Differential Geometry, Lie Groups, and Symmetric Spaces [Reprint ed.] 0821828487, 9780821828489

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Selected Titles in This Series 34 Sigurdur H e l g a s o n , Differential geometry, Lie groups, and symmetric spaces, 2001 33 D m i t r i B u r a g o , Yuri B u r a g o , and Sergei Ivanov, A course in metric geometry, 2001 32 R o b e r t G. B a r t l e , A modern theory of integration, 2001 31 Ralf K o r n and Elke K o r n , Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. M c C o n n e l l and J. C. R o b s o n , Noncommutative Noetherian rings, 2001 29 Javier D u o a n d i k o e t x e a , Fourier analysis, 2001 28 Liviu I. N i c o l a e s c u , Notes on Seiberg-Witten theory, 2000 27 Thierry A u b i n , A course in differential geometry, 2001 26 Rolf B e r n d t , An introduction to symplectic geometry, 2001 25 T h o m a s Friedrich, Dirac operators in Riemannian geometry, 2000 24 H e l m u t K o c h , Number theory: Algebraic numbers and functions, 2000 23 A l b e r t o Candel and Lawrence C o n l o n , Foliations I, 2000 22 Giinter R. K r a u s e and T h o m a s H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 J o h n B . C o n w a y , A course in operator theory, 2000 20 R o b e r t E. G o m p f and A n d r a s I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and M a r t i n Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 H e n r y k Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V . K a d i s o n and J o h n R. R i n g r o s e , Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V . K a d i s o n and J o h n R. R i n g r o s e , Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N . V . K r y l o v , Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 J a c q u e s D i x m i e r , Enveloping algebras, 1996 Printing 10 B a r r y S i m o n , Representations of finite and compact groups, 1996 9 D i n o Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried J u s t and M a r t i n W e e s e , Discovering modern set theory. I: T h e basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 J e n s C a r s t e n J a n t z e n , Lectures on quantum groups, 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. G o r d o n , The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 W i l l i a m W . A d a m s and P h i l i p p e L o u s t a u n a u , An introduction to Grobner bases, 1994 2 Jack Graver, B r i g i t t e Servatius, and H e r m a n Servatius, Combinatorial rigidity, 1993 1 E t h a n A k i n , The general topology of dynamical systems, 1993

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Differentia l Geometry , Lie Groups , an d Symmetri c Space s

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Differentia l Geometry , Lie Groups , an d Symmetri c Space s Sigurdu r Helgaso n

Graduate Studies in Mathematics Volume 34

\\| American Mathematical Society Providence, Rhode Island v ^ ..

Editorial Board Steven G. Krantz David Saltman (Chair) David Sattinger Ronald Stern

2000 Mathematics Subject Classification. Primary 22E15, 22E46, 22E60, 22F30, 32M15, 43A85, 43A90, 53B05, 53B20, 53C35. ABSTRACT. This book is a textbook and a reference work on the three topics in the title. The book begins with a self-contained exposition of differential geometry, with emphasis on Riemannian geometry. This is then applied to a treatment of the basic theory of Lie groups and Lie algebras. The structure theory of semisimple Lie algebras is developed in considerable detail, ending with a complete classification. This theory is intertwined with the theory of symmetric spaces which originated with the work of Elie Cartan in the late 1920s. While the classification of symmetric spaces had been done in at least two different ways we use here the method of V. Kac which at the same time gives a classification of finite order automorphisms of simple Lie algebras over the complex numbers.

Library of C o n g r e s s Cataloging-in-Publication D a t a Helgason, Sigurdur, 1927Differential geometry, Lie groups, and symmetric spaces / Sigurdur Helgason. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 34) Originally published: New York : Academic Press, 1978, in series: Pure and applied mathematics (Academic Press) ; 80. Includes bibliographical references and index. ISBN 0-8218-2848-7 (alk. paper) 1. Geometry, Differential. 2. Lie groups. 3. Symmetric spaces. I. Title. II. Series. QA641 .H464 516.3'6—dc21

2001 2001022205

C o p y i n g and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. © 1978 held by Sigurdur Helgason. All rights reserved. Reprinted with corrections by the American Mathematical Society, 2001, 2012. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2

17 16 15 14 13 12

To Artie

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CONTENTS PREFACE

xiii

PREFACE TO THE 2001 PRINTING

xvii

SUGGESTIONS TO THE READER

xix

SEQUEL TO THE PRESENT VOLUME

xxi

GROUPS AND GEOMETRIC ANALYSIS CONTENTS

xxiii

GEOMETRIC ANALYSIS ON SYMMETRIC SPACES CONTENTS

xxv

CHAPTER I Elementary Differential Geometry 1. 2.

