http://dx.doi.org/10.1090/surv/039
Geometric Analysis on Symmetric Spaces
Second Edition Mathematical Surveys and Monographs Volume 39
Geometric Analysis on Symmetric Spaces
Second Edition
Sigurdur Helgason
American Mathematical Society Providence, Rhode Island ^VDED Editorial Committee Jerry L. Bona Michael G. Eastwood Ralph L. Cohen Benjamin Sudakov J. T. Stafford, Chair
2000 Mathematics Subject Classification. Primary 43A85, 53C35, 22E46, 22E30, 43A90, 44A12, 32M15; Secondary 53C65, 31A20, 43A35, 35L05, 14M17, 17B25, 22F30.
For additional information and updates on this book, visit www.ams.org/bookpages/surv-39
Library of Congress Cataloging-in-Publication Data Helgason, Sigurdur, 1927- Geometric analysis on symmetric spaces / Sigurdur Helgason. — 2nd ed. p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 39) Includes bibliographical references and index. ISBN 978-0-8218-4530-1 (alk. paper) 1. Symmetric spaces. I. Title.
QA649.H43 2008 516.3/62—dc22 2008025621
Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected].
First Edition © 1994 by the American Mathematical Society. Reprinted with corrections 1997. Second Edition © 2008 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 13 12 11 10 09 08 To my Danish mathematical friends
past and present Contents
Preface to Second Edition xiii Preface xv
CHAPTER I A Duality in Integral Geometry. 1
§1. Generalities 3 1. Notation and Preliminaries 3 2. Principal Problems 5 §2. The Radon Transform for Points and Hyperplanes. 8 1. The Principal Results 8 2. The Kernel of the Dual Transform 13 3. The Radon Transform and its Dual on the K-types 17 4. Inversion of the Dual Transform 20 5. The Range Characterization for Distributions and Consequences 25 6. Some Facts about Topological Vector Spaces 29 §3. Homogeneous Spaces in Duality. 30 1. The Radon Transform for a Double Fibration 30 2. The Radon Transform for Grassmannians 39 3. Examples. 44 A. The d-Plane Transform 45 B. The Poisson Integral as a Radon Transform 49 C. Hyperbolic Spaces and Spheres 50 Exercises and Further Results. 52 Notes. 57
CHAPTER II A Duality for Symmetric Spaces. 59 §1. The Space of Horocycles. 60 1. Definition and Coset Representation 60 2. The Isotropy Actions for X and for E 62 3. Geodesies in the Horocycle Space 65
vn CONTENTS
§2. Invariant Differential Operators. 70 1. The Isomorphisms 70 2. Radial Part Interpretation 74 3. Joint Eigenspaces and Eigenspace Representations 75 4. The Mean Value Operators 77 §3. The Radon Transform and its Dual. 82 1. Measure-theoretic Preliminaries 82 2. Integral Transforms and Differential Operators 84 3. The Inversion Formula and the Plancherel Formula for the Radon Transform 89 4. The Poisson Transform 99 5. The Dual Transform and the Poisson Kernel 102 §4. Finite-dimensional Spherical and Conical Representations. 105 1. Conical Distributions. Elementary Properties 105 2. Conical Functions and Finite-Dimensional Representations 113 3. The Finite-dimensional Spherical Representations 119 4. Conical Models and Spherical Models 120 5. Simultaneous Euclidean Imbeddings of X and of E. Horocycles as Plane Sections. 122 6. Restricted Weights 127 7. The Component H(n) 131 §5. Conical Distributions. 134
1. The Construction of #'A s 134 2. The Reduction to RankOne 137 3. The Analytic Continuation of \£A,s 141 4. The Determination of the Conical Distributions 151 §6. Some Rank-One Results. 157 1. Component Computations 157 2. The Inversion of W 159 3. The Simplicity Criterion 165 4. The Algebra T)(K/M) 167 5. An Additional Conical Distribution for A = 0 169 6. Conical Distributions for the Exceptional A 171 Exercises and Further Results. 181
Notes. 192 CONTENTS
CHAPTER III The Fourier Transform on a Symmetric Space. 197 §1. The Inversion and the Plancherel Formula 198 1. The Symmetry of the Spherical Function 198 2. The Plancherel Formula 202 §2. Generalized Spherical Functions (Eisenstein Integrals.) 227 1. Reduction to Zonal Spherical Functions 227 2. The Expansion of $>x,s 233 3. Simplicity (preliminary results) 241 §3. The Q^-matrices. 243
1. The if-finite functions in £X(E) 243 2. Connections with Harmonic Polynomials 244 3. A Product Formula for det QS(X) (preliminary version) 248 §4. The Simplicity Criterion. 255 §5. The Paley-Wiener Theorem for the Fourier Transform on X = G/K. 260 1. Estimates of the T-coemcients 261 2. Some Identities for Cs 264 3. The Fourier Transform and the Radon Transform. if-types 266 4. Completion of the Proof of the Paley-Wiener Theorem. The Range £'(X)~ 268 5. A Topological Paley-Wiener Theorem for the if-types 273 6. The Inversion Formula, the Plancherel Formula and the Range Theorem for the 5-spherical Transform 279 §6. Eigenfunctions and Eigenspace Representations. 282 1. The if-finite Joint Eigenfunctions of D(X) 282 2. The Irreducibility Criterion for the Eigenspace Representations on G/K 284 §7. Tangent Space Analysis. 285 1. Discussion 285 2. The J-polynomials 286 3. Generalized Bessel Functions and Zonal Spherical Functions 292 4. The Fourier Transform of if-finite Functions 293 5. The Range V(p)~ inside W(a* x K/M) 298
§8. Eigenfunctions and Eigenspace Representations on XQ. 300 1. Simplicity 300 2. The if-finite Joint Eigenfunctions of D(G0/if) 303 3. The Irreducibility Criterion for the Eigenspace Representations of G0/K 309 CONTENTS
The Compact Case. 310 1. Motivation 310 2. Compact Symmetric Spaces 313 3. Analogies 316 4. The Product Decomposition 316 Elements of T>(G/K) as Fractions. 322 The Rank-One Case. 327 1. An Explicit Formula for the Eisenstein Integral 327 2. Harmonic Analysis of if-finite Functions 332 The Spherical Transform Revisited. 335 1. Positive Definite Functions 335 2. The Spherical Transform for Gelfand Pairs 339 3. The Case of a Symmetric Space G/K 346 Exercises and Further Results. 352 Notes. 358
CHAPTER IV
The Radon Transform on X and on X0. Range Questions. 363 The Support Theorem. 363 The Ranges £>(Xf, £'(Xf and £(~)v. 365 The Range and Kernel Determined in terms of if-types. 369 1. The General Case 369 2. Examples: H2 and R2 379 The Radon Transform and its Dual for K-invariants. 381
The Radon Transform on X0. 387 1. Preliminaries 387 2. The Support Theorem 392 3. The Range and Kernel for the if-types 394 f v 4. The Ranges £ {X0) and £(E0) 395 Exercises and Further Results. 397 Notes. 398
CHAPTER V Differential Equations on Symmetric Spaces. 401 Solvability. 401 1. Fundamental Solution of D 402 2. Solvability in £{X) 403 CONTENTS
3. Solvability in S'(X) 406 4. Explicit solution by Radon transforms 407 §2. Mean Value Theorems. 413 1. The Mean Value Operators 413 2. Approximations by Analytic Functions 416 3. Asgeirsson's Mean Value Theorem Extended to Homogeneous Spaces 418 §3. Harmonic Functions on Symmetric Spaces. 421 1. Generalities 421 2. Bounded Harmonic Functions 421 3. The Poisson Integral Formula for X 425 4. The Fatou Theorem 430 5. The Furstenberg Compactification 439 §4. Harmonic Functions on Bounded Symmetric Domains. 442 1. The Bounded Realization of a Hermitian Symmetric Space 442 2. The Geodesies in a Bounded Symmetric Domain 444 3. The Restricted Root Systems for Bounded Symmetric Domains 445 4. The Action of G0 on D and the Polydisk in D 451 5. The Shilov Boundary of a Bounded Symmetric Domain 453 6. The Dirichlet Problem for the Shilov Boundary 460 7. The Hua Equations 461 8. Integral Geometry Interpretation 466 §5. The Wave Equation on Symmetric Spaces. 468 1. Introduction. Huygens' Principle 468 2. Huygens' Principle for Compact Groups and Symmetric Spaces X = G/K (G complex) 471 3. Huygens' Principle and Cartan Subgroups 477 4. Orbital Integrals and Huygens' Principle 482 5. Energy Equipartition 486 6. The Flat Case Revisited 490 7. The Multitemporal Wave Equation on X = G/K 493 8. The Multitemporal Cauchy Problem 497 9. Incoming Waves and Supports 506 10. Energy and Spectral Representation 511 11. The Analog of the Priedlander Limit Theorem 524 §6. Eigenfunctions and Hyper functions. 527 1. Arbitrary Eigenfunctions 527 2. Exponentially Bounded Eigenfunctions 531 Exercises and Further Results. 532 Notes. 537 CONTENTS
CHAPTER VI Eigenspace Representations. 539 §1. Generalities. 539 1. A Motivating Example 539 2. Eigenspace Representations on Function- and Distribution-Spaces 540 3. Eigenspace Representations for Vector Bundles 541 §2. Irreducibility Criteria for a Symmetric Space. 543 1. The Compact Case 543 2. The Euclidean Type 545 3. The Noncompact Type 546 §3. Eigenspace Representations for the Horocycle Space G/MN. 547 1. The Principal Series 547 2. The Spherical Principal Series. Irreducibility 548 3. Conical Distributions and the Construction of the Intertwining Operators 554 4. Convolution on G/MN 557 §4. Eigenspace Representations for the Complex Space G/N. 562 1. The Algebra T>(G/N) 562 2. The Principal Series 564 3. The Finite-Dimensional Holomorphic Representations. 565 §5. Two Models of the Spherical Representations 567
Exercises and Further Results. 569
Notes. 571
SOLUTIONS TO EXERCISES 573
BIBLIOGRAPHY 599
SYMBOLS FREQUENTLY USED 627
INDEX 633 PREFACE TO SECOND EDITION
This book has been unavailable for some time and I am happy to follow the publisher's suggestion for a new edition. While a related forthcoming book, "Integral Geometry and Radon Transforms" (here denoted [IGR]) deals with several examples of homoge neous spaces in duality with corresponding Radon transforms, the present work follows the direction of the first edition and concentrates on analy sis on Riemannian symmetric spaces X = G/K. We develop further the theory of the Fourier transform and horocycle transform on X, also taking into account tools developed by Eguchi for the Schwartz space S(X). These transforms provide the principal methods for analysis on X, existence and uniqueness theorems for invariant differential equations on X, explicit solu tion formulas, as well as geometric properties of the solutions, for example the harmonic functions and the wave equation on X. On the space X there is a canonical hyperbolic system on X, introduced by Semenov-Tian- Shansky, which is multitemporal in the sense that the time variable has dimension equal to the rank of X. The solution has remarkable analogies to the classical wave equation on Rn, summarized in a table in Chapter V, §5. My intention has been to make the exposition easily accessible to readers with some modest background in Lie group theory which by now is rather widely known. To facilitate self-study and to indicate further developments each chapter concludes with a section "Exercises and Further Results". Solutions and references are collected at the end of the book. The harder problems are starred. Occasionally results and proofs rely on material from my previous books "Differential Geometry, Lie Groups and Symmetric Spaces" abbreviated [DS] and "Groups and Geometric Analysis", abbreviated [GGA]. Once again I wish to express my gratitude to my friends and collab orators, Adam Koranyi, Gestur Olafsson, Frangois Rouviere and Henrik Schlichtkrull and especially to my long-term colleague David Vogan for significant help at specified spots in the text. Finally, I thank Brett Coon- ley and Jan Wetzel for their invaluable help in the production and the editor Dr. Edward Dunne for his interest in the work and his patient and accommodating cooperation. I would also like to express my thanks for the following permissions of partial quotations: (i) To Academic Press concerning my papers [1970a], [1976], [1980a], [1992b], and [1992d] quoted at the end of the Preface to the first edition. (ii) To Elsevier concerning my paper [2005]. (iii) To John Wiley and Sons concerning my paper [1998a] and my paper with Schlichtkrull [1999].
