).

Use this result for s = 9, bi-invariance of t', D(s *

ss

=

t) =

t =

F ( ~ x - ' )d(Ds)(x) df(y)

p, D E I(l(p)). Then, using (13) $5 and the

D(s * 6 * t ) = D(s * t') = s * Dt'

=s

* (D*Yt'

=s

* (D*)-t'

=

D S* th = Ds * t.

F. The Wave Equation

F . l . By Eq. (63) of $5 we have for xo E R", r 2 0,

s

lo,0) d w ( x ) =

Sr(r0)

J

4 x 0 , y) d d y ) .

sr( 0 1

The function u(xo, y) is constant in y on hyperplanes perpendicular to the y , axis. Proceeding as with Lemma 2.30 in Chapter I we therefore obtain (1)

Rnr"-'(M'uo)(x) = R,,-' r

- t 2 ) ( 1 ' Z ) ( n - 3 ) ~ (t~) 0d,t .

Chapter I1

575

Case 1. n odd. Divide Eq. (1) by r, differentiate +(n respect to r2, and then apply a/&. This gives

-

3) times with

u ( x 0 , r ) + 4x09 - r )

(2)

Now the function w(x, t) = u,(x, t) satisfies the wave equation with initial data wo = ul, w 1 = u,,(x, 0) = L,,u0. We can replace u by w, uo by wo = t i 1 in the last formula and get by integration (3)

tI(X0,

r) - 4x0,

-4

Adding (2) and ( 3 ) an explicit solution formula for n odd is obtained. Case 11. n even. Again divide (1) by r and differentiate &n with respect to r2. This gives

-

2) times

The integral on the left is inverted as in Eq. (19) of 52 in Chapter I. Again we can use the identity replacing u by u, and uo by ul. Here one gets the solution formula

-

t

[t"-2(M'uo)(xo)]dt

F.2. In accordance with Chapter I, Exercise C1, we use the half-space model for H" and have ds2 = y i 2 ( d Y ?

+ .. . + dy:),

Then the Laplace-Beltrami operator is given by

y, > 0.

576

SOLUTIONS TO EXERCISES

which by the substitution y, = es can be written

With y, = es put u(x, y) = u(x,,.

. . ,x,,

y,, . ..,y,) = e("2)fn-')Su(xl,.

. . ,x,,

s).

Then, using the differential equation for u,

Y)).

= L,(u(x,

Now use Theorem 5.28 for the point (xo, yo) = (x, o), where o is the origin (0,. . . ,0, 1) in H". Since u(x, 0 ) = u(x, 0) we obtain

s

N Z , 0) d

44=

s

u(x, Y) d d y ) .

SAO)

The non-Euclidean sphere S,(o) is a Euclidean sphere with center (0,. . . ,0, ch r ) and radius sh r. O n the plane y, = es the metric is

+

e-"(dy:

+ dy,Z-l)

so the intersection with the sphere S,(o) is a non-Euclidean sphere of Euclidean radius [2es(ch r - ch s)]'l2; its non-Euclidean (n - 2)dimensional area is R,-l[2e-s(ch r

- ch s)](li2)("-')

and the function y + u(x, y) is constant on it. The arc-element on the circle y, = (2es(ch - ch s))'/',

y, = . . . = y,-

, = 0,

y,

=

es

is

y,-'(dy:

+ dyi)"'

=

sh r[2es(ch r - ch s)]"'

ds.

The integral relation above thus becomes (4)

,

R, sh"- 'r(Mruo)(x) = 0,- sh r

Jl

u(x, s)(2 ch

r - 2 ch s ) ( ~ / ~ ) ( " -ds. ~)

r

The function w(x, s) = tls(x, s) satisfies the same differential equation as u(x, s). This we can in the last formula replace u(x, s) by us(x, s) and uo(x) by ul(x). Using now the method of the last problem we obtain:

Chapter I11

577

Case I. n odd. The solution u(x, s) is given by

1 (5)

u(x, s)

=

Iz,,[? - 3))! 0,as

2(2n

(

a

)

(n-3)/2

a(2 ch s)

[shn-2s(Msu,)(x)]

d (m) [sh"- ' s ( M s u , ) ( x ) ] ] . (n-3)/2

+

In particular, Huygens's principle holds. Case 11. n even. From (4) we obtain

n-3 n-5 -.-...2

Q,

=-

jj

ch s - 2 ch r)-'/'[u(x,

(

a

r)

+ u(x, -r)]

dr

)(l/Z)(n-2)

~

d(2 ch s)

[sh"- 2s(Msuuo)(x)].

This Abel-type integral equation can be solved for u(x, s) + u(x, -s). Using (4) on ti, we get an analogous equation for u(x, s ) - u(x, -s), so a solution formula for u(x, s) follows. For relevant references see Holder [1938] (for the pure equation Lu us, = 0 with Huygens's principle absent), Gunther [1957, p. 231 (giving ( 5 ) for n = 3, using a different method), Helgason [1964a, 1977c, 19841, Lax and Phillips [1978], Qrsted [1981b], and Kiprijanov and Ivanov [19813.

CHAPTER Ill 1. Since the invariants of O(p,q) are the polynomials in Q (compare Exercise A3, Chapter 11) this follows from Theorem 1.2 [for (p, q) # (1, l)]. For ( p , q) = (1, 1) use substitution y , = xl, y , = ix, and observe that a polynomial identity P(cosh (, sinh c) = 0 implies P(cos 8, i sin 19)= 0. 2. (i) If 1 is singular then K, has a Lie algebra larger than that of M so K, # M and s,L = 1 for a certain Weyl reflection s, # e in W. If L is regular then K, and M have the same Lie algebra. If k E K, and 1 = + iq (t, q E a * ) then k5 = 5, kq = q. For each restricted root a either a(AJ or a(A,) is nonzero. Thus the centralizers 3{ and 3, of 5 and q, respectively, in p have intersection a. Since k leaves each of these centralizers invariant we have k E M', the normalizer of a in K. Then the Weyl group element s = Ad(k)la fixes the set RA, RA, pointwise. But

<

+

Ia(AJl

+ la(A,)I

>0

5 78

SOLUTIONS TO EXERCISES

for each restricted root a so this set contains a regular element. Hence s = e so k e M . (ii) The statement f ’ &,(p) ~ is proved just as was Lemma 3.14. (iii) This follows from (ii) because each M-harmonic polynomial on p is K,-harmonic. 3. Let h , , . . . , h , be a homogeneous basis of H . As in Exercise D1 of Chapter I1 consider the normal field extension

~(s)/c(r)(s = s(aC), I

= I(aC)),

which has Galois group W. Because of Theorem 3.4 the hi (1 I i I r ) are linearly independent over C(I). Each element of C(S) can be written as a quotient of a polynomial and an invariant polynomial. Thus the hi form a basis of C(S) over C(I) so the representation 0 --* (aij(a)) of W given by D

*

hj =

1U i j ( 0 ) h i i

is equivalent to the regular representation. 7. Let a, ,... ,al be the simple restricted roots; H I , . .. , H , the dual basis of a,; a = exp(Ci t i l l i ) any member of A . It is now easy to see that a2 = e e t j E niZ for each j . Thus (0, 1)’ is in a one-to-one correspondence with F . 8. Because of the results quoted it suffices to prove a . j = j for a E F and ,j E I(a). Since a acts trivially on a this is obvious.

CHAPTER IV A. Representations

A.1. (i) The proof of Corollary 1.4 gives the result. A.2. Let n, n’ be two unitary representations of G on Hilbert spaces V and V’, respectively. Assume the linear homeomorphism A : V + V’ satisfies An(x) = n‘(x)A. Passing to adjoints one finds

A*Aa(x) = n(x)A*A.

Let P be a positive bounded self-adjoint operator such that F 2= A * A . Then A P sets up the desired unitary equivalence (cf. Bore1 [1972], $5). A.3. (i) Let u + 0 be the canonical mapping of V into (V’)’. Consider the functional XI: A + (d/dr)(x,, A) on V’, which does belong to (V’)’ (Dunford and Schwartz [1958], Theorem I1 1.18). The function t + X,(A) is C’ for each I E V.Fix t E I and an E > 0 such that t s ~ forl 1st I;E.

+

Chapter IV

579

Let Vk = { A E V ' : l(l/s)(X,+s - X,, A)[ I n for Is1 IE } .

Un

Then V ; is closed for n = 1, 2, and V' = V ; so by the Baire Category Theorem there exists an m E Z + such that V k contains a ball Bs(Io).Thus

11111 < 6 => l(l/s)(X,+s - X,, I

+ A,)(

rn

for Is1 I

E,

which implies that the mapping t + X, is continuous from I to (V')'. Since

we see that 1 (A[+h

- a,) - x, =

[

t+h

( x , - XI) ds

--f

0

in the norm on (V')' as h 40. But P is closed in (V')', so X,E p. Since the injection u -, 8 of V in (V')' is isometric there is a unique continuous mapping t + 5, of I into V such that 2, = X,. This shows that t + x, is of class C'. (ii) Let X l , . . . , X , be a basis for the Lie algebra g of G. If N , is a suitable neighborhood of e in G the inverse of the mapping ( t l , . . . , t,,) + exp(tlX 1) . . exp(t,X,) is a local coordinate system on N , . Assuming x + (n(x)u, A) C" on G for each I E V' we see from (i) that the map (tl, . . . , t,) -, n(exp(tlXl) ... exp(t,X,))zi of R" into V has continuous partial derivatives of any order. A.5. (i) As an orbit n(G).u the image I ( G / K ) is a submanifold of V and the injectivity is clear from the maximality of K ( [ D S ] , Chapter VI, Exercise A3). (ii) Let 4 be a G-finite function on G / K , V' the space of complex linear combinations of translates 4r(8), T the natural representation of G on V,, and TC the extension of T to a representation of gc and GC on V,. We have T(g)ll/ = ll/T(g), ll/ E V,. Define 4c on GCby 4c(x) = (Tc(x-'>4)(o) where o is the origin in G / K . Then @(g) = &gK) for g E G . Also 4' is a @-finite function on GC. [If 4 = e l , . . . ,en is a basis of V+ and Tc(x)ej = T j ( x ) e i then @(x) = T1(x-')ei(o).]. Let k E K . Then if x E GC,

xi

xi

$C(Xk) =

( T C ( k - I x - 1 )4)(o)

=

( T @ -' ) T C ( x -l)4)(0),

580

SOLUTIONS TO EXERCISES

so @ ( x k ) = &x). In particular, we can define 4, on U/K by 4,(uk) = $c(u). Then 4, is K-finite. Denoting by I, the injection uK -+ n(u)u of U / K into V, we have by Introduction, Exercise A1 4, = po 0 I . where po is a polynomial function on V,. If p is the canonical extension of po to a (holomorphic) polynomial on V put $ = p o I . Since $ is a G-finite function on G / K we can repeat the above process, replacing 4 by $. Then 4, = $, so 4‘ = $‘ by holomorphic continuation. A.6. If D E D ( K ) let as in Chapter 11, $5 cD denote the distribution with support { e } defined by c D ( f )= (D{)(e). The mapping D --* eD then identifies D ( K ) with a subspace of .9‘(K). Each element 4 in the space d ( K ) of analytic functions on K gives a linear functional T ~ T: + T ( 4 ) on 9 ’ ( K ) ; we assign to .9’(K) the topology c@’(K), d ( K ) ) , i.e., the weakest topology making all these functionals continuous. From general theory it is then known that the dual of 9 ‘ ( K ) for this topology consists just of the functionals T& (4 E d ( K ) ) . Since no such functional annihilates the subspace D ( K ) c Y(K) (by Taylor’s formula [DS], Chapter 11, (6) $1) we conclude that D ( K ) is a dense subspace of .9’(K). Second, by the definition of the topology &Y(K), d ( K ) ) , the convolution T + T * za is a continuous mapping of Y(K) into itself. Thus D(K)za is dense in 9 ‘ ( K ) * ~6 which, however, by Corollary 1.8, equals the finite-dimensional space H a .