3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Manifolds Tensor Fields L Vector Fields and 1-Forms 2. Tensor Algebra 3. The Grassman Algebra 4. Exterior Differentiation Mappings 1. The Interpretation of the Jacobian 2. Transformation of Vector Fields 3. Effect on Differential Forms Affine Connections Parallelism The Exponential Mapping Covariant Differentiation The Structural Equations The Riemannian Connection Complete Riemannian Manifolds Isometries Sectional Curvature Riemannian Manifolds of Negative Curvature Totally Geodesic Submanifolds Appendix 1. Topology 2. Mappings of Constant Rank Exercises and Further Results Notes

2 8 8 13 17 19 22 22 24 25 26 28 32 40 43 47 55 60 64 70 78 82 82 86 88 95

CHAPTER II Lie Groups and Lie Algebras 1.

The Exponential Mapping / . The Lie Algebra of a Lie Group

98 98 ix

X

2. 3. 4. 5. 6. 7. 8.

CONTENTS

2. The Universal Enveloping Algebra 3. Left Invariant Affine Connections 4. Taylor's Formula and the Differential of the Exponential Mapping Lie Subgroups and Subalgebras Lie Transformation Groups Coset Spaces and Homogeneous Spaces The Adjoint Group Semisimple Lie Groups Invariant Differential Forms Perspectives Exercises and Further Results Notes

.

100 102 .104 112 120 123 126 131 135 144 147 153

CHAPTER III Structure of Semisimple Lie Algebras 1. 2. 3. 4. 5. 6. 7. 8.

Preliminaries Theorems of Lie and Engel Cartan Subalgebras Root Space Decomposition Significance of the Root Pattern Real Forms Cartan Decompositions Examples. The Complex Classical Lie Algebras Exercises and Further Results Notes

155 158 162 165 171 178 182 186 191 196

CHAPTER IV Symmetric Spaces 1. 2. 3. 4. 5. 6. 7.

Affine Locally Symmetric Spaces Groups of Isometries Riemannian Globally Symmetric Spaces The Exponential Mapping and the Curvature Locally and Globally Symmetric Spaces Compact Lie Groups Totally Geodesic Submanifolds. Lie Triple Systems Exercises and Further Results Notes

198 201 205 214 218 223 224 226 227

CHAPTER V Decomposition of Symmetric Spaces 1. 2. 3. 4.

Orthogonal Symmetric Lie Algebras The Duality Sectional Curvature of Symmetric Spaces Symmetric Spaces with Semisimple Groups of Isometries

229 235 241 243

CONTENTS 5. 6.

Notational Conventions Rank of Symmetric Spaces Exercises and Further Results Notes

XI 244 245 249 251

CHAPTE R V I Symmetric Spaces of the Noncompact Type 1. 2. 3. 4. 5. 6.

Decomposition of a Semisimple Lie Group Maximal Compact Subgroups and Their Conjugacy T h e Iwasawa Decomposition Nilpotent Lie Groups Global Decompositions The Complex Case Exercises and Further Results Notes

252 256 257 264 270 273 275 279

CHAPTE R VI I Symmetric Spaces of the Compact Type 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

T h e Contrast between the Compact Type and the Noncompact Type . The Weyl Group and the Restricted Roots Conjugate Points. Singular Points. T h e Diagram Applications to Compact Groups Control over the Singular Set The Fundamental Group and the Center The Affine Weyl Group Application to the Symmetric Space UjK Classification of Locally Isometric Spaces Geometry of UjK, Symmetric Spaces of Rank One Shortest Geodesies and Minimal Totally Geodesic Spheres . . Appendix. Results from Dimension Theory Exercises and Further Results Notes

.281 283 293 297 303 307 314 318 325 327 .334 344 347 350

CHAPTE R VII I Hermitian Symmetric Spaces 1. 2. 3. 4. 5. 6. 7.