xiii PREFACE
Among Riemannian manifolds the symmetric spaces in the sense of E. Cart an form an abundant supply of elegant examples whose structure is particularly enhanced by the rich theory of semisimple Lie groups. The simplest examples, the classical 2-sphere S2 and the hyperbolic plane if2, play familiar roles in many fields in mathematics. On these spaces, global analysis, particularly integration theory and partial differential operators, arises in a canonical fashion by the require ment of geometric invariance. On Rn these two subjects are related by the Fourier transform. Also harmonic analysis on compact symmetric spaces is well developed through the Peter-Weyl theory for compact groups and Car- tan's refinement thereof. For the noncompact symmetric spaces, however, we are presented with a multitude of new and natural problems. The present monograph is devoted to geometric analysis on noncompact Riemannian symmetric spaces X. (The Euclidean case and the compact case are also briefly investigated in Chapter III, §§7-9, and Chapter IV, §5, but from an unconventional point of view). A central object of study is the algebra D(X) of invariant differential operators on the space. A simultaneous diagonalization of these operators is provided by a certain Fourier transform / —• /~ on X which is the subject of Chapter III. Just as is the case with Rn the symmetric space X turns out to be self-dual under the mentioned Fourier transform; thus range questions like the intrinsic characterization of (C£°(X))~ in analogy with the classical Paley-Wiener theorem in Rn become natural and their answers useful. Chapters II and IV are devoted to the theory of the Radon transform on X, particularly inversion formulas and range questions. The space E of horocycles in X offers many analogies to the space X itself and this gives rise to the study of conical functions and conical distributions on El which are the analogs of the spherical functions on X. They have interesting connections with the representation theory of the isometry group G of X, discussed in Chapter II, §4, and in Chapter VI, §3, where the conical dis tributions furnish intertwining operators for the spherical principal series. In Corollary 3.9, Ch. VI, these intertwining operators are explicitly related to the above-mentioned Fourier transform on X. While the Fourier transform theory in Chapter III gives rise to an ex plicit simultaneous diagonalization of the algebra D(X), the Radon trans form theory in Chapter II is considered within the framework of a general integral transform theory for double fibrations in the sense of Chapter I, §3. This viewpoint is extremely general: two dual integral transforms arise whenever we are given two subgroups of a given group G. In the intro duction to Chapter I we stress this point by indicating five such examples
xv XVI PREFACE arising in this fashion from the single group G = SU(1,1) of the conformal maps of the unit disk, namely the X-ray transform, the horocycle trans form, the Poisson integral, the Pompeiu problem, theta series, and cusp forms. When range results are considered, this viewpoint of the Poisson integral as a Radon transform offers a very interesting analogy with the X-ray transform in JR3 (Chapter I, §3, No. 5). With the tools developed in Chapters I-IV we study in Chapter V some natural problems for the invariant differential operators on X, solvability questions, the structure of the joint eigenfunctions, with emphasis on the harmonic functions, as well as the solutions to the invariant wave equation on X. In Chapter VI we consider in some detail the representations of G which naturally arise from the joint eigenspaces of the operators in the algebra D(X) and the algebra D(E). The length of this book is a result of my wish to make the exposition easily accessible to readers with some modest background in semisimple Lie group theory. In particular, familiarity with representation theory is not needed. To facilitate self-study and to indicate further developments each chapter is concluded with a section "Exercises and Further Results". Solutions and references are given towards the end of the book. The harder problems are starred. Occasionally, results and proofs rely on material from my earlier books, "Differential Geometry, Lie groups, and Symmet ric Spaces" and "Groups and Geometric Analysis". In the text these books are denoted by [DS] and [GGA]. Some of the material in this book has been the subject of courses at MIT over a number of years and feedback from participants has been most beneficial. I am particularly indebted to Men-chang Hu, who in his MIT thesis from 1973 determined the conical distributions for X of rank one. His work is outlined in Chapter II, §6, No. 5-6, following his thesis and in greater detail than in his article Hu [1975]. I am also deeply grateful to Adam Koranyi for his advice and generous help with the material in Chapter V, §§3-4, as explained in the notes to that chapter. Similarly, I am grateful to Henrik Schlichtkrull for beneficial discussions and for his suggestions of Proposition 8.6 in Chapter III and Corollary 5.11 in Chap ter V, indicated in the text. I have also profited in various ways from expert suggestions from my colleague David Vogan. I am grateful to the National Science Foundation for support during the writing of this book. Many people have read at least parts of the manuscript and have fur nished me with helpful comments and corrections; of these I mention Ful ton Gonzalez, Jeremy Orloff, An Yang, Werner Hoffman, Andreas Juhl, Frangois Rouviere, Sonke Seifert, and particularly Frank Richter. I thank them all. Finally, I thank Judy Romvos for her expert and conscientious T^jX-setting of the manuscript. A good deal of the material in this monograph has been treated in earlier papers of mine. While subsequent consolidation has usually led to a rewriting of the proofs, texts of theorems as well as occasional proofs PREFACE xvii have been preserved with minimal change. I thank Academic Press for permission to quote from the following journal publications of mine, listed in the bibliography: [1970a], [1976], [1980a], [1992b], [1992d], as well as the book [1962a]. SOLUTIONS TO EXERCISES
CHAPTER I
A. Radon Transform on Rn. £ A. 1. By §2 (27) each Ek
l
A. 2. For the Fourier transform R By the definition (Yk®Vy(x) = ± J R Vs^-1 / On the other hand, we have the classical formula (see e.g. [GGA], p. 25) iX (A) j e ^«)Yk(uj)du = Cn9kYk(ri)- A(n/2)-l ' gn-1 n 2 k where cn^k = (2ir) / i and Jr is the Bessel function. Here we replace k by 0 and n by n + 2k. Then we obtain k iX iA J e ^Yk(V)dr, = (^j nH y e ^dC Sn-1 Sn+2fc-l Finally, we put x = ru and get V 1 fc k isr<1 (yfc®V') M = ^ (r/27r) nH | ±d<; J $(s)(is) e ds gn + 2/e-l R 573 574 SOLUTIONS TO EXERCISES as desired. A. 3. We know from §2, No. 3 that Pk(2,cos0) = Hk(sm6,cos6) where 2 Hk(xi,x2) is the unique harmonic polynomial on R which is homogeneous — of degree fc,i s invariant under (#i, x2 —> ( xi,x2) and satisfies Hk(0,1) = 1. k k th Since {xi+ix2) and (x\ —ix2) span the space of homogeneous k degree harmonic polynomials we have k Hk(xux2) = Re((x2 + ixi) ), which gives the desired result. A. 4. If dfc is the normalized Haar measure on K we have (/)*(*) = j ^Jtf(x + k>y)dm(y)>jdk K Co = j dm(y) J f(x + k- y)dk = J (M^f)(x)dm(y), Co K Co r where (M f)(z) is the average of / over Sr(z). Hence 00 r a (fy(x) = ndJ(M f)(x)r -'dr, 0 so, using polar coordiantes around x {f)y{x) = ^j\x-y\d-nf{y)dy and now the inversion formula follows from the standard inversion of the Riesz potential, ([GGA], Ch. I, Prop. 2.38). The statement (i) amounts to that if V is a fc-dimensional vector sub- n space of C then V — k -£0 for some k G U(n). This is obvious by choosing a basis of V orthonormal with respect to the standard Hermitian inner product ( , ) on Cn. Statement (ii) amounts to proving that if W is a Lagrangian vector 2n subspace of R then W = k • £0 for some k G U(n). It is well known that dim W = n. Writing z = x + iy, w = u + iv with x, y, u, v G Rn we have (z,w) — x - u-\- y - v — i(x • v — y • u) = (x, y) • 0, v) - i{(x, y), (u, v)} so the action of U(n) on Cn ~ R2n preserves both the standard inner 2n product on R and the skew symmetric form { , }. If ei,... ,en is an orthonormal basis of W over R then the formula above shows, W being Chapter I 575 isotropic, that (e^e^) = 5{j so the e* form a complex orthonormal basis of Cn. Viewing the standard orthonormal basis of Rn x 0 as a complex n orthonormal basis of C we see that W = k • £0 for a suitable k G U(n). A. 5. Prom [GGA], Ch. I, Theorem 2.20 we have supp(T) c BA(0). For e > 0 let / G V{X) have supp(/) C BA_e(0). Then supp(/) C /?A_e where [3R = iZ: d(o,t) A. 6. We have with a constant c {$*f)(x)=c ( / / ( / ^(M™,x)-p)/(w,p)dpjdw S"-1 R i R v = c / ((p*f)(w, (w,x))dw = (ip*f) (x) gn-1 (Natterer, [1986], p. 14). B. Homogeneous Spaces. Grassmann Manifolds. B. 1. For (ii) we may take X2 = x0 and write x\ = g\K, £ = ^H. Then x0, £ incident <=$• ^h — k ( some h € H, k £ K) xi,£ incident «<=> g\k\ — yhi ( some hi G H, k\ £ K). 1 1 Thus if XQI X\ are incident to £ we have #i = kh h\k1 . Conversely, if g\ — k'h'k" we put 7 — k!h! and then x0, #i are incident to £ = 7#. For (hi) suppose first XiJ D UK — K U H. Let x\ 7^ #2 in X. Suppose £1 7^ £2 hi H both incident to x\ and #2- Let a^ = (^if, £j = 7?#. Since x^ is incident to £j there exist kij G X, /iij G # such that #;fc;j = 7?^j * = 1» 2; ,7 = 1,2. By eliminating gi and 7^ we obtain &22 ^21^21 ^11 = ^22 ^12^12 &11 • 576 SOLUTIONS TO EXERCISES This being in KH n HK it lies in K U H. If the left hand side is in K, then h^hn G K so we get g2K = jih2iK = ~iihiiK = gxK which is a contradiction. Similarly, if the mentioned left hand side is in H 1 we have k2~2 k2i G H which gives the contradiction 72 # = 71 i?. Conversely, suppose KHnHK ^ KUH. Then there exist ^1,^2,^1? &2 such that /ii/ci = A/2^2 and /iifci ^ KUH. Put xi = /ii-K", £2 — &2#- Then %o 7^ #i5 £0 7^ £2? yet both £0 and £2 are incident to both x0 and #i. B. 2-3. For the first statement see [GGA], Cor. 4.10, Ch. II. For the other suppose the generators Di = Dpi were not algebraically independent. Let Y n l P = Eani ... ntxl ...x t be a nonzero polynomial such that P(D±,..., Di) = 0. Let di = degree (Pi) and AT = max(E^n^), the maximum taken over the set of ^-tuples (ni,..., ni) for which ani ... n^ ^ 0. We write the polynomial ni i S = Eani...n,P1 ...P; < as the sum S = Q + R, where 1 Q= Yl ani...niP? ...P?< T,diTii=N and degree (R) < N. Also Q ^ 0 by assumption. Consider the operator Vani...niD?...D?