B. Spherical Functions B . l . We have by Lemma 6.5 p ( H ( a k ) ) I p(1og a ) for a E A + so - 40(a). Next put $(a) = ep(loga)4,(a)(a E A). Let b E A. Then < using Lemma 5.19 of Chapter I we find [since H(abk)=H(ak(bk))+H(bk)] ,-p(loe4

=

dk =

/:-p(H(.kII

so

(1)

s,

,-p(H(obk)),-p(H(bk))

dk

if U E A , b e p .

$(a) I $(ab)

Select H, E a+ such that a,(H,) = 1 (1 I i 5 1) and put for r > 0 F ( t ) = $(exp tH,) = ceP(rHo) {m,(~(n)4,(exP~ ~ O ) ) } A = ‘O

Then F(t) =

1 pp(t)e-‘”(HO), CEA

where p,, is a polynomial of degree I d . Using the uniform convergence for 1 If < co; this implies IF(t)l I ao(l + t)” for t 2 0, a, being a constant.

Chapter IV Let H E a + and put P ( H ) = maxi ai(H). Then fl(H)H, - H inequality (1) above implies Nexp H)IF(P(H)) Ia,(l

58 1 E

2,so

+ P(H)Id Ia(1 + IHl)d

as desired. 9.2. I f f ' € 9 ( a ) we can write it as a Fourier transform f(H)=

s

g(l)e-i"'H) d l

a*

and then our formula comes from Eq. (38) of $6 if we ignore problems of convergence. The problem is that the integral for c,(A) is not absolutely convergent for 1 E a*. One way to overcome the difficulty is to retrace the steps in the proof of the product formula (33). A simpler way is t o shift the integration in the formula for f and write f(H) =

1

g(l)e-'"'') d l , '

IP

because then the absolute convergence of (36) is guaranteed by (37).

B.3. We know from Eq. (34) of $6 that p(A,) = p,(A,) if a is one of the simple roots a,, . . . ,a,. For fl E C l put as in $5, fl = C: miai and m(P) = Ci mi. Assume now fl not simple and select an i such that (fl, ai) > 0. Then the root y = sK,p= fl - ciai (ciE Z') satisfies m(y) < m(P) and y E Xi. We have p(A,) 2 p(A,). After finitely many steps we get p(A,) 2 p(A,,) where S E W and s p is simple. By the first part of the proof, p(A,,) = pss(A,,) = p,(A,) (cf. Kostant [1975a]).

B.4. Using B2 for B3,

m,= fl, p = d , f ( H ) = [l + p(H)Id'& we have by

582

SOLUTIONS TO EXERCISES

xsEw

Here we can replace e-q(A(ke)) by the average w-' e-s'J(A(kg)), However, if H E a, s - H is fixed under each s, and hence is 0. By convexity of ex, eai 2 n whenever ai = 0. Thus C e-sq(A(k8)) > - w,

csew c:='

xi

ssw

giving the desired result. B.8. We have 2p = (ma + 2m2,)a and (a, a ) = &ma Eq. (8) of $5 we have 2(m,

+ 4m2,)-'.

From

d2 + 4m2,)A(Lx) = + {m,coth t + 2m2, coth 2 t ) -.dtd dt2

Also

( A I > + (P, P>

= 4Cl/(ma

+ 4ma)I[2 + (+ma + m2a)'I

and the rest follows by direct computation. B.9. (i) Let f*(g) = f@-'). Then since (f *)-(A) = for I E a*. Thus if I E a*

= +n(g-')

fn

we have

is positive definite. B.10. Considering the group Ad,,(M) acting on n we see from Chapter 11, Theorem 4.9, that with the identification MN/M = N the algebra D(MN/M) is generated by the two commuting operators 13' P2 and 2 (p being the left invariant vector field on N generated by V E n). Putting for F E d(N), so

+

f (x, y, z) = F(exp(xX

+ yY + z Z ) )

we have, by [DS], Chapter 11, Lemma 1.8, since

[X,Y ] = 2 2 ,

[X,Z ]

=

[Y, 21 = 0,

whence

a2f + 2 a2f + (2+ y 2 ) a'f + 2x a'f - 2y-.a'f (R2+ P2)F(n) = 7 ax ay az2 ay a Z axaZ Thus the spherical functions on N = MN/M are given by

f (x, y, z) = +(x2

+ y2)eaz,

Chapter IV where a E C and

583

4 satisfies the differential equation

for some A E C . This example and its generalization to G of real rank one are given in Koranyi [1980]; see also Kaplan and Putz [1977]. 8.12. Let A, = 5, + iq, (to,qo E a*) be such that - A q o ~ pand is bounded. We have to prove qo + p nonnegative on a t . Let U c V c W be the subgroups of the Weyl group W leaving fixed A, and q o , respectively. We can assume that for the lexicographic ordering of a* defined by means of the simple roots al,. . . ,al we have 5, 2 sto for s E V. In particular, a&,) 2 0 for a E X ' satisfying a&,) = 0. Lemma. The subgroup U of W leaving A, j x e d is generated by the rgections s,, where ai is a simple root vanishing at AAo.

For this let U' denote the subgroup of U generated by these sa,. The group U is generated by s, as a runs through the roots, vanishing at A,, ([DS],Chapter VII, Theorem 2.15). For each such a > 0 we shall prove a = sap where s E U' and ap is a simple root vanishing at A,,,. We prove this by induction on 1mi if a = miai (mi E 2 ' - (0)). The statement is clear if mi = 1 so assume mi > 1. Since (a, a ) > 0 we have (a, ak)> 0 for some k among the indices i above. Then SaraEX' so s,a = m;aj (mi E Z') and mi < mi. Now a@,,) = 0 so ai(Aq0)= 0 for each i above, whence ai(Aco)2 0 by our convention. But then a(A,,,) = 0 implies ai(At,) = 0. In particular, s, E U.Thus the induction assumption applies to saka,giving a s' E U' such that saka = slap. Hence a = sap with s E U'.But then s, = SS,,S-~, proving the lemma. Let K' denote the product of the positive roots vanishing on A,, K'' the product of the remaining positive roots. We multiply the formula (Theorem 5.7) for +,(a) by n'(A), then apply the differential operator a(n') on a: and evaluate at A = I,. This gives the formula

cc

1

c

c (det s)esP(loga)+,,(a) c ps(log a)eisAo('oga), =c

SEw

ssw

with c E C,the ps being certain polynomials. Now write the sum on the right in the form

584

SOLUTIONS TO EXERCISES

Using the lemma above we see that n'(sH) = (det s)n'(H) for s E U. Examining the definition of p s , i.e., ")I}'= l o e - is'O(lOgO), ps(log a) = {d(d),[(det s)n"(A)we see that the highest-degree term in CsE,,ps(loga) is a nonzero constant multiple of n'(1og a). Hence CPt(l0g a> ICU

+ 0.

Thus the expression q(a) inside the large parentheses above is nonzero (cf. Chapter I, Exercise D5).We have

l SCEW

ps(log a)eisAo('Oga)

and

Ic SE

I

(det S)esP(loga) 5

w

eP(Io.ai(

1+

c

e(sp-p)(loga)

S#C

.

)

Now (qo - sqo)(H) c 0 for H E a+, and s E W - V, and of course sp p < 0 on a+ if s # e. Fix a unit vector H E a + such that ( s ( , ) ( H ) # <,(If) for s E I/ - U and write q(exp t H ) = C qj(t)eirJ1, j

where the r j are different real numbers and the q j are polynomials. The q j which corresponds to rj = t,(H) equals CsEu ps(exp t H ) whose leading term is n'(tH) (up to a constant factor). By Chapter I, Exercise D5, the function t -,q(exp t H ) is unbounded. The inequalities above, together with the boundedness of c$',~,thus imply that e - ( q l l + p ) ( l His) bounded for t 2 0; thus qo + p 2 0 on a dense subset of a + , hence on all of a + . B.13. For x E G, consider the mean value operator M" given by (MX1')(g) =

J f'(gkx) dk K

for 1'in the space

C,(G) = { f ' C(G) ~ :J'(gk)= f'(g)

for g

E

G, k

E

K}.

Let g be the Lie algebra of G. For X ~ g m, E Z + , let Df denote the operator Df =

JK

Ad(k)X" dk E D,(G).

Chapter IV

585

Let f’E CK(G) be analytic. Then if X E g is sufficiently small, the formula r

n

q

holds on a neighborhood of e in G (cf. Helgason [1957a, 1962a], Chapter X). Assuming (i) we thus derive the relation (M”Myf’)(e)= ( M y M 7 ) ( e )

for x , y near e, hence for all x , y E G by analyticity. Using approximation by analytic functions (loc. cit. Chapter X, Lemma 7.9) the relation is seen to hold for all J’E C,(G). It follows that i f f € C;(G), then

JK

f ( x k y - ’) dk =

JK

f(y- ‘ k x ) dk.