Almost Complex Manifolds Complex Tensor Fields. The Ricci Curvature Bounded Domains. The Kernel Function Hermitian Symmetric Spaces of the Compact Type and the Noncompact Type Irreducible Orthogonal Symmetric Lie Algebras Irreducible Hermitian Symmetric Spaces Bounded Symmetric Domains Exercises and Further Results Notes

352 356 364 372 377 381 382 396 400

xu

CONTENTS

CHAPTE R IX Structur e o f Semisimpl e Li e Group s 1. 2. 3. 4. 5. 6. 7.

Cartan, Iwasawa, and Bruhat Decompositions The Rank-One Reduction The SU(2, 1) Reduction Cartan Subalgebras Automorphisms The Multiplicities Jordan Decompositions Exercises and Further Results Notes

401 407 409 418 421 428 430 434 436

CHAPTE R X Th e Classificatio n o f Simpl e Li e Algebras and of Symmetric Spaces 1. 2.

3.

4. 5. 6.

Reduction of the Problem The Classical Groups and Their Cartan Involutions L Some Matrix Groups and Their Lie Algebras 2. Connectivity Properties 3. The Involutive Automorphisms of the Classical Compact Lie Algebras . Root Systems 1. Generalities 2. Reduced Root Systems 3. Classification of Reduced Root Systems. Coxeter Graphs and Dynkin Diagrams 4. The Nonreduced Root Systems 5. The Highest Root 6. Outer Automorphisms and the Covering Index The Classification of Simple Lie Algebras over C Automorphisms of Finite Order of Semisimple Lie Algebras . . . . The Classifications J. The Simple Lie Algebras over C and Their Compact Real Forms. The Irreducible Riemannian Globally Symmetric Spaces of Type II and Type IV 2. The Real Forms of Simple Lie Algebras over C. Irreducible Riemannian Globally Symmetric Spaces of Type I and Type IV 3. Irreducible Hermitian Symmetric Spaces 4. Coincidences between Different Classes. Special Isomorphisms . . . Exercises and Further Results Notes

438 444 444 447 451 455 455 459 461 474 475 478 481 490 515 515 517 518 518 520 535

SOLUTIONS TO EXERCISES

538

SOME DETAILS

586

SUPPLEMENTARY N O T E S

599

ERRATA

603

BIBLIOGRAPHY

605

L I S T OF NOTATIONAL CONVENTIONS

635

SYMBOLS FREQUENTLY USED

638

INDEX

641

REVIEWS FOR THE F I R S T EDITION

647

PREFACE T h e present book is intended as a textbook and a reference work on the three topics in the title. Together with a volume in progress on " G r o u p s and Geometric Analysis" it supersedes my "Differential Geometry and Symmetric Spaces," published in 1962. Since that time several branches of the subject, particularly the function theory on symmetric spaces, have developed substantially. I felt that an expanded treatment might now be useful. This first volume is an extensive revision of a part of "Differential Geometry and Symmetric Spaces." Apart from numerous minor changes the following material has been added: Chapter I, §15; Chapter II, §7-§8; Chapter III, §8; Chapter VII, §§7, 10, 11 and most of §2 and of §8; Chapter VIII, part of §7; all of Chapter I X and most of Chapter X. Many new exercises have been added, and solutions to the old and new exercises are now included and placed toward the end of the book. T h e book begins with a general self-contained exposition of differential and Riemannian geometry, discussing affine connections, exponential mapping, geodesies, and curvature. Chapter II develops the basic theory of Lie groups and Lie algebras, homogeneous spaces, the adjoint group, etc. T h e Lie groups that are locally isomorphic to products of simple groups are called semisimple. These Lie groups have an extremely rich structure theory which at an early stage led to their complete classification, and which presumably accounts for their pervasive influence on present-day mathematics. Chapter III deals with their preliminary structure theory with emphasis on compact real forms. Chapter IV is an introductory geometric study of symmetric spaces. According to its original definition, a symmetric space is a Riemannian manifold whose curvature tensor is invariant under all parallel translations. T h e theory of symmetric spaces was initiated by £ . Cartan in 1926 and was vigorously developed by him in the late 1920s. By their definition, symmetric spaces form a special topic in Riemannian geom-