-Ds whose order is < N ([GGA], p. 287). This operator equals 0 - DQ — DR which by the definition in Exercise B2 has order N. This gives the desired contradiction. B. 10. Method of Helgason [1957] or [GGA], Ch. V, Lemma 2.6. First show that it suffices to compute 2 2 j \vl3\ \vM\ dV U(n) and that this integral is given by (i) (n(n + 1))_1 if (i,j) and (k,£) are either in the same row or the same column (not both). (ii) 2(n(n+l))-1if(i,j) = (M) Chapter II 577 (hi) (n2 — 1) x if (i,j) and (fc, £) are neither in the same row nor the same column. See also Faraut-Koranyi [1993], p. 237. B. 11. The proof is obtained by expanding in a Fourier series on T2 (also observed by Gindikin). B. 12. If U/K has rank one see [GGA], Ch. I, Cor. 4.19. If U/K has higher rank the result is immediate from Exercise 11 as pointed out by Grinberg. B. 13. d is a if-orbit containing (1,0) so equals B. Also H • o is two- dimensional so equals D. CHAPTER II A. The Spaces X = G/K and S = G/MN. A. 1. If kN C NK then k • £0 C £0 so k G M by text. If nK c KN then n • o belongs to each horocycle through o. If n ^ e, n • o = ka • o (a ^ e). But fc • £0 does not contain ka - o = n - o. Let g = ! + a + n be the usual Iwasawa decomposition of g = sC(2, R) (as before Lemma 4.9). Let g = m + n + q where q is MA^-invariant. Let H e a have the component Hi in q. Then [#i,n] C n is a contradiction. A. 2. Use (4) §3. A. 3. Recall proof of Lemma 4.9 (ii). A. 4. Consider V = Cn+1 with the Hermitian form (y, w) = y0w0 - yiwi ynwn and put V+ = {y G Cn+1 : (y, y) > 0}. The Hermitian hyperbolic space + can be taken as F /C*. With non-homogeneous coordinates zi = yi/y0, 1/+/C* is identified with the ball n B+ = {zeC : |Zl|2 + ... + |^n|2<1} and the unitary action U(l,n) = U(V) on V induces the action of the projective group PU(V) on B+ (SU(l,n) mod its center, cf. [DS], X, Exercise Dl). Let n : V —> V/C* be the natural map. Choose t G dB+ and choose y* ^Oonf. The Iwasawa subgroup N (the unipotent radical of the isotropy group PXJ(V)i*) viewed as a subgroup of SU(1, n) fixes y* and hence also the function \{y,y)\2 578 SOLUTIONS TO EXERCISES Thus the equation dy* — c, that is, \(y*,y)\2 = \(y,y)\c2 is a horocycle. In non-homogeneous coordinates this is r2 |1 - z\zx z* z \ = (1 - \z \ \z \ ) —c n n x n 2\ 12 Wo which is an ellipsoid in the Euclidean metric. A PU(V)-invariant metric on £+ is given by (cf. Mostow [1973], p. 136) \{w,w)2{y,y)2>(w,w)2(y,y)l J so the sphere Sr(ir(w)) is r — |(w, w/|2cn r. I(J/,J/>I Let it? —> y*,r —-> oo with (w,w)*ch. r = c (where (y*,y*) = 0). Then the sphere converges to the horocycle above. Another verification in terms of the notation of [DS], IX, (§3 and Ex ercise B4). The horocycle N • o is given by 2it-\z\2 -2z 2 K2(l - it)+ \z\ ' 2(1-it) and therefore equals the ellipsoid 2 2 2K + i| + |W2| = i. Similarly the horocycle TV • o equals (*) 2K-||2 Let ^chr 0 shr^ o 1 0 ^shr 0 chry Then the sphere Sr(o) equals Kar • o which is given by 2 \Zl\* + \Zi\ =\tfr. The image ar • 5r(0) is by [DS], IX, Exercise B4 given by Chapter II 579 so the equation for ar • Sr(o) is 2 2 2 (1 + th r)|it;i| - thr(>i + w{) + \w2\ = 0. Thus as r —~> oo the sphere ar5r(0) converges to the horocycle (*). 2 A. 5. First reduce the problem to the case X ~ H as follows. Let Xa be a root vector in the Lie algebra of N and let Ga denote the analytic subgroup of G with Lie algebra KXa + H0Xa -f R[Xa, 0-Xa], Then Ga - o is a totally geodesic submanifold of X isometric to H2 and the horocycle exptXa • o in Ga • o equals (Ga - o)n(N • o). This reduces the problem to H2 with metric y2 where the geodesies are the semicircles 7u,r • x — u-\-rcos6, y~rsin#, 0 < 0 < TT, We have 7T f(lu,r)= / f (u + r cos Q,r sin 0) (sin 0)~1d6 o so taking £ as the line y = 1 our assumption amounts to H*£dw = 0, r (*) / F(s)dw(s) =0 x e R5 r < 1. Sr(a;) This implies for the corresponding disk Br(x) I F(u, v)dudv = 0, whence / (d1F)(x + u,v)dudv = Q Br(o) with #i ~ d/du. Using the divergence theorem on the vector field F(x + u, v)d/du we get F(x + u, i>)u dw(u, v) = 0. / Sr(o) 580 SOLUTIONS TO EXERCISES Combining this with (*) we deduce (**) / F(s)sidw(s) s = (s1,s2). Sr(x) Iterating the implication (*) => (**) we obtain f F(s)P{Sl)dw(s) = 0 Sr(x) where P is any polynomial so we get the desired conclusion / = 0 on the strip 0 < y < 1. A. 7. Because of Theorem 2.9 it suffices to prove that the convolution algebra C^(MN) of M-bi-invariant functions in CC(MN) is commutative. This result from Koranyi [1980] follows (for m2Q ^ 1) from Kostant's theo rem (Exercise D3 below) which implies that for each n £ N there exists an me M such that mum,-1 = n"1. Thus f(n) = fin'1) for / G C\(MN) which implies the commutativity. For the case rri2a — 1 see [GGA], Ch. IV, Exercise BIO. A. 8. With the customary notation we have (as m*k(n)M = fc(n(ra*n))M), 2p{H )) J F{k{n)M)e- ^ dn= f F(kM)dkM N K/M J F(k(n{m*n))M)e-2piH{fl))dfi, N and since by §6, H(n) = H{n{m*n)) + B(m*n), this integral equals /F(fc(Jn)M)e-2^H(Jn))e-2^B(m*n»3^d(Jn), J d(Jn) N proving the result. A. 9. The vector v is in the center of t0 so is fixed under Adc0(K0); also e is in the highest root space so, Adc0 being spherical, e is M0-fixed. By computation sht i 0 —cht £\ 0 0 0 ( cht i 0 —sht ij (i 0 -^ 2t Ad(at)e = e 0 0 0 \i 0 -ij Chapter II 581 Put -\ o o v0 = I 0 % 0 vi v2 0 0 Then the curve t —• Ad(at)v = v0 + §ch2t v\ + |sh 2t v2 > v lies in the intersection of X0 with the plane (si,S2) — o + si^i + £2^2- A. 10. Consider a0 as in [DS], Cor. 7.6, Ch. VIII. The geodesic Ad(exp £(Xy+ X-^))v is easily computed and lies in the plane (si,52) —• v + 5i(X7 - X_7) + s2H1. B. Conical Functions. Part (i) is immediate from Theorem 4.8. For (ii) recall that by Corollary 4.13, —s*/x is the highest weight of the contragredient TTL . For m* we choose /o 0 e\ 1 0 V 0 0/ where e = ±1, the sign determined by det(m*) = 1. Also M consists of the diagonal matrices m with diagonal elements ±1 satisfying det(m) = 1. If g G NMAN, g = n(g)m(g) expB(g)nB[g) then by [DS], IX, Exercise A2, the diagonal matrix expB(g) has entries |Aifo)| expB(g)u = iA*_i(s)r where A;(#) = det((#m)i<*,m ip(m*g • £o) = ip(m*n(g) expB(g) • &,) 1 1 / =(^(exp(-JB(^))n(^)- (m*)- )e,e ) = (^((m*)-1)e,^(exp(B(^))e/) =^(£*)e(~sV)(£?0?)). Now if h = ra*# so g = (m*)_1/i then |Aj(g)| = |A(^)| so the desired formula for \j;(h - ^0) follows. The conical functions in this case are related to "conical polynomials" studied in the book by Faraut and Koranyi [1993]. 582 SOLUTIONS TO EXERCISES C. Hyperbolic Space; Inversion and Support Theorems. C. 1. (i) By orthogonality with the geodesies, the horocycles are the (n — l)-spheres tangential to the boundary \x\ = 1. The induced metric on the horocycle is flat. This is obvious for example in the upper half-space model where N - o is a horizontal plane. (ii) We see that if £0 == N • o and dq the volume element on £0 then (/)VG/ >o) = Jdkj f(gk • q)dq = j [M<°^ f] (p) dq, r where (M f)(p) is the average of / over Sr(p), Thus oo where r = d(o,q) (d = distance in Hn) and p = d'(o,q) {d' = distance on horocycle). It suffices to prove p = sinh r when q is in the xixn-plane so we are in the two-dimensional case. From [GGA] p. 36 (R and N • o are isometric under x —• -~) we see that HlogGqlt9 P = N' The first formula means , , — tanh r or p(l + p2)~ * = tanh r \x + %\ so p = sinh r. Hence oo v r n 2 (/) (p) = Qn_! J(M f)(p)sh - rchrdr. O n_1 (v) (vi) Since the area of Sr(p) is proportional to sh (2r) the formula in (v) follows from [GGA], Ch. II, Prop. 5.26. For (vi) we can write oo r n 3 (/)» = iOn_! j(M f)(p)sh(2r)sh - (r)dr. 0 r (vi)-(vii) Let F(r) = (M /)(p), let Ar = A(£) and assume k even Chapter II 583 > 0. Then k k fsh rsh(2r)&rF(r)dr=(k + 2)(k-2n + 4) f F(r)sh rsh(2r)dr o oo +fc(fc - n + 2) / F(r)shk~2rsh(2r)dr. o If k = 0 £/iis should be -2(n - 2) I / 2F(r)sh(2r)dr + F(0) J . Proof. By the Darboux equation, L applied to (/)v(p) amounts to the application of A(L) = Ar to F(r). Now fshkrsh(2r) (^ + 2(n - 1) coth(2r)^ ) dr = \shkr sh(2r)F/ oo - f F'[shkr ch(2r)2 + fcsh^rch rsh(2r) - 2(n - l)shfcreh(2r)]dr o oo oo = 2(n - 2) fshkrch(2r)F'dr - | fshk-2r sh2(2r)F'dr o o = 2(n - 2) { [sh/srch(2r)Fl °° IL Jo oo - f F[2sh(2r)shkr + A; shfc-Vchrch(2r)]dr} o fc 2 2 -^{[Sh - rsh (2r)Fl°° 2 IL Jo oo - />F[sh/c~2r4sh(2r)ch(2r) + (k - 2)shk~3rch rsh2(2r)]dr} o oo = -2(n - 2) / F[2sh(2r)sh*y + |sh*-2rsh(2r) + k shfc r sh(2r)]dr o oo +| / F[4shfe_2rsh2r + 8shfersh(2r) + (k - 2)(2shfe_2r + 2shkr)sh(2r)}dr 584 SOLUTIONS TO EXERCISES oo = [ F shkrsh(2r)dr {(k + 2)(fc - 2n + 4)} dr o oo fc_2 + / F(r) sh rsh(2r) {ife(fc - n + 2)} dr 0 This means oo (L + (A; + 2)(2n - k - A)) j F(r) shkr sh(2r)dr o oo -(n - fe - •2)Jf2)Jfc //F( F{r)r shk-2rsh(2r)dr. By iteration, A; = n — 3, n — 5, • • • , we obtain (L + (n - l)(n - 1)) ... (L + 2(2n - 4))(/)v = (-l^Q^n - 2)!/. For a different inversion method see Gelfand, Graev and Vilenkin [1966], Ch. V, §2. C. 2. Use [GGA], Ch. IV, Exercise C3 (for the case of a hyperbolic space) and combine with [GGA], Ch. I, Lemma 4.4. (For full details see Helgason [1980b]). D. Conical Distributions. D. 1. (Sketch) To see first that the theorem is local let {Va}aeA be a locally finite covering of V by coordinate neighborhoods and 1 = ^2 ipa a a partition of 1 subordinate to this covering. Then T = Y,(pa(T\Va) where each restriction T\Va is assumed to have the indicated representation with distributions Tniv..np)a on Va. In order to move the ipa past the Xi over on Tni...n a we repeatedly use the formula (p(Xf) = X(f (X1X2^)(v) = | A(X2(/p)(exp(-tX1) • V)| ^ = ( e 1 -jr-rr P( M-t2X2)ex.p(-t1X1) • v) \ [dt1dt2 Jt!=t2=o and if di = d/dti, ((X?...X?)( a a D. 2. Let g be the subalgebra generated by ga and 0_Q,. Then g is semisimple of real rank one and ga = 0-2a + 0-a +Qa+ 02c* + (0°% ([DS], IX, §2). Let ej G 0^. Then Cj,6ej and w = [ej,6ej] span sl(2, C) a c 0 0 which operates on (0 ) . By [GGA], Appendix, Cor. 1.5, gja C [(s ) ,^] so [(fla)o,ej] =9ja- A fortiori [m + a, e^] = Qja so the orbit M • ej has codimension 1 so if the sphere is connected it must be M • ej (cf. Kostant [1975], Ch. II). D. 3. (Sketch following Wallach [1973] and Lepowsky [1975].) (a) 0 = 02c* + 0-a + 0o + Qa + 02a 0o = m + a. Select X G ga, y = —#X Gg_Q, such that the vector H = [X, FJGa satisfies [H,X] = 2X, [ff,y] = -2y. The algebra s = HX + Ry + R# is isomorphic to $£(2, R) and 7r = adg \s is a representation of 5 on 0. Deduce from [GGA], Appendix, Lemma 1.2 (ii) that since the eigenvalues of ad H on 0 are 0, ±2, ±4 the dimensions of the irreducible components of TT can only be 1,3 or 5. (b) Let g1 denote the sum of all the (2i + 1)-dimensional irreducible components of 0 and put fl^jfriflja (0 Then 0* = ®jtfj, Q±2a = 0±2>S±a = 9±1 © 5±2>9o = 0o ® So ® 9o> and the decomposition 9 = 0° 0 01 0 02 is both 5- and f?0-orthogonal. (c) Using [XaiX-a] — B(XOCiX-a)Aa G m, show that 0o Cm, 0O = RAa 0 (QI n m), 0° c m. Let m* = 0* n m (i = 0,1, 2). The m0 is the Lie algebra of M0. (d) For Z G 0 put Z* = [X, Z], Z** - (Z*)*, Z* = [y, Z], Z** = (Z*)*. Prove that if Z G m2, (Z**)* = 4Z*, (Z**)* = 4Z*, (Z*)* = 6Z, (Z*)* = 6Z 586 SOLUTIONS TO EXERCISES and deduce for Y, Z G tri2 [Y,Z**] = [Y*\Z] = l[Z*,Y*\, [Y,Zr = j[Y*,Z*}. (e) Given Z G 0, let Z^ be the component in gl in the decomposition 1 2 0 = 0° 0 0 0 0 . Then if y, Z G m2, [y, z]x = o, [[r, z]0 + 2[y, z]2, z**] - o. (f) Suppose y, Z G m2 and ^(y**, Z**) - 0. Then [y**,z**] - -[z**,y**] Gm, [y**,z**]x = o and [y*,z„] = -6[y,z]*, [[y?z]0z**] = i^5(y,^y)y**. (g) Let [/ G 02a: and select Z G tri2 such that f/ = Z**. Let V be in the orthocomplement (for Be) of U in 02a and select Y G rri2 such that y** = V. Deduce from (f) that [W, U] = V for some W G m0 and consequently M0 • J7 fills up a sphere in 02a- D. 4. For the existence of Sy one can just repeat the proof of Prop. 4.4. Part (a) is obvious. For Part (b) we have by the definition of \I>0, Lemma 3.1 and Cor. 6.2, *o(y>) = y,(^-^o)(0ep(logfl(0)de = / ( NA p oga Now take tp of the form (p(na • £0) = f(n)g{a) where f g(a)e ^ ^da —. 1. Then (b) follows. D. 5. (i) Use Theorem 4.1 and Corollary 6.2. (ii) Use Cor. 6.2. (iii) Use the M-invariance of S and S^. (iv) Prove 2 2 (u +v )£W®T0eCon(Z>£) as an intermediary result, (vi) With the particular g chosen one finds (with r~\n) = f(n(nn))) *((f®g)n~1) = (S + cA5)r~1 and for the particular choice of /, (AS)(fn~1) = 0. Thus h(s) = 5(/n_1) - S(f) and the contradiction h'(0) ^ 0 is obtained by an elementary compu tation. Chapter III 587 D. 6. Solution is similar to that of Exercise D5. For (iv) it is useful to remark the following. Let 9n 912 9i3\ 921 922 923 € SU(2, 1) ( 931 932 933/ and fp -Q 0\ a=\q p 0\eK \0 0 l) such that k(g)M = aM> Then V = {911 + 913)1'{931 + £33), Q = (921 + 923)/{921 + #33) and k(n(nn))M = k{nn)M. E. The Heisenberg Group. E. l.