Integrating this relation against g ( y ) (g E C:(G)) and using the unimodularity we get f’ * g = g * 1; proving (ii). Conversely, assume (ii) holds and let D, E E D,(G), f,g E @(G). Then since D is left invariant, D ( f * g) = f * Dg = Dg * f,so D E ( f * g ) = D(Eg *f)= Eg *Of= Dj’* E g , ED(f’ * 9 ) = ED(g * f) = Eg * Of. Thus E and D commute on the subspace @ * 9 ‘of gq,hence, by density, on all of 9‘. Now let F E 9(G)n CK(G) and consider F*(x, Y ) = (M”F)(y). Let subscript 1 denote differentiation with respect to the first variable. Since F* is bi-invariant under K in the first variable, we have

so

Hence (DEF)(Y) = ( D l w - ) * k Y ) = (D,E,F*)(e, Y ) = ( E , D , F * ) ( e , y ) = (EDF)(y), so (i) holds. Part (ii) 3 (i) is from Helgason [1979]; part (i) =-(ii) was noticed independently by Thomas [19843. B.14. The adjoint representation of U is irreducible and the multiplicity of the 0 weight (Chapter V, 9;l) equals dim T. Now the result follows from B.13 and Theorem 3.5, Chapter V.

586

SOLUTIONS TO EXERCISES

B.16. Assuming L: C:(G) + C to be a continuous surjective homomorphism we have, by the definition of a measure, a measure p on G,biinvariant under K such that

L(f)=

1

f Ecm.

f(x) dcl(x),

G

Take g E C:(G) such that L ( g ) = 1. Then

so p = p * 8 is a continuous function which by Lemma 3.2 is spherical. B.17. Let r be the maximal rank of the matrices (Qi(xj))i, for all sets of x j E S, and choose xl, . . . ,x, such that (Qi(xj))i, has rank r. If r < w there exists a nonzero w-tuple (Al,. . ., A,) depending on xl,. . . ,x, such that if uj = (Ql(xj),. . . ,@,(xi)) then W

C Aiai(xj)= 0,

1 Ij I r.

i= 1

Let x E S be arbitrary and put u = (Q1(x), . . .,Q,(x)). Since the matrix formed by the column vectors ul, . . . , u,, u has rank r (by the maximality) there exist numbers akx) such that u = a,(x)uj. Then, by the above,

c:

W

i= 1

i=1

j=1

Since x E S is arbitrary, this is a contradiction. C. Spherical Transforms

C.l. (i) Let I = 5 F E L'(G) we have

+ iq (5, q E a*)

co > JGIF(g)I dg =

1

KAN

be such that 4A is bounded. For

IF(kan)leZP(*"*")dk da dn

587

Chapter IV whence by Fubini, whenever

is bounded,

E L’(A); Replacing F by IF1 and taking 1 = 0 in (*) we conclude that 41F., hence & E L’(A). The assumption F(1) = 0 (1E a*) and the injectivity of the Fourier transform on L’(A) then imply & = 0 almost everywhere on A. But then (*) together with Prop. 3.8 implies that F = 0 almost everywhere on G. The argument also gives part (ii). C.2. Clearly is W-invariant. Also since T is K-invariant

T(1) = T*(A,) where T * ( Y ) = / e - i ” ( r . ” )dT(Z). P

By the Paley-Wiener theorem for &(p) (cf. Hormander [1963]) T* and therefore satisfies the desired inequality. Conversely, suppose 0 satisfies the stated conditions. Consider the K-invariant distribution T on p defined by (cf. (2) 49) T(f) =

1 f””(1)WW

dA

f E

.m),

Q’

where f‘(2)= f(k . Z) dk. Let I], E 9&) have support in B,(O). Then @(A)ijc(l) is entire of exponential type I R E so by Theorem 9.2, T * I], has support in B , + ,. Letting E -,0, we see that T E &(p). C.3. (i) Writing a, = exp r H , a ( H ) = 1, we have as in [DS], p. 417, iia, = u1 exp A+(iia,)u2 (ul, u2 E K), whence

+

a,e(ii)- %a, = u,

exp 2~+(na,)u,,

giving Tr(arO(ii)-lii~r)= 2 ch(2a[A+(iia,)])+ 1, and (i) follows by direct computation. (ii) If a is the root in C+ the formula reads, replacing ch x by 2 ch2(ix) - 1, ch2(3a[A+(iia)]) = ch2($x(loga)) + (1/16m,)ea(10gQ)IX12.

588

SOLUTIONS TO EXERCISES

If 2a is the root in Z + the formula reads, writing

p = 2a,

ch2(#[A+(iia)]) = ch2(ip(log a ) ) + (1/16rn2Jeac10g")I Y 12. which is the same formula. (iii) See solution to B8. (iv) The change of variables X factor eP(log0),so (*) follows. (v) Putting

+ e(1/2)rX,Y + e'Y

cancels out the

@(u) = i - 2 m 4 ( u + 2 II Y1I2) dY, we have $(u) =

j--w+ 11x112)dX.

Introducing polar coordinates in g - 2 a and then putting we find @(u) = c1 and similarly

r

$ ( u ) = c2 p

rf$(t2)(t2

q s f ) ( s 2

-

u2)(1/2)(m2=2)

-

y2)(1/2)(m=2)

t 2 = u2

+ IIY112,

nt,

ds,

and c2 being constants. (vi) These two integral transforms can be inverted by the method used in the proof of Theorem 2.6 of Chapter I. This gives the conclusions

c1

$(u) = 0

=-@(s2) = 0 =-4(t2)= o

for u > A, for s > A , for t > A 2 .

Now assume F&) = 0 for )/loga,ll > R. Since IIlog arll = r we have = 0 for r > R, i.e., > By the above, &t2) = 0 for t > ch R, that is, f(g) = 0 if 6(0, g . o ) > R. C.4. Select J'E 9 ' ( G ) such that the function H + f(exp H) on a + has support inside the set

f i JchR.

$(Jchr)

a+(M) = {H E a + : a ( H ) > M for a E C'},

M > 0 being fixed. With 6 ( H ) = by

no,,2 sinh a(H), consider the function f,

~ ; ( H > S ' / ~ ( H=) ~ ( l / ~ ) f l ( c H ) d ~ / ~ ( c HH) ,E a+.

E .9"(C) given

Chapter IV

589

Then

On the other hand,

Jl(4 = J

.+4 - L

j&(H)4-'W)KH)dH,

and, using the expansion for theorem on a, it can be proved that (2) Since

l*

[Jl(A)c(-A)-'

c

s,

and the Euclidean Paley-Wiener

- F , ( A ) ~dA ~ +o

I f ( 9 ) I2 dg

=

1 Jlm(4I

as

E + 0.

l2 d l

a*

(1) and (2) would imply c = I W I. This proof, with further details, was given by Rosenberg [1977]; in Harish-Chandra [1958b] ($15) the measures d A and da are regularly normalized, the measure dg is normalized by sf(g) dg = ~KAN/.(kan)e2r(10sa) dk da dn,

and the Plancherel formula (Theorem 7.5) is proved with the same constant c = I W I. In particular, the two normalizations of dg considered above must agree. See also Mneimnt [1983]. C.5. From the inversion formula we have

1

1 f'(exp H) = - f(W,(exp H)I c ( 4 I- dA. I W I a* Using Theorem 5.7 we get for H E a, E ( S ) = det(s),

h ( p )

1 e(s)esp"'H'f'(exp H) = ssw

f(A)n(A)eiA(cH) dH,

590

SOLUTIONS TO EXERCISES

and in this last integral we can shift the integration to the extremes of the tube a* + iC(p) where C(p) is the convex hull of the orbit W . p . This gives i-dn(p)

c e(s)e"P'H'(Aof')(exp H) SE

w

=

s..

$*(I) n(l)e'"'' d I

where

$*(I)

C f ( I + isp)n(I + isp).

= n(n)-'

sow

Reintroducing the formula for

+A

this gives the desired formula.

CHAPTER V A. Representations

A.l. Apply Eq. (16) of 41 to the representation n = Ad,; then ~ ( t=, ) cl, I = 6 and Tr(n(L,)) = - c dim u so c = - 1. A.2. (i) Use Proposition 6.10, Chapter IV. For (ii) suppose I E A and 111 = I p I . By Chapter I, Proposition 5.15(ii)

1(det s)e"' saw

=

(c SE

(det s)esp) 1 apeP

w

P

By the W-invariance we have a,, # 0 for some p with H, E p.But then if I,, is the highest among SI (s E W) we have I, = p + p. Then ( p , p ) 2 0, so 1IOl22 IpI2 lpI2, which implies p = 0. A.3. While the formula for dim(7c) (Theorem 1.8) is obtained by comparing the lowest-degree terms in x, the present formula is obtained by considering the next power of x. A.4. Clear from A1 and A3 if we also recall

+

( p , v > = C ( P ? U > ( h a> asA

([DS], Chapter 111, Exercise C4). A.5. As in the proof of Eq. (25), Chapter IV, $5, we have

C P(p)e-P = fl (1 + e-' + e-" + aeA+

n

(1 - e-')-' - asA+

= ep

. . a )

Chapter V

59 1

Hence

1 (det s)es('+p) 1B(v)e-'-,, so (i) follows by putting v =-s(A + p ) - ( p + p ) (cf. Cartier [1961]). =

VEA

S

For (ii) note that by Eq. (16) of 91 and Prop. 3.12 of Chapter I1 we have (1)

NL,)

(c.*.)

1 m,e'

= ((P,

P> - ( P

+ 4 P +A>) C

m,ec.

c E A(n)

By Theorem 3.7 of Chapter I1 A(L,) = LT -k grad(1og 6) and

6

=

e2p

fl ( 1 - e-a)2.

aaA+

For v E A let H, be determined by ( H , H,) = v(H). Then as a differential operator on T, H,e* = ( p , v)eP. Let H I , . . .,H,be a basis of f such that - ( H i , H j ) = Sij. Then grad(1og S) =

c H,(log 6)Hi i

= 2p(Hi)Hi

+ 2 C 1(1 - e-")-'e-"u(Hi)Hi

= - H Z p-

2

i asA+

1 e-a(l - e-')-lH,, acA+

so 4 L U )= L ,

- H 2 , - 2(

e-*")H,. asA+,k>l

Now we have

592

SOLUTIONS TO EXERCISES

Substituting this into (1) we obtain the desired formula (ii) in analogy with Eq. (12), Chapter IV, 85. For the last statement put Iz = 0 in (i). A.6. (i) Consider the mapping A from the proof of Theorem 1.8. Let I , E Z + be determined by xlxr = Ivxv. Then multiplying by A(eP)we get

Eve,,(+)

=

(

C C (det s ) m g + p - s ( r + p ) CEA

sew

1

Now H v + pE t + whereas Hs(,+p,4 t+ for s # e in W. Thus I, is the coefficient to ev+pon the left. Hence

and now the result follows from AS(i). (ii) We have xl = m,ev and a similar formula for x,,. Since m, = mp = 1 the result follows by multiplication. (iii) Let n denote the restriction of n, to the algebra g" + g-" + CH,, which is isomorphic to eI(2, C). Now apply Lemma 1.2 of the Appendix. A.8. In fact xjaj = (xJ)' so by Lemma 2.6 for A = I,

and the statement follows. A.9. The simple roots at (1 I i In) are given by (cf. [DS],Chapter 111, §8)

ai(H) = e i ( W

- ei+ i(W,

where H is a diagonal matrix of trace 0 with diagonal elements e,(H). The fundamental weights w j satisfy (mi,dsi) = dij so since ( e i , e i ) = &n + 1)- ' we obtain w j = el + . . . + ej. If u l , . . . ,on+ is the standard basis of 'L = C"" the elements uil A < i,) form a basis of A' V. A ui, (il < For t E T [maximal torus formed by the diagonal elements in SU(n + l)] we have for t = exp H

(A'J)(t)(ui, A . . . A ui,) = (ei, + . . . + ei,)(H)ui, A . . . A

ui,.