Xlll

XI V

PREFACE

etry; their theory, however, has merged with the theory of semisimple Lie groups. This circumstance is the source of very detailed and extensive information about these spaces. They can therefore often serve as examples on the basis of which general conjectures in differential geometry can be made and tested. T h e definition above does not immediately suggest the special nature of symmetric spaces (especially if one recalls that all Riemannian manifolds and all Kahler manifolds possess tensor fields invariant under parallelism). However, the theory leads to the remarkable fact that symmetric spaces are locally just the Riemannian manifolds of the form Rn X G/K where Rn is a Euclidean ra-space, G is a semisimple Lie group that has an involutive automorphism whose fixed point set is the (essentially) compact group K, and GjK is provided with a G-invariant Riemannian structure. E. Cartan's classification of all real simple Lie algebras now led him quickly to an explicit classification of symmetric spaces in terms of the classical and exceptional simple Lie groups. On the other hand, the semisimple Lie group G (or rather the local isomorphism class of G) above is completely arbitrary; in this way valuable geometric tools become available to the theory of semisimple Lie groups. In addition, the theory of symmetric spaces helps to unify and explain in a general way various phenomena in classical geometries. T h u s the isomorphisms that occur among the classical groups of low dimensions are geometrically interpreted by means of isometries; the analogy between the spherical geometries and the hyperbolic geometries is a special case of a general duality for symmetric spaces. In Chapter V we give the local decomposition of a symmetric space into Rn and the two main types of symmetric spaces, the compact type and the noncompact type. These dual types are already distinguished by the sign of their sectional curvature. In Chapter VI we study the symmetric spaces of noncompact type. Since these spaces are completely determined by their isometry group, this chapter is primarily a global study of noncompact semisimple Lie groups. In Chapter I X this study is carried quite a bit further in the form of Cartan, Iwasawa, Bruhat, and Jordan decompositions. In Chapter VII we derive topological and differential geometric properties of the compact symmetric space U/K by studying the isotropy action of K on U/K and on its tangent space at the origin. Chapter VIII deals with Hermitian symmetric spaces; we are primarily concerned with the noncompact ones and the Cartan-Harish-Chandra representation of these as bounded domains. T h e book concludes with a classification of symmetric spaces by means of the Killing-Cartan classification of simple Lie algebras over C

PREFACE

XV

and Cartan's classification of simple Lie algebras over R. T h e latter is carried out by means of Kac's classification of finite-order automorphisms of simple Lie algebras over C. Each chapter begins with a short summary and ends with references to source material. Given the enormity of the subject, I am aware that the result is at best an approximation as regards completeness and accuracy. Nevertheless, I hope that the notes will help the serious student gain a historical perspective, particularly as regards Cartan's magnificent papers on Lie groups and symmetric spaces, which are found in the two first volumes of his collected works. For example, he can witness Cartan's rather informal arguments in his climactic paper [10], written at the age of 58, leading him to a global classification of symmetric spaces; an example of a different kind is Cartan's paper in Leipziger Berichte (1893) where he indicates models of the exceptional groups as contact transformations or as invariance groups of Pfaffian equations and which, to my knowledge, have never been verified in print. 1 Being the first such models, they have distinct historical interest although simpler models are now known. In addition to papers and books utilized in the text, the bibliography lists many items on topics that are at best only briefly discussed in the text, but are nevertheless closely related, for example, pseudoRiemannian symmetric spaces, trisymmetric spaces, reflexion spaces, homogeneous domains, discrete isometry groups, cohomology and Betti numbers of locally symmetric spaces. This part of the bibliography is selective, and no completeness is intended; in particular, papers on analysis and representation theory related to the topics of " G r o u p s and Geometric Analysis" are not listed unless used in the present volume. This book grew out of lectures given at the University of Chicago in 1958, at Columbia University in 1959-1960, and at various times at M I T since then. At Columbia I had the privilege of many long and informative discussions with Harish-Chandra, to whom I am deeply grateful. I am also indebted to A. Koranyi, K. de Leeuw, E. Luft, H. Federer, I. Namioka, and M. Flensted-Jensen, who read parts of the manuscript and suggested several improvements. I want also to thank H.-C. Wang for putting the material in Exercise A.9, Chapter VI at my disposal. Finally, I am most grateful to my friend and colleague Victor Kac who provided me with an account of his method for classifying automorphisms of finite order (§5, Chapter X) of which only a short sketch was available in print. With the sequel to this book in mind I will be grateful to readers who take the trouble of bringing errors in the text to my attention. t See "Some Details," p. 586.