-E. 2. See Faraut and Harzallah [1987]. E. 3. The homogeneity and the left invariance are obvious. Since d(g,e) = \\g\\ only the inequality ||<7i#2|| < ||#i|| + ||#2|| remains to be proved and this just involves the Schwarz inequality (Cygan [1981], Koranyi [1983] or Faraut and Harzallah [1987]). For E. 4, E. 5. and E. 6. see Cowling [1982], Folland [1973] and Koranyi [1982b]. For an exposition of these results see Faraut and Harzallah [1987]. Much of the theory is generalized to N for G/K of rank one in Cowling, Dooley, Koranyi and Ricci [1992]. CHAPTER III A. Differential Operators. A. 1. We have for k G K, g e G, n e AT, a G A iX Vx(kgn) = r,x(g), rjx(ga) = e^ -^^^rjx(g). In the decomposition D(G) = (*D(G) + D(G)n) 0 D(A) let D —• DA denote the projection of D(G) onto D(A). If T G t, X G n and Di,D2 G D(G) we have DiXrix = 0, {TD2r]x){e) = 0, (Dr]X)(e) = (DArix)(e), 588 SOLUTIONS TO EXERCISES and if / G V{G) is right invariant under K, j(TD2Vx)(g)f(g)dg = J (D2rlx)(g)((-T)f)(g)dg = 0. Hence j(DVx)(g)f(g)dg = j (DAVx)(g)f(g)dg G G = (DArix)(e) J Vx(g)f(9)dg = (Drjx)(e) J Vx(g)f(g)dg. G G A. 2. See Helgason [1992a]. B. Rank One Results. B. 1. By the Fourier expansion for a F G £(K/M) (see e.g. [GGA], Ch. V, §3, (13)) we have r ~ d(6) F(e)= J^ d(S) F(k)Y,(S(k)vi,vl)dk 2=1 seKM K where F(k) = F(kM), (v^ is an orthonormal basis of V$ such that v — v\ M span VS . Replacing /c by /cm and integrating over M the sum over i can be restricted to i = 1. D. The Compact Case. D. 1. (i) By calculation (x^x_1)i = cos#. Alternatively, note that 1 3 u —> xux" is a rotation fixing to and tn. (ii) The area of a sphere in S of radius 6 is a constant multiple of sin2 6. (iii) Calculate lim Ff(6)/0. o—•() (iv) The basis zpwq(p + q — £) diagonalizes irt{te) giving the formula for Xi(to)- Then note that by (ii) and the fact that Ff is odd, ie ie Xe(f) = J f{u)xM)du = ^ J(e- - e )Ff{6)Xl{te)de u 2TT F = U ^e-W+WM. Part (v) follows from the fact that \e has L2 norm on U equal to 1 as a result of (ii) and (iii). Part (vi) follows from (iv). For (vii) suppose IT € U is not of the form ne; using (vi) on / = Trace(-7r) we get a contradiction. Chapter III 589 D. 2. likeK we have (lP*f)(u)= ¥>(uv 1)f(v)dv = (p(uv 1)f(vk)dv u u = / (f(uk~1v~1)f(v)dv u which by averaging over K becomes (p(u) / {p{v~l)}(y)dv. u The generalization follows from [GGA], Proposition 2.4 in Ch. IV. D. 3. The dual of the symmetric space G/K is now (U x U)/U* where the diagonal U* is isomorphic to K. Formula (24) in §9 gives *"-{n^}'- Here d(fi) is the degree of the irreducible representation r^ of U x U which has a fixed vector under the diagonal group U* and highest weight /i. The irreducible representations r of U x U are of the form T{UI,U2) = TTI(UI) (8)7r2(u2) where 7TI,7T2 G U (cf. Weil [1940], §17). Here r has a fixed vector under U* if and only if there is a nonzero vector A G V\ ® V2 such that 7Ti(u) V^ being the representation space of 71^. This means for the tensor product 7Ti ®7T2 (TTI 0 7r2)(u)A = A r / Because of the identification V\ 0 F2 = Hom(V 2 5 Vi) A is a linear transfor mation of V2 into Vi so this equation amounts to 7TI(U)AK2(U~1) — A which means TTI and 7r2 contragredient, i.e., iri ~ 7r, 7r2 ~ 7r. Thus /i = (1/, — sz/) where 1/ is the highest weight of n (relative to a maximal abelian subal- gebra t C u) and s is the "maximal" Weyl group element. Considering the relationship between the root system A(uc,tc) and the restricted root system of u x u with respect to t* - {(H, -H) : H G t} ([DS], Ch. VII, §4), where each restricted root has multiplicity 2. Note also for the Killing forms BUXU((H, -H), (H\ H')) = 2BU(H, H'). 590 SOLUTIONS TO EXERCISES Thus TT (M + A <*) = TT (V + Po,0) M+ <**> & bo,® where on the left (,) refers to -BUxu? on the right to Bu, (5 runs over the c c 2 positive roots in A(u ,t ) and p0 half their sum. Since d{n) = d(y) the formula above gives the formula for d(y) the degree of it. E. The Flat Case. E. 2. See Helgason [1980a], §6. E. 3. We have (MyMxf)(z)= J f f(z + £-x + k-y)dkd£ K K I f f(z + £-x + £k-y)dkd£ = f(Mx+kyf)(z)dk. K K K Here we take x = ren, y = sen where en = (0, ••• ,1). Then the last integral is constant for k in the subgroup fixing en so the integral equals L J (Mx+swf)(z)dw. s^-^o) Letting 0 denote the angle between en and w we integrate this last integral n_1 with w first varying in the section of S (0) with the plane (en, y) = cos 6. Since \x -f sw\2 = r2 + s2 — 2rs cos 6 this gives the second expression for (MyMxf)(z). The last is obtained by the substitution t — (r2 + s2 — 2rs cos 6) 2. (For a different proof see John [1955], p. 80; see also Asgeirsson [1937]). F. The Noncompact Case. F. 1. If A G a* then |c(A)|2 = c(A)c(-A) = c(sA)c(-sA). F. 2. The formula iX J f{g) Since the functions F.5. Let As(#) G a be given by geNsexpAs(g)K X and as in (3) §1 put As(gK, kM) = As{k- g). Then As(gK, kM) = sA(gK, kmsM). The p which corresponds to Ns is sp so the formula 1 fs(\,kM) = f{s- \,kmsM) follows easily. CHAPTER IV 1. Writing h in G as h = kan according to the Iwasawa decomposition and using the K-invariance of jfe we have 1 (A x f2)(g • o) = jhigh- . o)/2(ft • 6)dh G 1 2/)loga = I fiign^a- • o)f2(an • o)e ^adn. AN Hence (A x A)A(^i«i • &>) = / (A x AH^im • o)*i, f fiikKunxa,-1 • o)dni Wan • o)e2p{loga)dadn / AN N Interchanging ni and a-1 in the inner integral cancels out the factor e2p(loga") so the expression reduces to / A(*ittia_1 • £o)A>(a ' £o)cfo A 592 SOLUTIONS TO EXERCISES as desired. Since * is commutative whereas x is not the if-invariance condition cannot be dropped. 2. Because of the K-invariance of (/ x $)(x) = f f{g-o) j ^(Aig-1 • x, bVeWte-^Mdbdg. G B Using loc. cit. (47) and (51) this becomes J f(9 • o) j 2 kM = J e 'W*' »dkM J f(g • o) / F(kan • o)dadn = / F(kg • o)dgx = / F(g • o)dgx AN G/K G/K on the function F(y) = f(y)(p(A(x, kM) — A(y, kM)) whereby our integral over G/K becomes / f(kan • o)(p(A(x,kM) — loga)dadn — (f x cp)(kexpA(x1kM)) AN Substituting and using (56) again this gives (/ x $){x) = (/* vy as stated. 3. By definition p Hihk)) (Anh)(F) = fih(A*F) = fF(expH(hk))e- ( dk K and {H(hk) : k G K} = C(h) ([GGA], Ch. IX, Theorem 10.5). 5. (i) The Fourier series (20) §3 converges in the topology of £(R x S1) SO (ii) By Theorems 2.4 and 3.4, if a G £'(£) then the following conditions are equivalent: Chapter V 593 (a) aG (b) CT(I/J) = 0 for each ij; G £(H) satisfying (1) Dnie^n) is odd (n G Z) where Dn denotes (£> -f 1) • • • (D + 2|ra| - 1), D = d/dt. (c) cr(ip) = 0 for each ip G £(H) satisfying ± (2) eVn e (JD;^(R)) (n G Z) * denoting adjoint and subscript e indicating "even", and _L denoting an- nihilator. t If a G £'(H) is such that crn has the form in (ii) then e an = £^Tn where rn G £'e(R>)- If ^ G £(H) satisfies (1) then 2 (e Vn)(^_n) = (D;rn)(eVn) = 0 so a(^) = 0 by (i). Thus by (b) we have a G £'(H2)A. On the other hand, suppose a G £'(H) satisfies (c), that is lk0 Fix /cGZ and use this on the function ip(£,t,e) = ip-k(t)e~ • Then (T(I/J) = 0 implies (e2t CHAPTER V 1. By the symmetry of L X so the conditions are necessary. For the sufficiency, consider the Fourier transform /(A, 6)= f f(x)e(-iX+pKA(x>h»dx. x The conditions amount to /(±ip, b) = 0 so /(A, 6) is divisible by (A, A) + (p, p) and the quotient is holomorphic of uniform exponential type and satisfies (3) in Ch. Ill, §5. By the Paley-Wiener theorem, u exists. 3. (i) See Helgason [1976], (Theorem 8.1); another proof is in Dadok [1979]. (ii) See Helgason [1973b] and Eguchi [1979a]. 594 SOLUTIONS TO EXERCISES 5. See deRham [1955], Ch. V. 4.,6.-7. See Theorems 5.3, 6.1-6.3 in Helgason [1964a]. 8. One has to verify TV — 7T and using (d2/d02)(\0\) = 28 this is a simple matter. 10. (i) If T G u, [T, Ui] — ^2 cijUj where (c^) is skew symmetric. Hence 3 [7>] = ]T[T, JC/ill/i + JUi[T, Ui] i = J2cij(JUj)Ui + J2cij(JUi)Uj = ^Cij{JUj)Ui -Y,cv(Jui)ui = 0- i,j i>3 Similarly, [JT, u}} = E[JT, J^]^ + Yl JUilJT> Ui\ i = - E ^^+E ciiijuwuj) i,3 i,j = l^caiUiUj ~ UjUi) + lY^CiMJUiHJUj) - (JUj)(JUi)) i,3 i,j = \J2 *i v» ui\ +1E °n vu*> Jui\ = °- ij i,3 This proves (i). For (ii) observe that UJ annihilates all C°° functions / on G which are right invariant under K. Thus if um = f we find a contradiction by averaging over right translations by K. 11. (From a discussion with Schlichtkrull). Let v : D(G) —> E(X) be the homomorphism (from Ch. Ill, §10) given by the action of G on X. Then T commutes with each v{D) so by (1) loc. cit. TZ = ZT for each Z G Z(G/K). Let D e T>(G/K). By Theorem 10.1 in Ch III, DZX = Z2 for some Zx ± 0, Z2 e Z(G/K). Then TDZX = TZ2, DTZX = Z2T so (TD - DT)(Zif) = 0 for / e £(X). By the surjectivity of Zx (Theorem 1.4) we conclude TD = DT. 12. The first statement is immediate from the theorem quoted. For the necessity of the condition and for the compact case see Helgason [1992a]. Chapter V 595 13. The equation holds for all / of the form f(kan) = fi(k)f2(a)fs(n), hence for all /. 14. Suppose first / holomorphic on all of D. Since the rotations z —• e%e z belong to the center of K we have (replacing / by fr^) 2?r 2TT 27r /(0) = -^ I f(eiez)d6 =±j f(eiek • z)d0 = //(*• z)dk. 0 0 0 Applying this to the composite function fog(gEGo)we see that / satisfies the mean value theorem (Cor. 2.2) so is harmonic. This argument can be localized since Cor. 2.2 can. 15. We have by (38) in §4, AQ • &ri = \ ^ tanh?/7X_7 + bTl : y1 G R > Ser-r! J proving (i). Part (ii) follows from the fact that the Weyl group consists of all signed permutations. 17. See Proposition 5.2 in Helgason [1987]. The flat case is proved in Menzala and Schonbeck [1984] on the basis of the spherical support theorem [GGA], Ch. I, Lemma 2.7. 18. It suffices to prove this for b = eM and then the function v is TV- invariant. If D G *D(G/K) then AN(D), the AT-radial part of D, is given by &N(D) = ePT(D)oe-P ([GGA], Ch. II, Cor. 5.19); the statement about v is then easily verified. 19. For this we use the transmutation property (1) A(D)D(p = T(D)A(p, (p K - invariant, (Ch. IV, Theorem 4.1) and the Darboux equation (2) DgK ( j f(gk • x) dk\ =DX J f(gk -x)dk. K K Putting f\x) = Jf(k.x)dk, K we have F(gK,\oga) = ep(loga) f (/^'^(an - o) dn. 596 SOLUTIONS TO EXERCISES Applying T(D)a and using (1) this becomes 1 eP(log a) J D(fT(g ))^an.0)dn N which by (2) becomes (\oga) J f J f( . ) dk\ dn = (DF)(gK, a). e P DgK gkan 0 N ' K CHAPTER VI 1. Using a if-invariant Laplace-Beltrami operator on K/M we see that each joint eigenspace E is finite-dimensional. Let E = 0 Ei be the direct decomposition into irreducible subspaces. Pick fi G Ei such that fi(eM) = 1 and fi is M-invariant. Then each fi is a spherical function and Dfi — x(D)fi, where the homomorphism \ : "D(K/M) —> C is the same for all i. Using [GGA], Ch. IV, Cor. 2.3 we find that all fi coincide so E is irreducible. Taking K = SU(2), M = e, each joint eigenspace has to contain a character \ °f K- If T is a maximal torus with Lie algebra spanned by a vector H it is easily seen that H\ is not a constant multiple of \ (°f- e.g. [GGA], Ch. V, Ex. A7). 2. This is a basic step in Bruhat's analysis [1956] §6) of the principal series for G. By Schur's lemma (for unitary rpresentations) (i) is equivalent to the statement that all bounded linear operators A : JC\ —> JC\ commuting with all r\(g) (g G G) are scalars. Let A be one such operator, consider the sesquilinear form B( and the form B(f,g) = B{f\g*) f,geV(G), where f\xMAN)= I f{xamn)e{-iXJrp)^aUmdadn. MAN Then B(fL(x)R(pi) qL(x)R(p2)\ _ e-(2A+p)(logai)e(iA-p)(loga2)jg/Y „\ Chapter VI 597 and by the Schwartz kernel theorem (Hormander [1983], Ch. V) B(f,g)= J f(x)con](g{y))df(x,y), GxG where f eV'(GxG). Then jiL(x,x)R(plip2) _ e(iA+/9)(logai)e(-iA+p)(loga2)^ where L(x,x)R(pi,p2) denotes the diffeomorphism (u,v) —> {xup\,xvp2) of G x G. Consider the diffeomorphism cp : (x,y) —> (y~1x,y) of G x G. Then, if h G V{G x G), we have by the left invariance of T, fv(h) = f (h^1) = T((h However (/^-y^Ocy) = h^iz-'x^z-'y) = h(y-lx,z-ly) SO, ^hcp-^L(z,z) = (hL(e,z)y-\ Thus Ttp(h)=T(p(hL^z)) so T*(/i) = y j h{x,y)dS{x)dy, G G where S G V(G). Since (^_1(x,?