Chapter V

593

A‘J

Thus e,, + + eir is a weight of and w, = el + + e, is the highest one. The reflection s,, interchanges e, and e,+, so the Weyl group W consists of all permutations of the e,. Thus W permutes transitively the weights of so it must be irreducible. The construction of the fundamental representations of the simple Lie algebras goes back to Cartan [1913]. For Sp(n) the results are similar; one has to replace A‘ V by the subspace of primitive elements (in the sense of Corollary 2.2 of Chapter 111); for Spin(2n + 1) [the double covering of SO(2n + l)] one needs in addition to the exterior powers the spin-representation; for Spin(2n) one needs in addition to the exterior powers J, the two semi-spin-representations. For detailed description see Weyl [1939]; Boerner [1955]; Bourbaki, Groupes et algebres de Lie, Chapter 8. A.lO. With notation from Q1 let V = V,, where p runs through A(II). If u‘ E V’ is such that u’(V,) = 0 for all p E h ( n ) except po then ii(H)u’ = -po(H)u’. Thus - p o E A(ii) and dim VLp0= dim Q 0 . This proves (i). Part (ii) follows from Theorem 1.3. Part (iv) follows since - s permutes the “co-roots” 6,. A . l l . For G = SL(2, R) we have Ad(g)X = gXg-’. Since M = fl, N = ,!,( y ) we find quickly

KJ,

KJ,

1

K

=

(9 E G : Ad(g)Xl = Xl},

M N = (9E G : Ad(g)(X,

+ X3) = Xi + X3},

so the two maps are injections. If

then

(-.

d -b

9-’ =

so

+ bZ + C’ + d2)X1 - (bd + UC)XZ + $(az + bZ - C’ - dz)X3, Ad(gHXl + X3) = &az + c2)Xl - acX, + &az - c2)X3, so (i) and (ii) follow easily. For (iii) consider the horocycle N - X, = Ad(N)(X,), which by the above is the section of G / K with the plane x l = x3 + 1. Writing 2, = &Xl + X3) this has the form B(X, 2,) = - I . Ad(g)X, = &az

Since G permutes the horocycles transitively, (iii) follows.

594

SOLUTIONS TO EXERCISES

6. Fourier Series

6.1. Part (i) is clear from Lemmas 2.4 and 2.4 and then (ii) follows from Theorem 2.9. 8.2. Let A be an operator on X,. Then A = PU where U is unitary and P is the unique nonnegative square root of AA*. Then P = ciPi where ci 2 0; the c: are the eigenvalues of AA* and the Pi are mutually P i = I . Let 2c = max ci. Then we can write orthogonal projections, ci = ai + pi (1 I i I d) where lai[ = lpil = c for all i (cf. Example to Theorem 2.3). Put

1;

xi

Then V and W are unitary and since IlAll = IIAA*II’12 = 2c we have ’4 = +11’4ll(V

+ W).

This shows that the Fourier series of f can be multiplied on the left by any hyperfunction B, bounded in the sense of Eq. (44) of $2, giving a function f; E L’. By the closed-graph theorem the map B + f B is continuous, i.e., llfB1l < C sup, llB,ll where C is a constant. Restricting B to Fourier coefficients of functions g E L’ we see that the mapping g - f * g is spectrally continuous so by Theorem 2.3, J E L2(K). Remark. This corollary of Theorem 2.3 (Helgason [1957]) has been extended by Figa-Talamanca and Rider [1967] so that the assumption is only required for a set of unitary hyperfunctions r of positive measure in the product group U(d,). They also proved [1966] a certain “converse” to Exercise B2, namely that if f € L 2 ( K ) then for any p < 00 there exists a unitary hyperfunction r such that fr E L P ( K ) .For the circle group such results are classical (Littlewood [19243; Zygmund [1959], Vol. 11, 58.14). 6.4. (Sketch) Since L = L , satisfies

nleK

W,)= ( ( P , P> - (1 + P , 1 + P > W , [because of Lemma 1.6 and (16)] we see from (3) $2 that the Fourier coefficients of f~ b(U)form a rapidly decreasing hyperfunction. Conversely, let {’4,},eA(n) be a rapidly decreasing hyperfunction and define f~ C(V) by f ( u )=

1 d , Tr(’4, U,(U)), ,€I?

the series being clearly uniformly convergent. Let d U , be the differential of U,. If H E t then d U A ( H )is a diagonal matrix With diagonal formed

Chapter V by p ( H ) as p runs over the weights of

595

U,.Thus by Theorem 1.3(iv)

Tr(dU,(H)(dU,(H))*) 5 d, II I 2 I H 12, and by the conjugacy theorem for u this holds for H replaced by any X E u. Since d, is majorized by a polynomial in I (Theorem 1.8) this inequality implies quickly that the series for f remains uniformly convergent after repeated differentiation. Hence f~ 8(U). 6.5. (i) By Exercise B1 we know that fs = f * p s , where ps is a measure with Fourier-Stieltjes series

to ps is a central idempotent. The converse is obvious. Under the assumption in (ii) ps is a positive measure so by a theorem of Wendel [1954], ps * ps = ps implies that ps is the Haar measure on a compact subgroup H c K . The Fourier-Stieltjes series above then shows that S equals the annihilator H 1 so by Proposition 2.11 we are finished, (cf. Helgason [19581). The central idempotent measures for abelian groups were determined (in increasing generality) by Helson [1953], Rudin [1959], and Cohen [1960]; for classes of nonabelian compact groups by Rider [1970, 1971) and Ragozin [1972] (this latter for all simple compact Lie groups).

This Page Intentionally Left Blank

APPENDIX In this Appendix we collect some scattered results which are used in the text but which are nevertheless somewhat extraneous to the main themes. In $4-2 we have primarily followed the treatment in Dixmier [1974], but we have also used some proofs from Chevalley [1955b], Jacobson [1953], and Iwahori [1959]. Proposition 2.12 is due to Jacobson [1951]. In 53 we follow to some extent the treatment in Zariski and Samuel [1958, 19601; the proof of Theorem 3.5(ii) was kindly communicated to the author by Arthur Mattuck.

$1. The Finite-Dimensional Representations of d(2. C )

In this section we write down for reference the finite-dimensional representations of sI(2, C); these are familiar from most books on Lie algebras. The elements

H

=

(I O), 0 -1

form a basis of sI(2, C ) satisfying

(1)

[ H , X] = 2X,

[ H , Y]

=

-2Y,

Let k E Z + .Consider the linear mapping nk:sI(2, c) 591

gl(k

+ 1, c)

[X,Y]

= H.

598

APPENDIX

given by

...

where l j = j ( k - j

(diagonal matrix) -k

+ I), 1 Ij Ik.

Proposition 1.l. The mapping of sI(2, C)on Ck+' .

nk

is an irreducible representation

To see that nk is a representation one just verifies directly that nk(X), and nk(Y) satisfy the bracket relations in (1). For the irreducibility we can take k > 0. Suppose 0 # V t Ck+' is an invariant subspace. Let u E I/ - (0). If el, . . . ,ek+ is the standard basis of Ck+' we have Proof.

Zk(H),

(2)

nk(H)ei = (k - 2i

+ 2)ei

(3)

nk(Y)ei= ei+l ( 1 Ii Ik),

(4)

nk(X)e, = 0,

(1 Ii

s k + I),

nk(Y)ek+1 = 0,

nk(X)ei+, = liei (1 I i Ik).

Thus, if n E Z + is suitably chosen, nk( Y)nu is a nonzero multiple of ek+1 , whence ek+lE V. Using (4) successively, we see that each ei E V , so 1/ = Ck+l. Lemma 1.2. Let n be a representation of sl(2, C) on a jnitedimensional uector space V. Let uo # 0 be an eigenuector of n(H), n(H)u, = Avo, AEC. Then i f k E Z + ,

+

(i) ~ ( H ) n ( x ) ~=u (A , 2k)n(x)'uo, n(H)n(Y)kuo = ( A - 2k)n( Y)ku,. (ii) Suppose n(X)uo = 0. Putting

ui = n(Y)'uo, i = 0, 1,2,

...,

we haue n(X)ui = i(l - i

+ l)ui-l

so {f uio = 0, uio- # 0, we haue 1 = i, - 1.

(i 2 l),

Appendix

599

Proof. (i)

n(H)n(X)uo = n(X)n(H)u,

+ n([H,X])UO

= (A

+ 2)n(X)u,

and the first formula follows by iteration; the other is proved in the same way. For (ii) we use induction on i. For i = 1, n(X)n(Y)uo = n(Y)n(X)uo + n(H)uo = Avo; assuming formula for i we have by the first part n(X)u,+1 = n(X)n( Y)u, = n( Y)n(X)u, n(H)u, = i(A - i + l)ui + ( A - 2i)ui = ( i + l)(A - i)ui.