XVI

PREFACE

In this third printing I have taken the opportunity to correct a few inaccuracies which have kindly been called to my attention by M. Brion, F. Gonzales, V. Kac, H. Marbes, D. Marcus, S. Marvisi, J. Orloff, R. Sulanke, and G. G. Wick. I have also added a few explanatory footnotes collected towards the end of the book in a section entitled "Some Details."

Preface to the 2001 Printing This book, published in 1978 by Academic Press, was originally a revision of my 1962 book "Differential Geometry and Symmetric Spaces". The 1978 book has been unavailable for some time and I am happy to have this new printing published by the American Mathematical Society in the series "Graduate Studies in Mathematics". This book has a sequel of two volumes: "Groups and Geometric Analysis" "Geometric Analysis on Symmetric Spaces" both of which have been published by the American Mathematical Society in its series "Mathematical Surveys and Monographs". I would like to express my gratitude to the editors and staff at the American Mathematical Society for a fine cooperation in this enterprise. In this ninth printing I have corrected a few inaccuracies and also added a number of footnotes which are collected toward the end of the book in a section entitled "Some Details". The placement of the footnotes is indicated by a dagger f- Most of these footnotes consist of alternative, more elementary proofs, and several are remarks which I have found to be helpful to readers.

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SUGGESTIONS TO THE READER Since this book is intended for readers with varied backgrounds, we give some suggestions for its use. Introductory Differential Geometry. Chapter I, Chapter IV, §1, and Chapter VIII, §l-§3 can be read independently of the rest of the book. These 120-odd pages, including the offered exercises, have on occasion served as the text for a one-term course on differential geometry, with only advanced calculus and some point set topology as prerequisites. Introduction to Lie Groups. Chapter I, §l-§6, Chapter II, and Chapter III could similarly be used for a one-term course on Lie groups, assuming some familiarity with topological groups. T h e chapters are rather independent after the fourth one and could for the most part be read in any order. Exercises. Each chapter ends with a few exercises. Some of these furnish examples illuminating the theory developed in the text, while others produce extensions and ramifications of the theory. With a few possible exceptions (indicated with a star) the exercises can be worked out with methods from the text. Since the exercises present additional material and since some exercise groups furnish suitable topics for student seminars, I felt that leaving out the solutions would be counterproductive and might turn the exercises into unnecessary obstacles. Accordingly, solutions are provided at the end of the book which each reader can use to the extent he wishes. S. Helgason

XIX

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Sequel to the Present Volume A. Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions. Mathematical Surveys and Monographs, vol. 83, American Mathematical Society (2000), xxii &; 667 pp. Originally published 1984 by Academic Press. B, Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs, vol. 39, American Mathematical Society (1994), xiv h 611 pp.

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GROUPS A N D GEOMETRIC ANALYSIS was reprinted with corrections in 2000 as Volume 83 in the series "Mathematical Surveys and Monographs". American Mathematical Society

CONTENTS I N T R O D U C T I O N . Geometric Fourier Analysis on Spaces of Constant Curvature 1. Harmonic Analysis on Homogeneous Spaces. General Problems 2. The Sphere and Spherical Harmonics 3. The Hyperbolic Plane. Non-Euclidean Fourier Analysis. Spherical Functions and Spherical Transforms. Eigenfunctions and Eigenspace Representations. Exercises and Further Results C H A P T E R I. Integral Geometry and Radon Transforms 1. Integration on Manifolds 2. The Radon Transform on Rn 3. A Duality in Integral Geometry. Orbital Integrals 4. The Radon Transform on Two-Point Homogeneous Spaces 5. Orbital Integrals on Lorentzian Manifolds and on

SL(2,R)