/) = (yx,y) this implies T(f®g) = J j f(yx)g(y)dS(x)dy. G G Using the homogeneity of T under R{p\,p2) we obtain the homogeneity condition for S in (ii). The converse follows by reversing the steps. All the commuting operators A are proportional if and only if the corresponding S are proportional and then they must be the example stated. 3. See Helgason [1970a], Ch. Ill, §6. 4. Using Lemma 3.10 and (39) we see quickly that ^\ e = \I/_A e- Thus if (3) (if x *Aie)(fcaMAr) = V-\,e( e(i\-p)(loS a) f ip{kcMN)e(-iX+pW°zc)dc. 598 SOLUTIONS TO EXERCISES Taking (f(kaMN) = (3(kM)j(a) the result follows. 5. In the solution below d and C[ denote compact sets and —» A denotes the interior of a set A. Let C\ C—> G2 C H, let 1)^(5) denote the set of (£) with support in Gi, and let G2' C G satisfy 7r(G|) = G^, 0 C[ C (C2) C G, n : G —• G/MN being the natural mapping. Let C0 be a compact neighborhood of e in MTV and put Ci — C^C0 (i = 1,2). Let /1 € £>(G) be > 0 on G, > 0 on Gi, and supp(/i) C G2. Then the function satisfies / = 99 (cf. (36) §3). Also y? —• / is a continuous mapping of G Thus b 37 in ©Ci(3) into T>c2( )' ^ ( ) §3, # x 77 is a distribution. For the last part one must show Jl>(t)( x r/)(0^(0^- Let /1 G D(G) satisfy fi = i[>. Then this last equation amounts to 1 J fi(g) j f(gh- )d(r]*r(h)dg = I f(g) J Mgh^drj^dg. G G G G However, (77*)~ = 77/ so this last equation is obvious. BIBLIOGRAPHY ABOUELAZ, A. 2001 Integral geometry in the sphere 5^ in "Harmonic Analysis and Integral Geometry". Chapman and Hall/CRC Res. Notes. Math. 422 (2001), Boca Raton, FL, 83-125. ABOUELAZ, A. and EL FOURCHI, O. 2001 Horocylic Radon transform on Damek-Ricci space. (English summary). Bull. Polish. Acad. Sci. Math. 49, no.2 (2001), 107-140. ADIMURTI, KUMARESAN, S. 1979 On the singular support of distributions and Fourier transforms on symmetric spaces. .Ann. Scuola Norm. Sup. Pisa CI. Sci. 6 (1979), 143-150. AGRANOVSKI, M. L. 1985 Invariant function algebras in symmetric spaces. Trans. Moscow. Math. Soc. 47 (1985), 175-197. AGRANOVSKI, M. L., KUCHMENT, P. and QUINTO, E.T. 2007 Range descriptions for the spherical mean Radon transform. J. Funct. Anal. 248 (2007), 344-386. AGRANOVSKI, M. L. and QUINTO, E.T. 1996 Injectivity sets for the Radon transform over circles and complete sets of radial functions. J. Funct. Anal 139 (1996), 383-414. AGRANOVSKI, M. L. and QUINTO, E.T. 2003 Stationary sets for the wave equation on crystallographic domains. Trans. Amer. Math. Soc. 355 (2003), 2439-2451. ANDERSON, A., and CAMPORESI, R. 1989 Intertwining operators for solving differential equations with applications to symmetric spaces. Comm. Math. Phys. 130 (1990), 61-82. ANKER, J-PH. 1990 Lp - Fourier multipliers on Riemannian symmetric spaces of the noncompact type. Ann. of Math. 132 (1990), 597-628. 1991a Handling the inverse spherical Fourier transform. In "Harmonic Analysis on Reductive Groups." (W. Barker and P.Sally eds.) pp. 51-56. Birkhauser, Basel and Boston, 1991. 1991b The spherical Fourier transform of rapidly decreasing functions - a simple proof of a characterization due to Harish-Chandra, Helgason, Trombi and Varadarajan. J. Funct. Anal. 96 (1991), 331-349. 1991c A basic inequality for scattering theory on Riemannian symmetric spaces of the noncompact type. Amer. J. Math. 113 (1991), 391-398. 1992 Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces. Duke Math. J. 65 (1992), 257-297. 599 600 BIBLIOGRAPHY ARTHUR, J. 1983 Paley-Wiener theorems for real reductive groups. Acta Math. 150 (1983), 1-89. ASGEIRSSON, L. 1937 Uber eine Mittelwertseigenschaft von Losungen homogener linearer partieller Differentialgleichungen 2.0rdnung mit konstanten Koefficienten. Math. Ann. 113 (1937), 321-346. ASTENGO, F., CAMPORESI, R. and DI BLASIO, B. 1997 The Helgason Fourier transform on a class of nonsymmetric harmonic spaces. Bull. Austral. Math. Soc. 55 (1997), 405-424. BADERTSCHER, E. 1990 Pompeiu transforms and Radon transforms on Riemannian symmetric spaces. Habilitationsschrift, Bern, 1990. BADERTSCHER, E., and KOORNWINDER, T. H. 1992 Continuous Hahn polynomials of differential operator argument and analysis on Riemannian symmetric spaces of constant curvature. Can. Math. J. 44 (1992), 750- 773. BADERTSCHER, E. and REIMANN, H.M. 1989 Harmonic analysis for vector fields on hyperbolic spaces. Math. Zeitschr. 202 (1989), 431-456. BAGCI, S., and SITARAM, A. 1979 Spherical mean periodic functions on semisimple Lie groups. Pacific. J. Math. 84 (1979), 241-250. BAN, VAN DEN, E.P. 1982 "Asymptotic Expansions and Integral Formulas for Eigenfunctions on a Semisimple Lie Group." Proefschrift, Utrecht, 1982. BAN, VAN DEN, E.P., and SCHLICHTKRULL, H. 1987 Asymptotic expansions and boundary values of eigenfunctions on a Riemannian symmetric space. /. Reine Angew. Math. 380 (1987), 108-165. 1989 Local boundary data of eigenfunctions on a Riemannian symmetric space. Invent. Math. 98(1989), 639-657. 1993 Convexity for invariant differential operators on semisimple symmetric spaces. Composittio Math. (1993). BARKER, W. H. 1975 The spherical Bochner theorem on semisimple Lie groups. J. Fund. Anal. 20 (1975), 179-207. BARLET, D., and CLERC, J. L. 1986 Le comportement a l'infmi des fonctions de Bessel generalisees. I Advan. Math. 61 (1986), 165-183. BARUT,A.D. andRACZKA,R. 1977 "Theory of Group Representations and Applications." Polish Scientific Publishers, Warsaw, 1977. BEERENDS, R.J. 1987 The Fourier transform of Harish-Chandra's c-function and inversion of the Abel transform. Math. Ann. Ill (1987), 1-23. 1988 The Abel transform and shift operators. Compositio Math. 66 (1988), 145-197. BENABDALLAH, A-L, and ROUVIERE, F. 1984 Resolubilite des operateurs bi-invariants sur un groupe de Lie semisimple. C.R. Acad. Sci. Paris. 298 (1984), 405-408. BERENSTEIN, C. and ZALCMAN, L. 1976 Pompeiu's problem on spaces of constant curvature. J. Analyse Math. 30 (1976), 113-130. 1980 Pompeiu's problem on symmetric spaces. Comment. Math. Helv. 55 (1980), 593- 621. BIBLIOGRAPHY 601 BERENSTEIN, C, and SHAHSHAHANI, M. 1983 Harmonic analysis and the Pompeiu problem. Amer. J. Math. 105 (1983), 1217- 1229. BERENSTEIN, C, and CASADIO-TARABUSI, E. 1991 Inversion formulas for the k-dimensional Radon transform in real hyperbolic spaces. Duke Math. J. 62 (1991), 613-631. 1992 Radon- and Riesz transform in real hyperbolic spaces. Contemp. Math. 140 (1992), 1-18. 1993 Range of the k-dimensional Radon transform in real hyperbolic spaces. Forum Math. 5 (1993), 603-616. BERENSTEIN, C, CASADIO-TARABUSI, E. and KURUSA, A. 1997 Radon transform on spaces constant curvature. Proc. Amer. Math. Soc. 125 (1997), 455-461. BERLINE, N., and VERGNE, M. 1981 Equations de Hua et integrales de Poisson. C.R. Acad. Sci. Paris Ser. A 290 (1980), 123-125. In "Non-Commutative Harmonic Analysis and Lie Groups," Lecture Notes in Math.No. 880, pp. 1-51, Springer Verlag, New York , 1981 . BETORI,W., FARAUTJ., and PAGLIACCI, M. The horocycles of a tree and the Radon transform (preprint). BIEN, F.V. " 1997 Helmholtz operators and symmetric space duality. Invent. Math. 129 (1977), 63- 74. BRANSON, T.P., OLAFSSON, G., and SCHLICHTKRULL, H. 1994 A bundle-valued Radon transform with applications to invariant wave equations. Quart. J. Math. Oxford AS (1994), 429-461. 1995 Huygens' principle in Riemannian symmetric spaces. Math. Ann. 301 (1995), 445- 462. BRAY, W. O. and SOLMON, D.C. 1990 Paley-Wiener theorems on rank-one symmetric spaces of noncompact type. Contemp. Math. 113 (1990), 17-30. BRUHAT, F. 1956 Sur les representations induites des groupes de Lie. Bull. Soc. Math. France 84 (1956), 97-205. BURACZEWSKI, D. 2004 The Hua system on irreducible Hermitian symmetric spaces of nontube type. Ann. Inst. Fourier (Grenoble) 54, no. 1 (2004), 81-127. CAMPOLI, O. 1977 The complex Fourier transform on rank one semisimple Lie groups. Thesis, Rutgers University. 1977. CAMPORESI, R. 1993 The spherical transform for homogeneous vector bundles over hyperbolic spaces. Preprint 1993. 1997 The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces. Pacific J. Math. 179 (1997), 263-300. 2005 A generalization of the Cartan-Helgason theorem for Riemannian symmetric spaces of rank one. Pacific J. Math. 222 (2005), 1-27. 2005 The Helgason Fourier transform for homogeneous vector bundles over compact Riemannian symmetric spaces - the local theory, J. Func. Anal. 220 (2005), 97- 117. 2006 The spherical Paley-Wiener theorem on the Grassmann manifolds SU(p+q)/S(UpxUp). Proc. Amer. Math. Soc. 134 (2006), 2649-2659. CARLEMAN, T. 1944 L'integrale de Fourier et les questions qui s'y rattachent. Publ. Sci. Inst. Mittag- Leffler, Uppsala, 1944. CARTAN, E. 1929 Sur la determination d'un systeme orthogonal complet dans un espace de Riemann symetrique clos. Rend. Circ. Mat. Palermo 53 (1929), 217-252. CARTAN, H., and GODEMENT, R. 1947 Theorie de la dualite et analyse harmonique des groupes abeliens localement compacts. Ann. Sci. Ec. Norm. Sup. 64 (1947), 79-99. CASSELMAN, W., and MILICIC, D. 1982 Asymptotic behavior of matrix coefficients of admissible representations. Duke Math. J. 49(1982), 869-930. CEREZO, A., and ROUVIERE, F. 1973 Operateurs differentiels invariants sur un groupe de Lie. Seminaire Goulaouic- Schwartz 1972-1973. Ecole Polytech., Paris., 1973. CHAMPETIER, C, and DELORME, P. 1981 Sur les representations des groupes de deplacements de Cartan. J. Funct. Anal. 43 (1981), 258-279. CHANG, W. 1979 Global solvability of the Laplacian on pseudo-Riemannian symmetric spaces. /. Funct. Anal. 34 (1979), 481-492. 1982 Invariant differential operators and P-convexity of solvable Lie groups. Advan. Math. 46 (1982), 284-304. BIBLIOGRAPHY 603 CHERN, S. S. 1942 On integral geometry in Klein spaces. Ann. of Math. 43 (1942), 178-189. CHEVALLEY, C. 1946 "Theory of Lie Groups." Vol. I. Princeton Univ. Press, Princeton, N.J. 1946. 1955 Invariants of finite groups generated by reflections. Amer. J. Math. 11 (1955), 778- 782. CLERC, J. L. 1976 Une formule de type Mehler-Heine pour les zonal es d'un espace riemannien symetrique. Studia Math. 51 (1976), 27-32. 1980 Transformation de Fourier spherique des espaces de Schwartz. /. Fund. Anal. 37 (1980), 182-202. 1987 Le comportment a l'infmi des fonctions de Bessel generalisees, II. Advan. Math. 66 (1987), 31-61. 1988 Fonctions spheriques des espaces symetriques compacts. Trans Amer. Math. Soc. 306 (1988), 421-431. CLERC, J. L., EYMARD,P., FARAUT, J., RAIS, M., and TAKAHASHI, R. 1982 "Analyse Harmonique." C.I.M.P.A Nice 1982. CLOZEL, L., and DELORME, P. 1984 Le theoreme de Paley-Wiener invariant pour les groupes de Lie reductifs. Invent Math. 11 (1984), 427-433. COHN, L. 1974 Analytic theory of Harish-Chandra's c-function. Lecture Notes in Math. No. 428. Springer-Verlag, Berlin and New York, 1974. CORMACK, A.M., and QUINTO, T. 1980 A Radon transform on spheres through the origin in Rn and applications to the Darboux equation. Trans. Amer. Math. Soc. 260 (1980), 575-581. COWLING, M. 1982 Unitary and uniformly bounded representations of some simple Lie groups. In "Harmonic Analysis and Group Representations" CIME (1980), Liguori Editore, 1982. COWLING, M., and KORANYI, A. 1984 Harmonic analysis on Heisenberg type groups from a geometric viewpoint. In "Lie Group Representations III." pp. 60-100. Lecture Notes in Math. No. 1077, Springer-Verlag, Berlin and New York, 1984. COWLING, M., DOOLEY, A. H., KORANYI, A, and RICCI, F. 1992 H-type groups and Iwasawa decompositions. Advan. Math. 87 (1992), 1-41. COWLING, M., SITARAM, A. and SUNDARI, M. 2000 Hardy's uncertainty principle on semisimple groups. Pacific. J. Math. 192 (2000), 293-296. CYGAN, J. 1981 Subadditivity of homogeneous norms on certain nilpotent Lie groups. Proc. Amer. Math. Soc. 83 (1981), 69-70. DADOK, J. 1976 Fourier transforms of distributions on symmetric spaces. Thesis MIT, 1976. 