+

Theorem I.3. Let n be an irreducible representation of sI(2, C ) on a complex uector space V of dimension k + 1. Then n is equivalent to x k ; that is, xk(X)Au = An(X)u (u E V, X E sI(2, C))for a suitable linear bijection A : V - , Ck+'. Proof. The endomorphism n(H) of the complex vector space V has an eigenvector L' # 0, say n(H)u = pu ( p E C).Then by Lemma 1.2(i) = ( p + 2k)~(X)~u II(H)~(X)~U

and since z(H) has at most finitely many eigenvalues there exists a ko E Z+ such that Z(X)~OU # 0, n(X)ko+' u = 0. Put uo = z ( X ) ~ O U , I = p 2k0, SO n(X)u, = 0. A(H)UO = Avo, Put ui = n(Y)b0(i 2 0) and let u, be the last nonzero ui. The vectors uo, . . . , u, have different eigenvalues for n ( H ) and the space Cui is invariant, hence equals V. Thus I = k. Since (k + l)(A - k)u, = n(x)uk+ = 0 we conclude A = k. Finally, using (2)-(4) we see that the linear mapping given by uo -,e l , u1 -, e,, . . . , u k + ek+ sets up an equivalence between n and nk.

+

zb

,

Corollary 1.4. Let n be any finite-dimensional representation of

sI(2, C).Then

(5) Range n(X) n Kernel n( Y) = 0 = Range n( Y) n Kernel n(X). For the representation n = nk this is obvious from (3) and (4). Since the representation n is semisimple ([DS], Chapter 111, Exercise B3) it decomposes into irreducibles which by Theorem 1.3 have the form 'Ilk. This proves (5) in general. Corollary 1.5. Let n be any finite-dimensional representation of sI(2, C)on a uector space V . Then (i) The endomorphism n(H) is semisimple and its eigenvalues are all integers. Let V, be the eigenspace of n(H),forthe eigenualue r .

600

APPENDIX

(ii) Kernel n(X)n Range n ( X ) c (iii) Kernel A( Y) n Range A( Y) c

x,,O cr
Again it suffices to verify this in the case mediate from (2)-(4).

c

Range n(X).

c Range n( Y). A

= nk and there it is im-

52. Representations and Reductive Lie Algebras 1. Semisimple Representations

Let k be a field, g a Lie algebra over k and V a vector space over k . We assume k = R or C and that dim g and dim V are both finite. We shall now recall some notions and results about representations of g on V, i.e., homomorphisms of g into gI(V). An injective representation is usually called a faithful representation. Let n1 and n, be representations of g on V, and V,, respectively; n1 and z, are said to be equivalent if there exists a linear bijection A : V, + V, such that n,(X)Au = Anl(X)u for u E V,, X E 9. If we form the direct sum V = Vl @ V, we can define a representation n of g on V by n(X)(ul + v,) = nl(X)ul + n,(X)u,; n is then called the direct sum of n1 and n,. This can obviously be extended to several representations ni of 9. Given the representations nl: g + gl( V,), A, :g -+ gI( V'), the tensor product n1 0 n2 is the representation of g on Vl @ V, given by

0 ~Z(X)(Ul@ h )= (nl(X)ol) 0 v2 + 01 (8(nz(X)h), X E 9, u1 E Vl, v2 E V,. If K is a Lie group with Lie algebra f and pi a representation of K on ( i = 1,2) then the representation p , 0 p2 as defined in Chapter IV, #I, No. 2 has differential 711

4Pl@

P2)

= 4%0 4%.

Let n be a representation of g on V ; A is said to be irreducible if the only n(g)-invariant subspaces of V are 0 and V ; n is said to be semisimple (or completely reducible) if for each n(g)-invariant subspace W c V there exists a complementary invariant subspace w',i.e., (1)

V=

w e w',

n(g)w' c w'.

Proposition 2.1. The representation n is semisimple it is the direct sum of irreducible representations.

if and only if

Proof. Suppose n is a semisimple representation of g on V. If W c V is an invariant subspace the subrepresentation of g on W given by X +

601

Appendix

n(X)IW is also semisimple. Using this remark on W' in (1) and iterating where for each i the repwe obtain a direct decomposition V =@i=l resentation X -+ n(X)I is irreducible. Conversely, suppose we have such a direct decomposition of V. Suppose W c V is any invariant subspace. Then either W = V or some F, say is not contained in W. Put W, = W + Fl. By the irreducibility, W n = 0 so we have the direct decomposition W, = W @ El. Repeating this argument with W, in place of W we obtain if F2$ W,, and if we put W, = W, +

v

v

v1,

v2,

w, = W Q &, Q &*. The process stops when each 6 belongs to a certain V

=

Wj = W @

y.,in which case

v, @ v2@ . .. Q hj.

vp

Then the sum W' = pi=, satisfies (1). Proposition 2.2. Let n be a semisimple representation of g on V. Let V9 denote the joint null space, i.e.,

V Q= { u : x(X)u = 0 for all X

E g},

and let [n(g)V] denote the subspace of V generated by the uectors n(X)u, X E g , U E V. Then and the subspace [n(g)V] is the only n(g)-inuariant subspace W complementary to V g . Proof. If V = VQQ W, x(g)W c W, we have for each u E V, X E g, u = uo + w (uo E Va, w E W),x(X)u = n(X)w E W.Thus [n(g)V] c W. Let W, c W be a x(g)-invariant subspace complementary to [n(g)V]. If w 1E W,, then x(X)wlE [ x ( g ) V ] n W, = 0 so w 1E V* whence w 1 = 0. Thus Wl = 0 and W = [n(g)V]. Proposition 2.3. Let n be a representation of the real Lie algebra 9 on the real uector space V. Let nc denote the induced representation of' the complexification gc on the complexiJed uector space Vc, i.e.,

nc(X,+ ix,)(u,+ iu,) XjE g, U ~ EV. Then

= x(Xl)ul -

n(X2)v2+ i[n(X2)u1 + n(X,)u,],

(i) n is semisimple o nc is semisimple. (ii) If n is irreducible then either xc is irreducible or is the direct sum of two irreducible components.

602

APPENDIX

Proof. (i) -=. Let 0 : V c + V c denote the conjugation of V c with respect to V. Let W c V be a K(g)-invariant subspace. Then Wc = W + iW is a nC(gc)-invariant subspace of Vc so W c has an invariant complement Z c Vc. Then the space = (1

+ o)[Z

n (1 - o)-'(iW)]

satisfies V = w Q3 z,

n(g)Z c 2 (cf. [DS], Chapter 111, solution to Exercise B3), so K is semisimple. (i) *. For this we may assume, by Proposition 2.1, that K is irreducible and nc not irreducible. Let Z c V c be any nc(gc)-invariant subspace, Z # {0},2 # Vc. Then

vC = z + OZ, Z n ( ~ 2=) (0). (2) In fact, since o(2 + 02) = Z + oZ we have 2 + 02 = Y + iY where Y = (2 + oZ) n V. Then Y # 0 is a n(g)-invariant subspace of I/ so Y = V and Vc = Z + oZ. Similarly, Z n (oZ)= (0) so (2) holds. Also 02 is a complex subspace of V c since ao(u) = u(iiu) for a E C , U E Vc. Finally, S ( X+, x2)au= onC(x1- ix2)u (3) so a 2 is nc(gc)-invariant. Thus (2) implies that nC is semisimple. (ii) It suffices to show that the space 2 in (2) above is irreducible under nC(gC).But if U c Z is an invariant subspace, U # 0, U # 2, then (2) holds for it, which is a contradiction.

2. Nilpotent and Semisimple Elements

Generalizing the notion of nilpotent and semisimple linear transformations we shall now define such notions in a semisimple Lie algebra. Lemma 2.4. Let V be a vector space over k and g = gl(V). Let X E g be nilpotent. Then ad, X is nilpotent. Proof.

Since (ad, X ) Y = X Y - Y X we have P

(ad, X)p(Y ) =

C (i=O

so the lemma follows.

l)i

Appendix Lemma 2.5.

603

Let g = gI(n, k) and let A E g be a diagonal matrix A = dlE11

+

*

a

*

+ d,E,,.

Then ad,A(Eij) = (di - dj)Eij so in the basis (Eij)of g, ad, A is given by a diagonal matrix. Lemma 2.6. Let V be a vector space over k , A an endomorphism of V, and A = S N its additive Jordan decomposition, S semisimple, N nilpotent, and S N = N S . Let g = gl(V). Then

+

ad, A = ad, S + ad, N (4) is the additive Jordan decomposition of ad, A. Proof. Passing to the complexifications in the case k = R we may assume k = C. Then S is diagonalizable and so is ad, S by Lemma 2.5. Also Lemma 2.4 implies that ad, N is nilpotent. Since in addition

[ad,, S, ad, N ] = ad,([S, N ] ) = 0 (4) is indeed the Jordan decomposition. Proposition 2.7.* Let g be a semisimple subalgebra of gl(V). Let A E g be arbitrary and A = S + N is additive Jordan decomposition. Then S, N E g.

Proof. In the case k E R we pass to the complexification so we may assume k = C. Let Y be the set of subspaces of V invariant under g. For each W E V let

g w = { X E ~ I ( V ) : X ( W W,Tr(XIW) )~ =O}. Then gw is a subalgebra of gI(V); it contains g because, by semisimplicity, g = [g, g]. Next consider the normalizer

n

=

{A E g W : [A, 91 c d

and put

Let A E g*, put

A = ad,,(,,

(A), and let

A=S+N, * Cf. [DS],Chapter IX, Exercise A6.

A=S+IV

604

APPENDIX

s

be their additive Jordan decompositions. By (4), = ad,,,,, (S), N = ad,,,,, ( N ) . Since and fl are polynomials in A’ we have S, N E n. Since S and N are polynomials in A they leave each WE Y invariant; also Tr(NIW)=O and Tr(S)W)=Tr(AIW)=O so S , N E g w . Thus S and N belong to g*. It suffices now to prove g* = g. Since g is an ideal in g* the Killing form B of g is the restriction of the Killing form B , of g*. Hence B , is nondegenerate on g so we have the direct decomposition g* = g @ g1 where the orthogonal complement g1 is an ideal in g*. Now since the representation of g on V is semisimple (Weyl’s theorem) we have V = 6 where each K is girreducible. Each X E g1 commutes elementwise with g so by Schur’s lemma it acts on K by a scalar multiplication. Since Tr(X1 6 ) = 0 we conclude XI 6 = 0 so X = 0 and g = g*.

s

Oi

Corollary 2.8. With the notation of Proposition 2.7, A E g is semisimple (nilpotent) if and only if ad, ( A ) is semisimple (nilpotent).

Let A = S + N be the additive Jordan decomposition. Since ad,(A) = ad,,(,, ( A )I g we deduce from Lemma 2.6 that ad, ( A ) = ad, ( S ) ad, ( N ) is the additive Jordan decomposition of ad,(A). Now A = S o a d , ( A ) = ad, (S), so the corollary is verified.