Exercises and Further Results C H A P T E R II. Invariant Differential Operators 1. Differential Operators on Manifolds 2. Geometric Operations on Differential Operators, Radial Parts. Transversal Parts and Separation of Variables xxii i

xxiv

GROUPS AND GEOMETRIC ANALYSIS: CONTENTS

3. Invariant Differential Operators on Homogeneous Spaces. LaplaceBeltrami Operator, the Darboux Equation and the Poisson Equation Generalized. Central Operators on G and Invariant Operators on the Symmetric Space G/K Exercises and Further Results C H A P T E R III. Invariants and Harmonic Polynomials 1. Decomposition of the Symmetric and the Exterior Algebra. Primitive Forms 2. Invariants of the Weyl Group, Harmonic Polynomials, Exterior Invariants, Eigenfunctions of the Weyl Group Invariant Operators 3. The Orbit Structure of p, Nilpotent, Semisimple and Regular Elements. Harmonic Polynomials on p Exercises and Further Results C H A P T E R IV. Spherical Functions and Spherical Transforms 1. Representations 2. Joint Eigenfunctions 3. Integral Formulas 4. Harish-Chandra's Expansion and the c-function. The Rank-One Reduction 5. The Paley-Wiener Theorem. The Inversion Formula. The Bounded Spherical Functions. The Plancherel Formula for the Spherical Transform. The Flat Case 6. Convexity Theorems Exercises and Further Results C H A P T E R V. Analysis on Compact Symmetric Spaces 1. Representations of Compact Lie Groups. Weights and Characters 2. Fourier Expansions on Compact Lie Groups and Symmetric Spaces 3. Finite-Dimensional Spherical Representations. Highest Weights 4. Eigenfunctions and Eigenspace Representations Exercises and Further Results SOLUTIONS TO EXERCISES BIBLIOGRAPHY

GEOMETRIC ANALYSIS ON SYMMETRIC SPACES was published in 1994 as Volume 39 in the series "Mathematical Surveys and Monographs". American Mathematical Society.

CONTENTS C H A P T E R I. A Duality in Integral Geometry 1. Generalities 2. The Radon Transform for Points and Hyperplanes 3. Homogeneous Spaces in Duality Exercises and Further Results C H A P T E R II. A Duality for Symmetric Spaces 1. The Space of Horocycles 2. Invariant Differential Operators 3. The Radon Transform and its Dual 4. Finite-Dimensional Spherical and Conical Representations 5. Conical Distributions 6. Some Rank-One Results Exercises and Further Results C H A P T E R III. The Fourier Transform on a Symmetric Space 1. The Inversion and the Plancherel Formula 2. Generalized Spherical Functions (Eisenstein Integrals) 3. The Q^-raatrices 4. The Simplicity Criterion 5. The Paley-Wiener Theorem for the Fourier Transform o n X = G/K 6. Eigenfunctions and Eigenspace Representations XXV

xxvi

GEOMETRIC ANALYSIS ON SYMMETRIC SPACES: CONTENTS

7. Tangent Space Analysis 8. Eigenfunctions and Eigenspace Representations on X0 9. The Compact Case 10. Elements of D{G/K)

as Fractions

11. The Rank One Case 12. The Spherical Transform Revisited Exercises and Further Results C H A P T E R IV. The Radon Transform on X and on X0. Range Questions 1. The Support Theorem 2. The Ranges V(X)A,

S'{X)* and £(~)v

3. The Range and Kernel for if-types •4. The Radon Transform and its Dual for if-invariants 5. The Radon Transform on X0 Exercises and Further Results C H A P T E R V. Differential Equations on Symmetric Spaces 1. Solvability 2. Mean Value Theorems 3. Harmonic Functions on Symmetric Spaces 4. Harmonic Functions on Bounded Symmetric Domains 5. The Wave Equation on Symmetric Spaces 6. Eigenfunctions and Hyperfunctions Exercises and Further Results C H A P T E R VI. Eigenspace Representations 1. Generalities 2. Irreducibility Criteria for a Symmetric Space 3. Eigenspace Representations for the Horocycle Space G/MN 4. Eigenspace Representations for the Complex Space G/N 5. Two Models of the Spherical Representations Exercises and Further Results SOLUTIONS TO EXERCISES BIBLIOGRAPHY