1979 Paley-Wiener theorem for singular support of K-finite distributions on symmetric spaces. /. Funct. Anal. 31 (1979), 341-354. 1980 Solvability of invariant differential operators of principal type on certain Lie groups and symmetric spaces. /. dAnalyse 37 (1980), 118-127. DAVIDSON, M.G., ENRIGHT, T. J., and STANKE, R. J. 1991 Differential operators and highest weight representations. Mem. Amer. Math. Soc. 94(1991). DEBIARD A., and GAVEAU, B. 1983 Formule d'inversion en geometrie integrate Lagrangienne. C.R. Acad. Sci. Paris, 296(1983), Serie I, 423-425. 604 BIBLIOGRAPHY DEITMAR, A. 2000 Differential operators on equivariant vector bundles over symmetric spaces. Electron. J. Differential Equations 59 (2000), 8 pp. (electronic). DE RHAM, G 1955 "Varietes Differentiates. " Hermann, Paris, 1955. DELORME, P. 1982 Theoreme de type Paley-Wiener pour les groupes de Lie semisimples reels avec une seule classe de conjugaision de sons-groupes de Cartan. J. Funct. Anal. 47 (1982). 26-63. DELORME, P., and FLENSTED-JENSEN, M. 1991 Towards a Paley-Wiener theorem for semisimple symmetric spaces. Acta Math. 167 (1991), 127-151. DIEUDONNE, J. 1978 "Treatise on Analysis," Vol VI, Academic Press, New York, 1978. DIJK, VAN, G. 1969 Spherical functions on the p-adic group PGL(2). Indag. Math. 31 (1969), 213-241. DIXMIER, J. 1964 "Les C* algebres et Leurs Representations," Gauthier-Villars, Paris, 1964. DORAN, R.S. and VARADARAJAN, V.S. (Eds.) 2000 The mathematical legacy of Harish-Chandra. Proc. Symp. Pure Math. 68, Amer. Math. Soc. 2000. DUFLO, M. 1977 Operateurs differentiels bi-invariants sur un groupe de Lie. Ann . Set Ecole Norm. Sup. 10 (1977), 265-288. 1979 Operateurs differentiels invariants sur un espace symetrique. C. R. Acad. Sci. Paris Ser. A 289 (1979), 135-137. DUFLO, M., and RAIS, M. 1976 Sur l'analyse harmonique sur les groupes de Lie resolubles. Ann. Sci. Ec. Norm. Sup. 9(1976), 107-144. DUFLO, M. ,and WIGNER, D. 1979 Convexite pour les operateurs differentiels invariants sur les groupes de Lie. Math. Zeitschr. 167 (1979), 61-80. DUISTERMAAT, J. J. 1984 On the similarity between the Iwasawa projection and the diagonal part. Mem. Soc. Math. France, 15 (1984), 129-138. DUISTERMAAT, J. J., KOLK, J. A. C, and VARADARAJAN, V. S. 1983 Functions, flows and oscillatory integrals on flag manifolds and conjugacy classes in real semisimple Lie groups. Compositio Math. 49 (1983), 309-398. EBATA, M., EGUCHI, M., KOIZUMI, S. and KUMAHARA, K. 2002 The Cowling-Price theorem for semisimple Lie groups. Hiroshima Math. J. 32 (2002), 337-349. EGUCHI, M. 1971 On the Radon transform of the rapidly decreasing functions on symmetric spaces. I, II. Hiroshima Math. J. 1(1971), 1-5, 161-169. 1979a An application of topological Paley-Wiener theorems to invariant differential equations on symmetric spaces. Lecture Notes in Math., No. 739 Springer-Verlag Berlin and New York 1979. 1979b Asymptotic expansions of Eisenstein integrals and Fourier transform on symmetric spaces. J. Funct. Anal. 34 (1979). 1980 Some properties of Fourier transform on Riemannian symmetric spaces. Lect. in Math. Kyoto Univ. No. 14, (1980), 9-43. EGUCHI, M., HASHIZUME, M., and OKAMOTO, K. 1973 The Paley-Wiener theorem for distributions on symmetric spaces. Hiroshima Math. 7.3(1973), 109-120. BIBLIOGRAPHY 605 EGUCHI, M., and KOWATA, A. 1976 On the Fourier transform of rapidly decreasing functions of Lp-type on a symmetric space. Hiroshima Math. J. 6 (1976), 143-158. EGUCHI, M., and KUMAHARA, K. 1982 An LP Fourier analysis on symmetric spaces. J. Fund Anal 47 1982), 230-246. EGUCHI, M. and OKAMOTO, K. 1977 The Fourier transform of the Schwartz space on a symmetric space. Proc. Japan Acad. 53 (1977), 237-241. EHRENPREIS, L. 1956 Solutions of some problems of division. Part III. Amer. J. Math. 78 (1956), 685- 715. 1973 The use of partial differential equations for the study of group representations. Proc. Symp. Pure Math. Vol. XXVI, Amer. Math. Soc.1973, 317-320. 2003 "The Universality of the Radon Transform'' Oxford University Press, Oxford 2003. EHRENPREIS, L., and MAUTNER, F. 1955 -1959 Some properties of the Fourier transform on semisimple Lie groups, I Ann. of Math. 61 (1955), 406-443; II, III. Trans. Amer. Math. Soc. 84 (1957), 1-55; 90 (1959), 431-484. EYMARD. P. 1983 Le noyau de Poisson et I'analyse harmonique non-Euclidienne. In "Topics in Modern Harmonic Analysis." Istituto Nat. Alta Mat. Roma, 1983. EYMARD, P. and LOHOUE, N. 1975 Sur la carre du noyau de Poisson dans les espaces symetriques et une conjecture de Stein. Ann. Set Ec. Norm. Sup. 8 (1975), 179-188. FARAH, S. B., and KAMOUN, L. 1990 Distributions coniques sur le cone des matrices de rang un et de trace nulle. Bull. Soc. Math. France 118 (1990), 251-272. FARAUT, J. 1979 Distributions spheriques sur les espaces hyperboliques. J. Math. Pures Appl. 58 (1979), 369-444. 1982a Analyse harmonique sur les pairs de Guelfand et les espaces hyperboliques. In J.-L Clerc, et. al.. "Analyse Harmonique" C.I.M.PA. Nice 1982, Ch. IV. 1982b Un theoreme de Paley-Wiener pour la transformation de Fourier sur un espace Riemannien symetrique de rang un. /. Funct. Anal. 49 (1982), 230-268. 2003 Analysis on the crown of a Riemannian symmetric space. Amer. Math. Soc. Transl. 210 (2003), 99-110. FARAUT, J., and HARZALLAH, K. 1984 Distributions coniques associees au groupe orthogonal 0(p,q). /. Math. Pure et Appl. 63 (1984), 81-109. 1987 "Deux Cours d'Analyse Harmonique. " Birkhauser, Basel and Boston, 1987. FARAUT, J., and KORANYI, A. 1993 "Analysis on Symmetric Cones." Oxford University Press, New York, 1993. 1990 Function spaces and reproducing kernels on bounded symmetric domains. /. Funct. Anal. 88 (1990), 64-88. FELIX, R. 1993 Radon Transformation auf nilpotenten Lie gruppen. Invent. Math. 112 (1992), 413- 443. FLENSTED-JENSEN, M. 1972 Paley-Wiener theorems for a differential operator connected with symmetric spaces. Ark. Mat. 10(1972), 143-162. 1977a Spherical functions on a simply connected semisimple Lie group. Amer. J. Math. 99 (1977), 341-361. 1977b Spherical functions on a simply connected semisimple Lie group., II. Math. Ann. 228 (1977), 65-92. 606 BIBLIOGRAPHY 1978 Spherical functions on a real semisimple Lie group. A method of reduction to the complex case. /. Func. Anal 30 (1978), 106-146. 1981 K-fmite joint eigenfunctions of U(g)K on a non-Riemannian semisimple symmetric space G/H In "Non-Commutative Harmonic Analysis and Lie Groups," Lecture Notes in Math. No. 880. Springer-Verlag, Berlin and New York, 1981. 1986 "Analysis on Non-Riemannian Symmetric Spaces." Conf. Board Math. Sci. No. 61. Amer. Math. Soc. Providence, RI 1986. FLENSTED-JENSEN, M. and KOORNWINDER, T. 1973 The convolution structure for Jacobi function expansions. Ark. Mat. 11 (1973), 245-262. 1979a Positive-definite spherical functions on a noncompact rank one symmetric space. In "Analyse Harmonique sur les Groupes de Lie II," Springer Lecture Notes No. 739. Springer-Verlag, Berlin and New York, 1979. 1979b Jacobi functions: the addition formula and the positivity of the dual convolution structure. Ark. Mat. 17 (1979), 139-151. FOLLAND, G. B. 1973 A fundamental solution for a subelliptic operator. Bull. Amer. Math. Soc. 79 (1973), 373-376. FUGLEDE, B. 1958 An integral formula. Math. Scand. 6 (1958), 207-212. FURSTENBERG, H. 1963 A Poisson formula for semisimple Lie groups. Ann. of Math. 77 (1963), 335-386. 1965 Translation-invariant cones of functions on semisimple Lie groups. Bull. Amer. Math. Soc. 71 (1965), 271-326. GANGOLLI, R. 1971 On the Plancherel formula and the Paley Wiener theorem for spherical functions on semisimple Lie groups. Ann. of Math. 93 (1971), 150-165. GANGOLLI, R and VARADARAJAN, V.S. 1988 "Harmonic Analysis of Spherical Functions on Real Reductive Groups. " Springer- Verlag, Berlin and New York, 1988. GARDING, L. 1960 Vecteurs analytiques dans les representations des groupes de Lie. Bull. Soc. Math. France 88 (1960), 73-93. GELFAND, I. M. 1950 The center of an infinitesimal group algebra,. Mat. Sb. 26 (1950), 103-112. 1960 Integral geometry and its relation to group representations, Russian Math. Surveys 15 (1960), 143-151. GELFAND, I. M., GINDIKIN, S. G, and GRAEV, M. I. 1982 Integral geometry in affine and projective spaces. /. Soviet Math. 18 (1982), 39- 164. 2003 "Selected Topics in Integral Geometry" Amer. Math. Soc. Transl. Vol. 220, Providence, RI, 2003. GELFAND, L M., and GRAEV, M. I. 1959, 1964 The geometry of homogeneous spaces, group representations in homogeneous spaces and questions in integral geometry related to them. Amer. Math,. Soc. Transl. 37 (1964) GELFAND, I. M., GRAEV, M. I., and SHAPIRO, SJ. 1969 Differential forms and integral geometry. Funct. Anal. Appl. 3 (1969), 24-40. GELFAND, I. M., GRAEV, M. L, and VILENKIN, N. 1966 "Generalized Functions." Vol. 5, Integral Geometry and Representation Theory. Academic Press, New York, 1966. BIBLIOGRAPHY 607 GELFAND, I. M. and NAIMARK, M. A. 1948 An analog of Plancherel's formula for the complex unimodular group. Dokl. Akad. Nauk USSR 63 (1948), 609-612. 1952 Unitary representation of the unimodular group containing the identity representation of the unitary subgroup. Trudy Moscov. Mat. Obsc. 1 (1952), 423- 475. GELFAND, I. M., and RAIKOV, DA. 1943 Irreducible unitary representations of locally compact groups. Mat. Sb. 13 (1943), 301-316. GELLER, D. and STEIN, E. M. 1984 Estimates for singular convolution operators on the Heisenberg group. Math. Ann. 267(1984), 1-15. GILBERT, J.E., and MURRAY, M.A.M. 1991 "Clifford Algebras and Dirac Operators in Harmonic Analysis." Cambr. Univ. Press, 1991. GINDIKIN, S. G. 1967 Unitary representations of groups of automorphisms of Riemannian symmetric spaces of null curvature. Funct. Anal. Appl. 1 (1967), 28-32. 2005a Horospherical Cauchy-Radon transform on compact symmetric spaces. ArXiv: math/0501022vl. 2005b Holomorphic horospherical duality "sphere-cone". Indag. Mathem. NS 16 (3-4) (2005), 487-497. GINDIKIN, S. G., and KARPELEVICH, F. I. 1962 Plancherel measure of Riemannian symmetric spaces of non-positive curvature. Dokl. Akad. Nauk. USSR. 145 (1962), 252-255. 1964 On a problem in integral geometry. Chebotarev Mem. Vol. Kazan Univ. 1964. SelectaMath. Sovietica 1 (1981), 169-184. GLOBEVNIK, J. 1992 A support theorem for the X-ray transform. /. Math. Anal. Appl. 165 (1992), 284- 287. GODEMENT, R. 1948 Les fonctions de type positif et la theorie des groupes. Trans. Amer. Math. Soc. 63 (1948), 1-84. 1952a Une generalisation du theoreme de la moyenne pour les fonctions harmoniques. C.R. Acad. Set Paris 234 (1952). 1952b A theory of spherical functions I. Trans. Amer. Math. Soc. 73 (1952), 496-556. 1957 Introduction aux travaux de A. Selberg. Seminaire Bourbaki, 144, Paris, 1957. GODIN, P. 1982 Hypoelliptic and Gevrey hypoelliptic invariant differential operators on certain symmetric spaces. Ann. ScuolaNorm. Pisa IX (1982), 175-209. GONZALEZ, F. 1984 Radon transforms on Grassmann manifolds. Thesis MIT , 1984. 1987 Radon transforms on Grassmann manifolds. J. Funct. Anal. 71 (1987), 339-362. 1988 Bi-invariant differential operators on the Euclidean motion group and applications to generalized Radon transforms. Ark. Mat. 26 (1988), 191-204. 1990a Bi-invariant differential operators on the complex motion group and the range of the d-plane transform on Cn. Contemp. Math. 113 (1990), 97-110. 1990b Invariant differential operators and the range of the Radon d-plane transform. Math. Ann. 287 (1990), 627-635. 1991 On the range of the Radon transform and its dual. Trans. Amer. Math. Soc. 321 (1991), 601-619. 