+

Corollary 2.9. Let g be a semisimple Lie algebra, A E ~ The . following conditions are equivalent:

(i) ad, ( A ) is semisimple (resp. nilpotent). (ii) There exists a faithful representation n of g such that n(A) is semisimple (resp. nilpotent). (iii) For each representation n of g n(A) is semisimple (resp. nilpotent). Proof. (iii) * (ii) is obvious; (ii) * (i) by Corollary 2.8. Finally, (i) * (iii) as follows: assuming ad, ( A ) semisimple, consider a representation n of g on V. Let g’ = g/Kernel(n), view g‘ as the subalgebra n(g) c gI(V), and put A‘ = p(A) where p : g + g’ is the natural mapping. If D c g‘ satisfies [A’, D] c o then [ A , p - ’ o ] c p - ’ o so there exists a u c g such that p - ’ o Q u = g, [A, u] c u. But then p(u) satisfies o Q p(u) = g’ and [A’, p(u)] c p(u). Thus ad,. (A’) is semisimple. By Corollary 2.8 used on n(g) = g‘ c gI(V) we conclude that A’ = n(A) is semisimple as desired. The nilpotent case of (i) * (iii) is proved in a similar way. Definition. Let g be a semisimple Lie algebra. An element A E g verifying the conditions of Corollary 2.9 is said to be semisimple (resp. nilpotent).

605

Appendix

Corollary 2.10. (i) Let g be a semisimple Lie algebra. Then each A E has ~ a unique decomposition (5)

A=S+N

where S E g is semisimple, N E g is nilpotent. . (ii) Let g1 c g be a semisimple subalgebra. If A E ~ then ~ , S, N € g l . An element A E g1 is semisimple (resp. nilpotent) in g1 if and only if it is semisimple (resp. nilpotent) in g. This follows directly from Proposition 2.7. Definition. Decomposition (5) is called the additive Jordan decomposition of A .

3. Reductive Lie Algebras Reductive Lie algebras are natural generalizations of semisimple Lie algebras; in this subsection we prove for them a few results which are useful primarily in Chapter 111, &A I.Lie algebra will mean a Lie algebra over R or C and a representation will mean a finite-dimensional representation. Let g be a Lie algebra, b c g a subalgebra. Definition. The subalgebra lj c g is said to be reductive in g if the representation H + ad, (H)of t, on g is completely reducible (i.e., semisimple). A Lie algebra g is said to be reductive if it is reductive in itself. Obviously, if t, is reductive in g then b is reductive. Proposition 2.11. direct sum

(6)

A Lie algebra g is reductive fi and only

g=a+s

of an abelian ideal a and a semisimple ideal

(7)

if it is the

a = center(g),

5

5.

=

I n that case,

[g, g],

Proof. Since the representation ad,, is semisimple g has a decomposition g = gi into subspaces on which ad, acts irreducibly. In particular, each gi is an ideal of 9. Each ideal of gi is an ideal in g so ([DS], Chapter 11, 86; Chapter 111, Exercise B8) gi is either a simple Lie algebra or one-dimensional. Decomposition (6) now follows by taking 5 as the sum of the simple gi and a the sum of the l-dimensional gi. The converse is clear since by the semisimplicity of 5, ad, (9) = ad, (4) is semisimple.

606

APPENDIX

Proposition 2.12. Let g be a reductive Lie algebra (over R or C ) and n a representation of g on a jinite-dimensional vector space V. Then n is semisimple if and only if the restriction nlcenter (9) is semisimple. Proof. We shall use the familiar fact from linear algebra that a commutative family of endomorphisms is semisimple if and only if each member of the family is semisimple. Assume first n is semisimple and consider the decomposition (6). Fix HE^ and write n(H) = N + S where N and S are the nilpotent and semisimple parts of n(H), respectively. Let W be the null space of N. Since N is a polynomial in n(H) ([DS], Chapter 111, $1) N commutes with each element in n(g). Hence n(g) leaves the subspace W c V invariant. Let W' c V be a complementary invariant subspace. Since n(H)W' c W' we have NW' c W' and the restriction N' = NIW' is nilpotent. But since W n W' = 0, N has no null vector x # 0 in W'. But, N' being nilpotent, this implies N = 0 ([DS], Chapter 111, $1). Thus N = 0, so n(H) is semisimple, whence by the initial remark, nla is semisimple. For the converse we may, by Proposition 2.3, assume that g is a Lie algebra over C.The commutative family n(a) now assumed semisimple, Then each V, is n(g)-invariant and n(g)IV, = n(e)lV,. Since 5 is semisimple this last family of operators is semisimple. Thus the representalet V = Oi be the direct decomposition into joint eigenspaces of n(a). tion x i : X --* n(X)lV, of g on V, is semisimple. Decomposing this into irreducibles we conclude from Proposition 2.1 that n is semisimple.

Proposition 2.13. Let g be a reductive Lie algebra, n1 and nz two semisimple representations of g. Then n1 @ nz is semisimple. Proof. Because of Propositions 2.1 and 2.3 we may assume g complex, n, and n2 irreducible. Using (6) we see that n,(a) commutes elementwise with nl(g) so by Schur's lemma consists of scalar multiples of the identity. The same holds for nz(a) and therefore also for (nl0 nz)(a), whence, by Proposition 2.12, ni @ a, is semisimple. Remark. The result holds also for g nonreductive (see, e.g., Dixmier [1974], Chapter I).

Proposition 2.14. Suppose the subalgebra and let n be a representation of g on V.

(i) Let W be the sum of the subspaces irreducible under n($). Then n(g) W c W. (ii) Zf n is semisimple, so is nlIj.

r,

of g is reductive in g

VC: of V which are invariant and

Appendix

607

w

c V be a subspace which is n(b)-invariant and irreProof. Let ducible and consider the representation p : X + ad,(X) of 9 on g. By Prop. 2.13 the representation p @ (nib) of lj on g @ y is semisimple. Consider now the linear map A : g @ W, + V given by A(X @ w ) = n(X)w. Then A commutes with the action of 6; in fact if Y Eb

A((p 0 (nlb))(Y)(X @ 4 )= AUY, XI @ w + x 0 n(Y)w) = n([Y,X ] ) o + n(X)n(Y)w = n(Y)A(X @ w). Hence we have a semisimple representation of

b

on A(g @ 4.).Thus

n(g)w = A(g @ q)c W. This proves (i). For part (ii) we may assume n irreducible. Then by (i) W = V. Thus V = and by irreducibility n y. = 0 unless = H$. Thus V is the direct sum of certain so nl t, is semisimple. Corollary 2.15. If t, is reductive in g and g reductive in f, then b is

w.

ciw

w.

reductive in f. Proposition 2.16. Let f be semisimple and g reductive in f. If X E ~ and ad, ( X ) nilpotent, then X E [g, g]. Proof. We may assume g and f complex. Let X = C A where C E center(g) and A E [g, g] and let A = S + N be the additive Jordan decomposition of A in [g, g]. Let n = adfig. Then n is a faithful semisimple representation of g which has a decomposition n = Oixi into irreducibles a,. By 2.9 for [g, g], n(N) is nilpotent, n(S) semisimple. By Prop. 2.12, n(C) is semisimple, so since n(S) and n(C) commute, n(S + C ) is semisimple. Since n ( X ) = n(N) + n(S + C ) is nilpotent and n ( N ) and n(S + C) commute it follows that n(S + C ) = 0. For each i ni(C) commutes elementwise with the irreducible ni(g), so is a scalar, and Tr(n,(S)) = Tr(n,(A)) = 0. Hence ni(C)= 0, so n(C) = 0 and C = 0, as desired.

+

53. Some Algebraic Tools We recall that "field" always means commutative field of characteristic 0. In this section we only consider rings and algebras which are commutative, have an identity element, and no divisors of 0. A module M over a ring A is called finite if there exist finitely many elements x l , . . . , x,,, E M such that M = Ax, + + Ax,,,. Definition. Let B be a ring and A a subring of B (with the same identity). An element X E B is called integral over A if it satisfies an equation (1)

X"+a,x"-'+**.+a,=O

608

APPENDIX

with leading coefficient 1, where a, E A. If each x E B is integral over A, the ring B is said to be integral over A. Lemma 3.1. Let A be a subring of B and let x E B . The following conditions are equivalent:

(i) x is integral over A. (ii) The ring A [ x ] (the subring of B generated by A and x ) is a,finite A-module. (iii) There exists an intermediate ring R, A [ x ] c R t B, such that R is ajnite A-module.

E':

Proof. (i)*(ii) There exists an integer n>O such that x n € i = o A x i . It follows that xn+' E X f Z d Axi+' so by induction on r, x"+'E i=o Ax' for each r > 0. (ii) +(iii) Take R = A [ x ] . (iii) =s (i) Select y,, . . . ,y , E R such that R = Ay,. Then we have for suitable elements aij E A, xyi = a i j y j (1 5 i In). Writing this system of linear equations as

,

n

C ( a i j x- aij)yj = 0

(1 Ii In),

j= 1

we conclude that the determinant d = det (dijx - aij) satisfies dy, = 0 for each i. Then dR = 0 so d = 0. But this is an equation of the form (1) so x is integral over A. By induction we conclude from Lemma 3.1, Lemma 3.2. Let x l r . ..,x , be elements of a ring B which are integral over a subring A. Then the ring A [ x , , . . .,x,] (the subring of B generated by A and x l , . . .,x,) is a finite A-module.

From this lemma and Lemma 3.l(iii) we obtain the following: Corollary3.3. Let A be a subring of a ring B. The set of elements in B which are integral over A form a subring A of B containing A. Definition. The subring

A is called the integral closure of A in B.

Definition. A ring A is called integrally closed if it coincides with its integral closure in the quotient field C ( A ) of A.

Lemma 3.4. A unique factorization domain A is integrally closed. Proof. Let x E C(A) be integral over A. Then x = a/P where we may assume that a and fl are relatively prime. Now x satisfies an equation

x"

+ a,x"-' +

+ a, = 0

609

Appendix so

+

a" a,a"-lfi + . - .+ anp" = 0. If fi has a prime factor y, then y must divide u" and therefore y divides a. This contradicts the fact that a and /? are relatively prime. Hence x E A. In particularly, the symmetric algebra S( V ) over a finite-dimensional vector space V is integrally closed. Theorem 3.5. Let K be an algebraically closed field and A and B finitely generated algebras (with identity) over K . Suppose that A c B and that B is integral over A. Then (i) Each homomorphism cp: A K extends to a homomorphism 11/ of B into K . (ii) I f each homomorphism q : A K extends uniquely to a homomorphism of B into K then the quotient fields C(A) and C ( B ) coincide. -+

-+

Proof of (i). If q(A) = (0)we define t,b by $(B) = (0).If cp(A) # (0) then the kernel of q is a maximal ideal m of A . We first prove the existence of a maximal ideal n of B such that n n A = m. For this purpose consider the set of all ideals p of B satisfying p n A c m. This set is partially ordered under inclusion and every (totally) ordered subset has an upper bound. By Zorn's lemma there exists a maximal element n of the set. If n nA is a proper subset of m, let x be an element in m which does not belong to n. Then n is a proper subset of the ideal n + B x ; by the choice of n, (n + Bx) n A is not a part of m. In other words, there exists an element z E B and an element y~ A, not in m, such that z x - y E n. Now z satisfies an equation z" + alz"-' + - + a, = 0, aiE A, which after multiplication by x" and use of the congruence z x = y (mod n) gives y" + a,xy"-' + - - - + a,x" = 0 (mod m).