CHAPTER I

ELEMENTARY DIFFERENTIAL GEOMETRY This introductory chapter divides in a natural way into three parts: § 1 - §3 which deal with tensor fields on manifolds, §4-§8 which treat general properties of affine connections, and §9-§14 which give an introduction to Riemannian geometry with some emphasis on topics needed for the later treatment of symmetric spaces. § 1 -§3. When a Euclidean space is stripped of its vector space structure and only its differentiable structure retained, there are many ways of piecing together domains of it in a smooth manner, thereby obtaining a so-called differentiable manifold. Local concepts like a differentiable function and a tangent vector can still be given a meaning whereby the manifold can be viewed ' 'tangentially," that is, through its family of tangent spaces as a curve in the plane is, roughly speaking, determined by its family of tangents. This viewpoint leads to the study of tensor fields, which are important tools in local and global differential geometry. They form an algebra T)(M), the mixed tensor algebra over the manifold M. The alternate covariant tensor fields (the differential forms) form a submodule $I(M) of £)(M) which inherits a multiplication from £>(M), the exterior multiplication. The resulting algebra is called the Grassmann algebra of M. Through the work of E. Cartan the Grassmann algebra with the exterior differentiation d has become an indispensable tool for dealing with submanifolds, these being analytically described by the zeros of differential forms. Moreover, the pair ($l(M), d) determines the cohomology of M via de Rham's theorem, which however will not be dealt with here. §4-§8. The concept of an affine connection was first defined by Levi-Civita for Riemannian manifolds, generalizing significantly the notion of parallelism for Euclidean spaces. On a manifold with a countable basis an affine connection always exists (see the exercises following this chapter). Given an affine connection on a manifold M there is to each curve y(t) in M associated an isomorphism between any two tangent spaces My(tl) and MY(ti). Thus, an affine connection makes it possible to relate tangent spaces at distant points of the manifold. If the tangent vectors of the curve y(t) all correspond under these isomorphisms we have the analog of a straight line, the so-called geodesic. The theory of affine connections mainly amounts to a study of the mappings Exp p : Mv —• M under which straight lines (or segments of them) through the origin in the tangent space Mp correspond to geodesies through p in M. Each mapping Exp p is a diffeomorphism of a neighborhood of 0 in Mv into My giving the so-called normal coordinates at p. Some other local properties of Exp p are given in §6, the existence of convex neighborhoods and a formula for the differential of Exp p . An affine connection gives rise to two important tensor fields, the curvature tensor field and the torsion tensor field which in turn describe the affine connection through E. Cartan's structural equations [(6) and (7), §8)]. 1

2

ELEMENTARY DIFFERENTIAL GEOMETRY

[Ch. I

§9-§14. A particularly interesting tensor field on a manifold is the so-called Riemannian structure. This gives rise to a metric on the manifold in a canonical fashion. It also determines an affine connection on the manifold, the Riemannian connection; this afrine connection has the property that the geodesic forms the shortest curve between any two (not too distant) points. The relation between the metric and geodesies is further developed in §9-§10. The treatment is mainly based on the structural equations of E. Cartan and is independent of the Calculus of Variations. The higher-dimensional analog of the Gaussian curvature of a surface was discovered by Riemann. Riemann introduced a tensor field which for any pair of tangent vectors at a point measures the corresponding sectional curvature, that is, the Gaussian curvature of the surface generated by the geodesies tangent to the plane spanned by the two vectors. Of particular interest are Riemannian manifolds for which the sectional curvature always has the same sign. The irreducible symmetric spaces are of this type. Riemannian manifolds of negative curvature are considered in §13 owing to their importance in the theory of symmetric spaces. Much progress has been made recently in the study of Riemannian manifolds whose sectional curvature is bounded from below by a constant > 0. However, no discussion of these is given since it is not needed in later chapters. The next section deals with totally geodesic submanifolds which are characterized by the condition that a geodesic tangent to the submanifold at a point lies entirely in it. In contrast to the situation for general Riemannian manifolds, totally geodesic submanifolds are a common occurrence for symmetric spaces.

§1. Manifolds Let Rm and Rn denote two Euclidean spaces of m and n dimensions, respectively. Let O and O' be open subsets, O C Rmy O' C Rn and suppose cp is a mapping of O into O'. T h e mapping cp is called differentiable if the coordinates yj(