1993 Range of Radon transform on Grassmann manifolds. In Proc. Conf. "75 Years of Radon Transform." , Vienna , Austria, 1992. International Press, Hong Kong, 1993. 608 BIBLIOGRAPHY 2000 A Paley-Wiener theorem for central functions on compact Lie groups. Radon transforms and tomography (South Hadley, MA, 2000), 131-136, Contemp. Math. 278, Amer. Math. Soc, Providence, RI, 2001. GONZALEZ, F., and HELGASON, S. 1986 Invariant differential operators on Grassmann manifolds. Advan. Math. 60 (1986), 81-91. GONZALEZ, F., and KAKEHI, T. 2006 Moment conditions and support theorems for Radon transforms on affine Grassmann manifolds. Adv. Math. 201 (2006), 516-548. GONZALEZ, F., and QUINTO, E. T. 1994 Support theorems for Radon transforms on higher rank symmetric spaces. Proc. Amer. Math. Soc. 122 (1994), 1045-1052. GONZALEZ, F., and ZHANG, J. 2006 The modified wave equation on the sphere. Integral geometry and tomography, 47- 58, Contemp. Math. 405, Amer. Math. Soc, Providence, RI, 2006. GOODEY, P. and WEIL, W. 1991 Centrally symmetric convex bodies and the spherical Radon transform. Preprint, 1991. GOODMAN, R. 2006 Harmonic analysis on compact symmetric spaces: the legacy of Elie Cartan and Hermann Weyl. (Preprint). GOODMAN, R. and WALLACH, N. 1980 Whittaker vectors and conical vectors. /. Funct. Anal. 39 (1980), 199-279. 1998 Representations and Invariants of the Classical Groups. Encycl. Math, and Appl. 68 Cambr. Univ. Pr., Cambr., 1998. GRINBERG, E. 1985 On the images of Radon transforms. Duke Math. J. 52 (1985), 939-972. 1987 Euclidean Radon transforms; ranges and restrictions. Contemp. Math. 63 (1987), 109-134. 1992 Aspects of flat Radon transform. Contemp. Math. 140 (1992), 73-85 1993a Integration over minimal spheres in Lie groups and symmetric spaces of compact type. Preprint (1993). 1993b Radon transform for maximally curved spheres. In Proc. Conf. "75 Years of Radon Transform." , Vienna, Austria, 1992. International Press, Hong Kong., 1993. GRINBERG, E. and QUINTO, E.T. 2000 Morera's theorems for complex manifolds. /. Funct. Anal. 178 (2000), 1-22. GRINBERG, E. and RUBIN, B. 2004 Radon inversion on Grassmannians via Garding-Gindikin fractional integrals. Ann. of Math. 159 (2004), 809-843. GROSS, K., and KUNZE, R. 1976 Bessel functions and representation theory. J. Funct. Anal. 22 (1976), 73-105. GUARIE, D. 1992 "Symmetries and Laplacians: Introduction to Harmonic Analysis and Applications." North Holland, Amsterdam , 1992. GUILLEMIN, V. 1976 Radon transform on Zoll surfaces. Advan. Math. 22 (1976), 85-99. 1985 The integral geometry of line complexes and a theorem of Gelfand-Graev. Asterisque, (1985), 135-149. 1987 Perspectives in integral geometry. Contemp. Math. 63 (1987), 135-150. GUILLEMIN, V. and STERNBERG, S. 1977 "Geometric Asymptotics." Mathematical Surveys, Amer. Math. Soc. 1977. 1979 Some problems in integral geometry and some related problems in microlocal analysis, Amer. J. Math. 101 (1979), 915-955. BIBLIOGRAPHY 609 GUIVARCH, Y. 1984 Sur la representation integrate des fonctions harmoniques et des fonctions propres positives dans un espace Riemannien symetrique. Bull. Sci. Math. 108 (1984), 373- 392. GUNTHER, P. 1988 "Huygens' Principle and Hyperbolic Equations." Academic Press, Boston, 1988. 1991 Huygens' Principle and Hadamard's conjecture. Math. Intelligencer 13 (1991), 56- 63. HARDY, GH. 1933 A theorem concerning Fourier transforms. J. London Math. Soc. 8 (1933), 227-231. HARISH-CHANDRA 1951 On some applications of the universal enveloping algebra of a semisimple Lie algebra. Trans. Amer. Math. Soc. 70 (1951), 28-96. 1953 Representations of semisimple Lie groups, I. Trans. Amer. Math. Soc. 75 (1953), 185-243. 1954 The Plancherel formula for complex semisimple Lie groups. Trans. Amer. Math. Soc. 76 (1954), 485-528. 1955 Representations of semisimple Lie groups IV, Amer. J. Math. 77 (1955), 743-777. 1956a Representations of semisimple Lie groups, VI. Amer. J. Math. 78 (1956), 564-628. 1956b The characters of semisimple Lie groups. Trans. Amer. Math. Soc. 83 (1956), 98- 163. 1958a Spherical functions on a semisimple Lie group, I. Amer. J. Math. 80 (1958), 241- 310. 1958b Spherical functions on a semisimple Lie group, II. Amer. J. Math. 80 (1958), 553- 613. 1959 Some results on differential equations and their applications. Proc. Nat. Acad. Sci. USA 45 (1959), 1763-1764. 1960 Differential equations and semisimple Lie groups. Collected Papers,Vol. Ill, pp. 6- 56. Springer -Verlag, New York, 1984. 1966 Discrete series for semisimple Lie groups, II. Acta Math. 116 (1966), 1-111. HARZALLAH, K. 1975 Distributions coniques et representations associees a SO0(l,q). In "Analyse Harmonique sur les Groupes de Lie." Lecture Notes in Math. No. 497, pp. 211-229, Springer-Verlag, Berlin and New York, 1975. HASHIZUME, M., MINEMURA, K., and OKAMOTO, K. 1973 Harmonic functions on hermitian hyperbolic spaces. Hiroshima Math. J. 3 (1973), 81-108. HECKMAN, G, and SCHLICHTKRULL, H. 1994 "Harmonic Analysis and Special Functions on Symmetric Spaces." Academic Press, Orlando, 1994. HELGASON, S. 1957 Topologies of group algebras and a theorem of Littlewood. Trans. Amer. Math. Soc. 86 (1957), 269-283. 1959 Differential operators on homogeneous spaces. Acta Math. 102 (1959), 239-299. 1962a "Differential Geometry and Symmetric Spaces," Academic Press, New York, 1962 1962b Some results in invariant theory. Bull. Amer. Math. Soc. 68 (1962), 367-371. 1963 Duality and Radon transforms for symmetric spaces. Amer. J. Math. 85 (1963), 667- 692. 1964a Fundamental solutions of invariant differential operators on symmetric spaces. Amer. J. Math. 86 (1964), 565-601. 1964b A duality in integral geometry; some generalizations of the Radon transform. Bull. Amer. Math. Soc. 70 (1964), 435-446. 1965a The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds. Acta Math. 113 (1965),153-180. 610 BIBLIOGRAPHY 1965b Radon-Fourier transforms on symmetric spaces and related group representations. BullAmer. Math. Soc. 71 (1965), 757-763. 1966a A duality in integral geometry on symmetric spaces. Proc. U.S. - Japan Seminar in Differential Geometry, Kyoto, 1965. Nippon Hyronsha, Tokyo, 1966. 1966b An analogue of the Paley-Wiener theorem for the Fourier transform on certain symmetric spaces. Math. Ann. 165 (1966), 297-308. 1969 Applications of the Radon transform to representations of semisimple Lie groups. Proc. Nat. Acad. Sci. USA 63 (1969), 643-647. 1970a A duality for symmetric spaces with applications to group representations. Advan. Math. 5 (1970), 1-154. 1970b Group representations and symmetric spaces. Actes Congr. Internat. Math. 2 (1970), 313-319. 1972 "Analysis on Lie Groups and Homogeneous Spaces". Conf. Board Math. Sci. Series, No. 14 Amer. Math. Soc, Providence, RI, 1972. 1973a The surjectivity of invariant differential operators on symmetric spaces. Ann. of Math. 98 (1973), 451-480. 1973b Paley-Wiener theorems and surjectivity of invariant differential operators on symmetric spaces and Lie groups. Bull. Amer. Math. Soc. 79 (1973), 129-132. 1974 Eigenspaces of the Laplacian; integral representations and irreducibility. /. Fund. Anal. 17(1974), 328-353. 1976 A duality for symmetric spaces with applications to group representations, II. Differential equations and eigenspace representations. Advan. Math. 22 (1976), 187- 219. 1977a Some results on eigenfunctions on symmetric spaces and eigenspace representations. Math. Scand. 41 (1977), 79-89. 1977b Invariant differential equations on homogeneous manifolds. Bull. Amer. Math. Soc. 83 (1977), 751-774. 1977c Solvability questions for invariant differential operators. In "Colloquium on Group Theoretical Methods in Physics." Academic Press, New York, 1977. 1978 [DS] "Differential Geometry, Lie groups and Symmetric Spaces," Academic Press, New York, 1978. 1979 Invariant differential operators and eigenspace representations. Pp. 236-286 in "Representation Theory of Lie Groups" (M. Atiyah, Ed.) London Math. Soc. Lecture Notes No. 34. Cambridge Univ. Press, London and New York, 1979. 1980a A duality for symmetric spaces with applications to group representations, III. Tangent space analysis. Advan. Math. 30 (1980) 297-323. 1980b Support of Radon transforms. Advan. Math. 38 (1980), 91-100. 1980c 'The Radon Transform" Birkhauser, Basel-Boston, MA, 1980, 2nd Ed. 1999. 1983a Ranges of Radon transforms. AMS Short Course on Computerized Tomography, Jan. 1982. Proc. Symp.Appl. Math. Amer Math. Soc. Providence , R.I. 1983. 1983b The range of the Radon transform on symmetric spaces, in Proc. Conf. Representation Theory of Reductive Lie Groups, Utah, 1982 (P. Trombi , Ed.), pp. 145-151. Birkhauser, Basel and Boston, Mass., 1983. 1983c Operational properties of the Radon transform with applications. In Proc. Conf. Differential Geometry with Applications. Nove Mesto, 1983, 59-75. 1984a [GGA] "Groups and Geometric Analysis; Integral Geometry, Invariant Differential Operators and Spherical Functions." Academic Press, New York, 1984. 1984b Wave equations on homogeneous spaces. In "Lie Group Representations III." Lecture Notes in Math. No. 1077, pp. 254- 287. Springer -Verlag, New York, 1984. 1987 Some results on Radon transforms, Huygens' principle and X-ray transforms. Contemp. Math. 63 (1987), 151-177. 1990 The totally geodesic Radon transform on constant curvature spaces. Contemp. Math.113 (1990), 141-149. BIBLIOGRAPHY 611 1991 Invariant differential operators and Weyl group invariants. In "Harmonic Analysis on Reductive Groups" (W. Barker and P. Sally, eds.) 193-200. Birkhauser, Boston, 1991. 1992a Some results on invariant differential operators on symmetric spaces. Amer. J. Math. 114 (1992), 789-811. 1992b The flat horocycle transform for a symmetric space. Advan. Math. 91 (1992), 232- 251. 1992c Radon transforms for double fibrations. Examples and viewpoints. In Proc. Conf. "75 Years of Radon Transform", Vienna, Austria 1992 (S.G. Gindikin , P. Michor, eds.), International Press , Hong Kong, 1993, 163-179. 1992d Huy gens' principle for wave equations on symmetric spaces. J. Fund. Anal. 107 (1992), 279-288. 1998a Radon transforms and wave equations. Lecture Notes in Math. No. 1684, pp. 99- 121, Springer-Verlag, New York, 1998. 1998b Integral geometry and multitemporal wave equations. Comm. Pure Appl. Math. 51 (1998), 1035-1071. 2000 Harish-Chandra's c-function. A mathematical jewel. In "The Mathematical Legacy of Harish-Chandra" Proc. Symp. Pure Math. 68, Amer. Math. Soc, 2000. 2005 The Abel, Fourier and Radon transforms on symmetric spaces. Indag. Mathem. NS 16 (3-4) (2005), 531-551. 2008 [IGR] "Integral Geometry and Radon Transforms" Springer-Verlag, Berlin and New York, 2008. HELGASON, S., and JOHNSON, K. 1969 The bounded spherical functions on symmetric spaces. Advan. Math. 3 (1969), 586- 593. HELGASON, S., and KORANYI, A. 1968 A Fatou-type theorem for harmonic functions on symmetric spaces. Bull. Amer. Math. Soc. 74 (1968), 258-263. HELGASON, S., RAWAT, R., SENGUPTA, J. and SITARAM, A. 1998 Some remarks on the Fourier transform on a symmetric space. Tech. Rep. Ind. Stat. Inst. Bangalore, 1998. HELGASON, S. and SCHLICHTKRULL, H. 1999 The Paley-Wiener space for the multitemporal wave equation. Comm. Pure Appl. Math. 52 (1999), 49-52. HERGLOTZ, G. 1911 Uber Potenzreihen mit positivem reellem Teil im Einheitskreis. Sitz. Ber. Sachs. Akad. Wiss. 63(1911), 501-511. HERMANN, R. 1964 Geometric aspects of potential theory in bounded symmetric domains, III. Math. Ann. 153 (1964), 384-394. HERTLE, A. 1983 Continuity of the Radon transform and its inverse in Euclidean space. Math. Zeitschr. 184(1983), 165-192. 1984 On the range of the Radon transform and its dual. Math. Ann . 267 (1984), 91-99. HILGERT, J. 1993 Radon transform on half planes via group theory. In Tanner and Wilson [1994]. 