This contradicts x E m, y i$ m. Hence n n A = m. If n were not a maximal ideal in B, suppose n' is a maximal ideal in B satisfying the proper inclusions n c n' c B. Then n' n A is an ideal in A properly containing m so A c n'. Let x E B. Since x is integral over n' we have x1E n' for some integer 1. Since B/n' is a field we conclude that x En'. Thus n' = B, which is a contradiction. This proves that n is a maximal ideal in B so B/n is a field. Let 1 denote the identity of A. Then the mapping a + a . 1 ( a E K ) is an isomorphism of K into A and the mapping a -+ a . 1 + n is an isomorphism of K into B/n. Now select b l , . . ., b,E B such that B = K [ b l , ..., b,]. Then B = A [ b , , ..., b,] so by Lemma 3.2, B is a finite A-module. Writing B = Ax, Ax, ( x i E B) and using A = K + m we see that B/n is a finite extension of K . Since K is algebraically closed we have B/n = K . The natural mapping t,b : B --+ K gives the desired extension of q.

+ +

610

APPENDIX

Proof of (ii). In order to prove (ii) we make use of the following theorem (the Noether normalization theorem, see, e.g., Zariski and Samuel [1958], p. 266).

Let R = k b , , . . . ,y,] be a finitely generated algebra over a field k and let d be the transcendence degree of' the quotient field k(y,, . . ., y,) over k. There exist d linear combinations z,, .. . ,z d of the yi with coeficients in k, algebraically independent over k, such that R is integral over k[z,, . . . ,z d ] . Combining this theorem with (i) we see that if k is algebraically closed there exists a homomorphism of kCyl, . . . , y,] onto k. Suppose now that the quotient fields C(A) and C(B) were different. Let a,, , , . , a , be a set of generators of A so A = K [ a l , ..., a,]. Pick any element a E B which does not belong to C(A). We shall find a homomorphism of K[a,, ..., a,,,] into K which has more than one extension to a homomorphism of K[a,, , . . ,a,, a ] into K . Since B is integral over K [ a l , .. ., a,, a], (ii) will then follow from (i). Let p,(x) = x"

+ fl(a)xn-' + -..+ f,(a)

=0

be the polynomial with coefficients in the field C(A) = K ( a l , . . . , a,) of lowest degree having CI as a zero and leading coefficient 1. [Here a stands for the rn-tuple ( a l , ..., a,)]. Let q(a) denote the product of all the denominators of all the fE(a) with the discriminant of the polynomial p,(x). From the remark above we see that there exists a homomorphism cp of K [ a l , ..., a,, l/q(a)] onto K. The image of the polynomial p,(x) under cp will then be a polynomial with coefficients in K having n distinct roots, say a l , . . .,a,. Fix one ai.We wish to extend cp (or more precisely, the restriction of cp to K [ a l , . . . ,a,]) to a homomorphism $i :K [ a l , . . . ,a,, a ] + K by putting t,+i(~) = ai. The condition for this being possible is that whenever a satisfies a polynomial equation p(a, x) = 0 with coefficients in K [ a l , , ,.,a,,,] then ai satisfies the corresponding equation p(cp(a), x ) = 0 with coefficients in K. Since the polynomial p,(x) has minimum degree it divides p(a, x): ~ ( aX), = pa(x)da, XI. Here the polynomial q(a, x ) can be found by long division; its coefficients gi(a) are rational expressions in a l , . . .,a, whose denominators divide the product of the denominators of the fi(a). Since cp does not map this product into 0 it is clear that p(cp(a), x) vanishes for x = M i . Now n > 1 and the homomorphisms $i(1 Ii I n) are all different. This concludes the proof.

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SYMBOLS FREQUENTLY USED Ad: adjoint representation of a Lie group, 87, 282 ad: adjoint representation of a Lie algebra, 282 A ( r ) :spherical area, 165 &): component in g = n exp A(g) k, 303 a': coroot, 192 A,,: root vector, 267 d ( B ) : space of analytic functions on B, 35, 278 d ' ( B ) : space of analytic functionals (hyperfunctions) on B, 35, 278 a, a', a*, a: abelian subspaces and their duals, 181, 303 'a: : subset of a:, 434 a+, a*+: Weyl chambers in a and a*, 181,430 B,(p): open ball with radius r, center p, 44, 118, 156 B : Killing form, 164 jA(0):ball in P,118 Card: cardinality, 188 Cl: closure, 3 conj : complex conjugate, 39 1 C :complex n-space, 15 C,: special set, 136 C(X):space of continuous functions of X, 3 C,(X):space of continuous functions of compact support, 3 C'"(X): space of functions with continuous partial derivatives of order < m , 3 Cm(X),C:(X): set of differentiable functions, set of differentiable functions of compact support, 3, 239 ~(1):Harish-Chandra's e-function, 39, 430, 434 C+,-C,'C,C - : closures of Weyl chambers and their duals, 459

L@(X):C:(X), 3, 239 9'(X): set of distributions on X, 3, 240 L@,(X): set o f f € d with support in K, 239 g H ( P )subspace : of 9(P),100 .9'(X), @'(x):space of K-invariant elements in g((~), o'(x), 43 O'(G), 9"(G): space of K-bi-invariant members of O(G), 9'(G), 292 D(G): set of left-invariant differential operators on G, 274 D,(G): subalgebra of D(G), 284 D(G/H): set of G-invariant differential operators on G/H, 274 d(6) or dd: dimension (=degree) of a representation, 391 A@): radial part of D, 259 6: density function, 260 E(M): set of all differential operators on M,241 646

SYMBOLS FREQUENTLY USED I ( X ) : Cm(X),3, 239 b ' ( X ) , dP"(X): space of K-invariant elements in b ( X ) , B'(X), 292 I, eigenspace, : 12, 35, 279 Ea,b: space, 4 E': space of entire functionals, 5 F(a, b ; c; 2): hypergeometric function, 50, 484 4A:spherical function, 38, 418 @A,,,:

generalized spherical function, 50

G(d, n), Gd,":manifolds of &planes, 124 GL(n, R), GL(n, C): classical groups, 222 X(2):space of holomorphic functions on Z, 15 H " ( C ) :exponential type, 15 Hom(V, W): space of linear transformations of V into W,392 H":hyperbolic space, 29, 152, 177 H (E): space of harmonic polynomials on E, 346 X : Hilbert transform, 113 H(g): component in g = k exp If(&, 277 Im: imaginary part, 2 I@): space of invariant polynomials on E, 346 Ip,q: matrix, 201

Iy:Riesz potential, 135 J,(z): Bessel function, 11 X ( X ) : C,(X), 396 xa: character of 6, 391 t: algebra in Cartan decomposition, 186 L ' ( X ) : space of integrable functions on X,414 Lp(X):space off with I f l p E L'(X), 510 L : diameter of X, 164 L = L,: Laplace-Beltrami operator on X,31, 244 L(g) = L,:left translation by g, 85 I: left regular representation, 386 I(6): dimension, 394 I(s): length function, 442 I: symmetrization, 282 A: set of integral functions (weights), 498, operator, 113, root lattice, 307 A( +): set of dominant integral functions, 498 A(n): set of weights of x, 498 A(E): Grassmann algebra over E, 354 M,: the tangent space to a manifold M at p, 84 M': mean-value operator, 103, 288 M(n): group of isometries of R", 4 n(s): cardinality of , : Z 441 N :set of nilpotent elements, 369 O(p, 4):orthogonal groups, 200 Q,: area of P-', 18 P": set of hyperplanes in R", 97 p: part of a Cartan decomposition, 181 R": real n-space, 3 R": set of reals 20, 2 Re: real part, 2

647

648

SYMBOLS FREQUENTLY USED

R , or R(g): right translation by g, 85 r: right regular representation, 386 Res: residue, 133 P : half sum of roots, 181 r : set of regular elements, 369 J ) : set of quasi-regular elements, 369 S": n-sphere, 3 S,(p): sphere or radius r and center p, 103, 312 Y(R"):space of rapidly decreasing functions on R" Y*(R"),Y,(R"):subspaces of Y(R"),104 Y(R"): space of tempered distributions, 131, 132 S ( V ) : symmetric algebra over V, 346 sgn(x): signum function, 114 YH(P"): subspace of Y(P"),99, 100 sh x: sinh x, 36 Z, Z', Z;, :X sets of restricted roots, 441 9 :set of semisimple elements, 369 th x: tanh x, 76 'A: transpose of A, 388 Tr(A): trace of A, 221 7(x): translation on G / H , 88 U(g): universal enveloping algebra of g, 503 U(n): unitary group, 515 V,: representation space of 6, 391 V r : space of fixed vectors under 6(M), 394 V ( 6 ) : space of K-finite vectors of type 6, 395 V " : space of differentiable vectors, 387 W :Weyl group, 194 2, 2' : the integers, the nonnegative integers, 2 Z(G): center of D(G) Z(g): center of the universal enveloping algebra U(g), 503 -: Fourier transform, 4, 33, 457; spherical transform, 39, 415, 449 -: Radon transform, 76, 97; incidence 140 -: Dual Radon transform, 97; incidence, 140 *, x : convolutions, 22, 43, 44, 99, 119, 132, 289, 290; adjoint operation, 241, 513; pullback, 83; star operator, 330 C 3I : direct sum, 601 @ : tensor product, 39 1 A : exterior product, 354 (, ): inner product, 18, 29, 32, 186, 347 ', E': space of K-invariants in E, 43, 292 11 :;1 norm on Cm(V), 234 0 : operator 98, 337; wave operator, 208 Set theory. Let A and B be sets. We use the usual symbols€, c ,n, u for being element of, inclusion, intersection, and union. The set A - B is the set of elements in A not in B. If 9 is a property and M a set, then {x E M :x has property 9 }denotes the set of elements x E M with property 8.The composite of two maps f:A + B and g:B + M is denoted g o f : A 4M. The sign * means "implies."

SYMBOLS FREQUENTLY USED

649

Algebra. If K is a subgroup of a group G, the symbol G/K denotes the set of left cosets g K , g E G. When K is considered as an element in G / K it will sometimes be denoted by { K } . If x E G, the mapping gK xgK of G / K onto itself will be denoted by T(x). Let A: V -+ W be a linear mapping of a vector space V into a vector space W.Let V* and W * denote the respective duals. The mapping 'A: W *+ V* defined by 'A(w*)(u) = w*(Au) is called the transpose of A. A linear map A: V + V will often be called an endomorphism of V. If V has finite dimension the determinant and trace of A will be denoted by det(A) and Tr(A), respectively. Topology. Let X be a topological space, A c X a subset. The closure of A is denoted A or Cl(A), the interior of A by A. Iff is a function on X its restriction to A is denoted f l A. The Hausdorff separation axiom for a topological space will always be assumed. A coset space G / H where H is a closed subgroup of the topological group G will always be taken with its natural topology which is characterized by the fact that the mapping g -+ g H of G onto G / H is continuous and open. A topological vector space V (over R or C) can be topologized by a family of seminorms if and only if it is locally convex. This family can be chosen countable if and only if V is metrizable. If in addition V is complete it is called a Frkchet space. Manifolds. If M is a manifold and p E M, the tangent space to M at p is denoted by M,. If 6:q -+ (x,(q), .. . ,x,(q)) is a system of coordinates near p the mappings -+

form a basis of M,. Let @: M -+ N be a differentiable mapping of M into a manifold N . Its diflerential at p. mapping M , into NO(,)is denoted d@,,. If q is a form on N its pullback by @ is denoted @*q; it is a form on M . Lie groups. Lie groups are usually denoted by capital letters and their Lie algebras by the corresponding lower case German letters. The exponential mapping of a Lie algebra into a corresponding Lie group is denoted exp. The identity component of a Lie group G is denoted Go. We use the standard notation for the classical groups: GL(n, C), GL(n, R), SL(n, C), S U n , R),U P ,4). SU@, q), SU*(2n), SO(n, C),O@,q), S0*(2n), Sp(n, C),S p h R),and Sp@, 4). See, for example, [DXJ, Chapter X, 52 for detailed descriptions.

INDEX Convex hull, 459 Convolution, 42, 44, 132, 289, 290, 407 mixed, 119 Cusp forms. 148

A

Abel's integral equation, 41, 106, 488 Adjoint representation, 282 Admissible, 419 Alternation, 192 Antiderivation. 354 Antipodal manifold, 164 Area, 312

D Darboux equation 159, 288 Decomposition Bruhat, 196 Cartan, 186 Iwasawa, 181 Density function, 260 Diagram, 190 Diameter, 164, 312 Differentiable vector, 387 Differential operator adjoint of, 241 characterization, 236 definition, 236, 239 globally solvable, 275 image of, 241 invariant, 241, 274 projection on a submanifold, 251, 300 radial part of, 259, 335 separation of variables, 255 solvable, 275 symbol of, 277 transversal part of, 255 Distinguished, 548 Distribution compact support, 3, 240 definition, 3, 240 derivative of, 131 support of, 3, 240 tempered, 13I Divergence, 243, 246 Double fibration. 141

B Bessel function, I I Boundary component, 463

C

Cartan motion group, 424 product, 545 subalgebra, 373 subspace, 374 Casimir operator, 331, 333 Cauchy principal value, I38 Cauchy problem 127, 130, 342, 343 Cayley hyperbolic space, 177 Cayley plane, 167 Central, 495 Cone light cone, 203, 205 null cone, 203 retrograde cone 139, 205 Conformal, 331 diffeomorphism, 332 vector field, 332 Conjugate point, 164 Convergent, 529 absolutely convergent, 529 650

lNDM

E Eigenfunction, 275 joint, 275, 402 Eigenspace, 275 representation, 2, 275 Elliptic space, 161 Euler angles, 225 Exponential type, 15, 43, 448 entire function of, 43 uniform, 33 Exponentially decreasing function, 179

F Finite module, 607 Flag manifold, 142 Fourier coefficients, 507. 508 Fourier series absolutely convergent, 519 for analytic functionals, 62 for entire functionals, 7 Fourier-Stieltjes series, 514 of a function, 507, 508 lacunary, 525 Fourier transform in R", 4 of distributions, 131 non-Euclidean, 33 on a sphere, 78 on a symmetric space, 457 Functional analytic, 35. 278 entire, 5

G Generalized sphere, 149 Generating function, 353 Geodesic polar coordinates, 313 Gradient, 243 Grassmann manifold, 124, 223 Group algebra, 407

H Harmonic analysis, 2

65 1

function, 14, 403 polynomial, 17, 345, 360, 380 Heisenberg group, 276 Hilbert transform, 113 Hilbert's Nullstellensatz, 18, 351, 380 Homogeneous spaces in duality, 141 polynomial, 17 subspace, 347 two-point homogeneous, 150 Horocycle, 3 I Huygens's principle, 130, 343 Hyperbolic plane, 29, 75 space, 152 Hyperfunction, 35, 278, 547 rapidly decreasing, 548 unitary, 547 Hypergeometric equation, 484 confluent, 67 Hypergeometric function, 60 Hypergroup, 523 normal subhypergroup, 523 subhypergroup, 523

I Incident, 140 Inductive limit, 5, 240 strict, 396 Integral, 607 dominant integral function, 498 integral function, 498 Integral closure, 608 Integrally closed. 608 Invariant differential operator, 2, 241, 274 exponential, 192 exterior, 363 polynomial, 345 quasi-invariant, 74, 332 Inversion formula Fourier transform, 4, 21, 33, 78 Radon transform, 96, 110, 112, 113, 120, 121, 159, 166, 178 spherical transform, 40, 454 Isotropic manifold, 150, 199 vector, 15, 203

652

INDEX

J Jacobi polynomial, 543 Joint eigenfunction, 2 eigenspace, 2 Jordan decomposition, 605

K

N Nilpotent element, 369, 604 principal, 369 Non-Euclidean Fourier transform, 33 Laplacian, 31 Plane, 29 Normed algebra, 409

K-finite, 72, 395, 278, 419 of type 6, 395

0 L Laplace transform, 7, 15 Laplace-Beltrami operator, 31, 242 invariance of, 246 Left-invariant differential operator, 280 form, 85 measure, 86 Legendre function, 406 polynomial, 404 Leibnitz rule, 234 Length function. 442 LF-space 398 Lie derivative, 243 Lipschitzean, 342 Lorentz group, 276 Lorentzian, 199

M Mean value operator, 103, 288, 319, 584 Measure, 81 equivalent to Lebesgue measure, 241 Haar, 89 Haar measure in canonical coordinates, 96 invariant, 90 positive, 81 Riemannian, 83, 243 Modular group, 148, 222 Multiplicities, 165, 181 Multiplier, 547

Orbital integrals, 149, 207

P Paley-Wiener theorem, 15, 33, 109, 450 Partition function, 544 Plancherel formula Fourier transform, 4, 21, 33 Radon transform, 116 spherical transform, 40, 454 Plane wave, 31, 97 Poincare group, 276 inequality 250 model, 29 Poisson equation, 135, 315 integral, 69 kernel, 69 transform, 279 Positive definite, 389 root, 271 strongly positive sequence, 474 Primitive, 355, 356 Projection of a differential operator, 251 N-projection. 454 Proper map, 469 Pseudo-Riemannian manifolds, I99 of constant curvature, 200 quadric models, 200 Pullback, 83

653

INDEX

Q Quasi-invariant, 74 Quasi-regular, 369

R Radial part, 259 distributions, 335 examples, 265, 334 Radon transform applications. 126 for distributions, 118, 146 for a double fibration, 143 for &planes, 123 dual transform. 97 for hyperplanes, 97 totally geodesic submanifolds, 152. 161 two-point homogeneous spaces, 166, 177 Rapidly decreasing function, 99, 179 Reduced, 442 Reductive homogeneous space. 284 Lie algebra. 605 subalgebra, 605 Reflection. group, 356 Regular. 369, 383 regular normalization, 449 Representation, 274 adjoint. 282 character of. 391 coefficient, 386 completely reducible, 600 contragredient. 388 differential of. 388 direct sum, 391 eigenspace, 2. 275 equivalent, 386 faithful, 600 fundamental, 500 irreducible. 2. 274 K-finite. 419 left regular. 386 quasisimple. 388 right regular, 386 scalar irreducible, 74 semisimple, 600 spherical, 410 tensor product. 391

topologically irreducible, 274 unitarily equivalent. 414 unitary, I , 386 weakly equivalent, 419 weights of, 496 Ricci curvature, 332 Riesz potentials, 135, 139, 21 I , 213 Right-invariant, 85 differential operator, 284 form, 85 measure, 87

S Scalar curvature, 332 irreducible, 74 Schur's orthogonality relations, 391 Schwartz space, 100, 483, 489 Semireflexive. 388 Semisimple element. 369. 603 endomorphism, 369 representation, 600 Simple, 46 Simply convex, 248 Singular, 383 Skew, 192. 271, 504 sl, triple. normal. 370 Solvable, 275 globally, 275 Spacelike. 203 Spectrally continuous, 51 1 Spectral norm, 51 1 Spherical bi-spherical function, 423 function, 38, 399 generalized spherical function. 53 harmonics, 19 spherical function of class, 6 , 424 transform, 38, 399 ultraspherical polynomial. 20, 78 zonal spherical function, 424 Stirling's formula. 7 Strong topology, 388 Support theorems for distributions, 119 for H". 153

INDEX

for R", 105

U

for X-ray transform, 179 Symmetric algebra, 280 operator, 245 pair, 291 space, 186 vector field, 333 Symmetrization, 282

Unimodular. 88 Unit sphere bundle, 337

V

Vector field, induced, 254

W

T Timelike. 203 Totally geodesic, 150 Transform, 83 dual transform, 97 Fourier transform, 4, 33. 78, 131 Radon transform, 97 X-ray transform. I78 Transpose. 388 Tranwersaliiy condition, 254, 259 Two-point homogeneous space, 150

Wave equation, 127, 130, 342 .343 operator, 208 plane, 97 Weight, 496 fundamental, 498 highest, 498 multiplicity of. 496

X

X-ray transform, 123, 178

Pure and Applied Mathematics A Series of Monographs and Textbooks Editors

Samuel Ellenberg and Hymen Bass Columbia University, New York

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