1993 Radon transform on Lie groups. Preprint 1993. HOLE, A. 1975 Representations of the Heisenberg group of dimension 2n + 1 on eigenspaces. Math. Scand. 37 (1975), 129-141. HORMANDER, L. 1963 "Linear Partial Differential Operators." Springer-Verlag, Berlin and New York, 1963. 1983 "The Analysis of Linear Partial Differential Operators I, II. Springer-Verlag, Berlin and New York, 1983. 612 BIBLIOGRAPHY 1991 A uniqueness theorem of Beurling for Fourier transform pairs. Ark. Mat. 29 (1991), 237-240. HOTTA, R. 1971 On realization of the discrete series for semisimple Lie groups. /. Math. Soc. Japan 23 (1971), 384-407. HOWE, R. 1980 On the role of the Heisenberg group in harmonic analysis. Bull. Amer. Math. Soc. 3 (1980), 821-843. HOWE, R., and TAN, E.C. 1992 "Non-Abelian Harmonic Analysis; Applications of SL(2,/?)," Springer-Verlag, Berlin and New York, 1992. HU, M-C. 1973 Determination of the conical distributions for rank one symmetric spaces. Thesis MIT 1973. 1975 Conical distributions for rank one symmetric spaces. Bull. Amer. Math. Soc. 81 (1975), 98-100. HUA, L. K. 1963 "Harmonic Analysis of Functions of Several Complex Variables in Classical Domains." Trans. Math. Monographs, Vol. 6, Amer. Math. Soc. 1963. HUANG, J.-S. 2001 Invariant differential operators and eigenspace representations on an affine symmetric space. Ann. of Math. (2) 154 (2001), 703-737. HUANG, J.-S., OSHIMA, T. and WALLACH, N. 1996 Dimensions of spaces of generalized spherical functions. Amer. J. Math. 118, no. 3 (1996), 637-652. ISHIKAWA, S. 1997 The range characterization of the totally geodesic Radon transform on the real hyperbolic space. Duke Math. J. 90 (1997), 149-203. 2003 Symmetric subvarieties in compactifications and the Radon transform on Riemannian symmetric spaces of the noncompact type J. Funct. Anal. 204 (2003), 50-100. JACOBSEN, J. 1982 Invariant differential operators on some homogeneous spaces for solvable Lie groups. Preprint No. 34, Aarhus Univ., 1982. 1983 Eigenspace representations of nilpotent Lie groups, II. Math. Scand. 52 (1983), 321-333. 1985 Eigenspace representations of exponential groups. Preprint, Aarhus Univ., 1985. JACOBSEN, J., and STETILER, H. 1981 Eigenspace representations of nilpotent Lie groups. Math. Scand. 48 (1981), 41-55. JAKOBSEN, H.P. 1985 Basic equivariant differential operators on hermitian symmetric spaces. Ann. Sci. Ec. Norm. Sup. 18 (1985), 421-436. JOHN, F. 1934 Bestimmung einer Funktion aus ihren Integralen iiber gewisse Mannigfaltigkeiten. Math. Ann. 109 (1934), 488-520. 1938 The ultrahyperbolic differential equation with 4 independent variables. Duke Math. J. 4 (1938), 300-322. 1955 "Plane Waves and Spherical Means." Wiley (Interscience), New York, 1955. JOHNSON, K. 1976a Composition series and intertwining operators for the spherical principal series II. Trans. Amer. Math. Soc. 215 (1976), 269-283. 1976b Differential equations and an analog of the Paley-Wiener theorem for semisimple Lie groups. Nagoya Math, J. 64 (1976), 17-29. 1977 Remarks on a theorem of Koranyi and Malliavin on the Siegel upper half plane of rank two. Proc. Amer. Math. Soc. 67 (1977). BIBLIOGRAPHY 613 1978 Differential equations and the Bergman-Shilov boundary on the Siegel upper half plane. Ark. for Mat. 16 (1978), 95-108. 1979a Paley-Wiener theorem for groups of split rank one. /. Funct. Anal. 34 (1979), 54-71. 1979b Partial differential equations on semisimple Lie groups. Proc. Amer. Math. Soc. 60 (1979), 289-295. 1980 On a ring of invariant polynomials on a Hermitian symmetric space. /. of Algebra 67 (1980), 72-81. 1984 Generalized Hua operators and parabolic subgroups. Ann. of Math. 120 (1984), 477- 495. 1987a A strong generalization of Helgason's theorem. Trans. Amer. Math. Soc. 304 (1987), 171-192. 1987b Differential operators and Cartan motion groups. Contemp. Math. 63 (1987), 205- 219. JOHNSON, K. and KORANYI, A. 1980 The Hua operators on bounded symmetric domains of tube type. Ann. of Math. Ill (1980), 589-608. JOHNSON, K. D., and WALLACH, N. 1972 Composition series and intertwining operators for the spherical principal series. Bull. Amer. Math. Soc. 78 (1972), 1053-1059. 1977 Composition series and intertwining operators for the spherical principal series I. Trans. Amer. Math. Soc. 229 (1977), 131-173. KAKEHI, T. 1992 Range characterization of Radon Transforms on complex projective spaces. J. Math. Kyoto Univ. 32 (1992), 387-399. 1993 Range characterization of Radon transforms on Sn and PnR, ibid. 33 (1993), 315- 228. 1995 Range characterization of Radon transforms on quaternionic projective spaces. Math. Ann. 301 (1995), 613-625. 1997 Range theorems and inversion formulas for Radon transforms on Grassmann manifolds. Proc. Japan Acad. Ser. S Math. Sci. 73 (1997), 89-92. 1998 Radon transforms on compact Grassmann manifolds and invariant differential operators of determinantal type. Harmonics analysis and integral geometry (Safl, 1998), Chapman & Hall/CRC Res. Notes Math. 422 Chapman & Hall/CRC, Boca Raton, FL, 2001. 1999 Integral geometry on Grassmann manifolds and calculus of invariant differential operators. /. Funct. Anal. 168 (1999), 1-45. KAKEHI, T. and TSUKAMOTO, C. 1993 Characterization of images of Radon transforms. Progress in differential geometry, 101-116, Adv. Stud. Pure Math. 22, Math Soc. Japan, Tokyo, 1993. KARPELEVICH, F. I. 1965 The geometry of geodesies and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces. Trans. Moscow Math. Soc. 14 (1965), 51-199. KASHIWARA, M., KOWATA, A., MINEMURA, K., OKAMOTO, K, OSHIMA, T., and TANAKA, M. 1978 Eigenfunctions of invariant differential operators on a symmetric space. Ann. of Math. 107(1978), 1-39. KASHIWARA, M. and OSHIMA, T. 1977 Systems of differential equations with regular singularities and their boundary value problems. Ann. of Math. 106(1977), 145-200. KASHIWARA, M. and SCHMID, W. 1994 Quasi-equivariant D-modules, equivariant derived category, and representations of reductive Lie groups. Lie theory and geometry, 457-488, Prog. Math. 123, Birkhauser Boston, Boston, MA, 1994. 614 BIBLIOGRAPHY KAWAZOE, T. 1979 An analog of Paley-Wiener theorem on rank one semisimple Lie groups I, II. Tokyo J. Math. 2 (1979), 397-407, 409-421. 1980 An analog of Paley-Wiener theorem on semisimple Lie groups and functional equations for Eisenstein integrals. Tokyo J. Math. 3 (1980), 219-248. KELLEY, J. L. 1963 "Linear Topological Spaces." Van Nostrand Co. Princeton, N.J. 1963. KNAPP, A.W. 1968 Fatou's theorem for symmetric spaces I. Ann. of Math. 88 (1968), 106-127. 1986 "Representation Theory of Semisimple Lie Groups. An Overview Based on Examples." Princeton Univ. Press., Princeton, NJ.,1986. KNAPP, A.W. and STEIN, E. M. 1971 Intertwining operators on semisimple groups. Ann. of Math. 93 (1971), 489-578. KNAPP, A.W., and WILLIAMSON, R. E. 1971 Poisson integrals and semisimple Lie groups. /. Anal. Math. 24 (1971), 53-76. KOLK, J. and VARADARAJAN, V. S. 1992 Lorentz invariant distributions supported on the forward light cone. Compositio Math. 81 (1992), 61-106. KOORNWINDER, T.H. 1973 The addition formula for Jacobi polynomials and spherical harmonics. SI AM J. Appl. Math. 25 (1973), 236-246. 1974 Jacobi polynomials II. An analytic proof of the product formula. SIAM J. Math. Anal. 5(1974),125-137. 1975 A new proof of a Paley-Wiener theorem for the Jacobi transform. Ark. Mat. 13 (1975),145-159. 1984 Jacobi functions and analysis on noncompact semisimple Lie groups. In "Special Functions : Group Theoretical Aspects and Applications ." Reidel Press, (1984). KORANYI, A. 1970 Generalizations of Fatou's theorem to symmetric spaces. Rice Univ. Stud. 56 (1970), 127-136. 1971 A remark on boundary values in several complex variables. Lecture Notes in Math. No. 185. pp. 1-6. Springer-Verlag, Berlin and New York, 1971. 1972 Harmonic functions on symmetric spaces. In "Symmetric Spaces." (W.M. Boothby and G. L. Weiss, eds.) Marcel Dekker, New York, 1972. 1976 Poisson Integrals and boundary components of symmetric spaces. Invent. Math. 34 (1976), 19-35. 1980 Some applications of Gelfand pairs in classical analysis. In "Harmonic Analysis and Group Representations. " CIME, Cortona, 1980. 1982a On the injectivity of the Poisson transform. /. Fund. Anal. 45 (1982), 293-296. 1982b Kelvin transform and harmonic polynomials on the Heisenberg group. /. Funct. Anal. 49(1982), 177-185. 1983 Geometric aspects of analysis on the Heisenberg group. In "Topics in Modern Harmonic Analysis" Istituto Nat. Alta. Mat. Roma 1983. 1985 Geometric properties of Heisenberg-type groups. Advan. Math. 56 (1985), 28-38. 2008 Cartan-Helgason theorem, Poisson transform and Furstenberg-Satake compactification. (Preprint). KORANYI, A. and MALLIAVIN, P. 1975 Posson formula and compound diffusion associated to an overdetermined elliptic system on the Siegel halfplane of rank two. Acta Math. 134 (1975), 185-209. KORANYI, A., and WOLF, J. A. 1965 Realization of Hermitian symmetric spaces as generalized half planes. Ann. of Math. 81 (1965), 265-288. KOSTANT, B. 1963 Lie group representations on polynomial rings. Amer. J. Math. 85 (1963), 327-404. BIBLIOGRAPHY 615 1969 On the existence and irreducibility of certain series of representations. Bull. Amer. Math.Soc. 75(1969), 627-642. 1975a On the existence and irreducibility of certain series of representations. In. "Lie Groups and Their Representations" (I. M. Gelfand, ed.), pp. 231-329. Halsted, New York, 1975. 1975b Verma modules and the existence of quasiinvariant partial differential operators. In ""Non-Commutative Harmonic Analysis." Lecture Notes in Math. No. 466. Springer-Verlag, New York, 1975. 1991 A formula of Gauss-Kummer and the trace of certain intertwining operators. In "Operator Algebras, Unitary Representations, Enveloping Algebras and Invariant Theory. " Birkhauser, Basel and Boston, 1991. KOSTANT, B., and RALLIS, S. 1971 Orbits and Lie group representations associated to symmetric spaces. Amer. J. Math. 93 (1971), 753-809. KOTHE, G. 1952 Die Randverteilungen analytischer Funktionen. Math. Zeitschr. 57 (1952), 13-33. KOUFANY, K. and ZHANG, G. 2006 Hua operators and Poisson transform for bounded symmetric domains. J. Funct. Anal 236 (2006), 546-580. KOWATA, A., and OKAMOTO, K. 1974 Harmonic functions and the Borel-Weil theorem. Hiroshima Math. J. 4 (1974), 89- 97. KOWATA, A., and TANAKA, M. 1980 Global solvability of the Laplace operator on a non-compact affine symmetric space. Hiroshima Math. J. 10 (1980), 409-417. KROTZ, B. and OLAFSSON, G. 2002 The c-function for non-compactly casual symmetric spaces. Invent. Math. 144 (2002), 647-659. KROTZ, B., OLAFSSON, G. and STANTON, R. 2005 The image of the heat kernel transform on Riemannian symmetric spaces of the noncompact type. Int. Math. Res. Not. 22 (2005), 1307-1329. KUCHMENT, P. A. 1981 Representations of solutions of invariant differential equations on certain symmetric spaces. Sov. Math. Dokl. 24 (1981), 104-106. 1986 On the spectral synthesis in the spaces of solutions of invariant differential equations. Lecture Notes in Math. No. 1214, pp. 85-100. Springer-Verlag, New York, 1986. KUNZE, R, and STEIN, E. 1967 Uniformly bounded representations, III. Amer. J. Math. 89 (1967), 385-442. KURUSA, A. 1991a A characterization of the Radon transform's range by a system of PDE's. /. Math. Anal. Appl. 161 (1991), 218-226. 1991b The Radon transform on hyperbolic spaces. Geom. Dedicata 40 (1991), 325-339. 1994 Support theorems for the totally geodesic Radon transform on constant curvature spaces. Proc. Amer. Math. Soc. 122 (1994), 429-435. LANGLANDS, R. 1963 The dimension of the space of automorphic forms. Amer. J. Math. 85 (1963), 99- 125. LASALLE, M. 1982 Series de Laurent des fonctions holomorphes dans la complexification d'un espace symetrique compact. Ann. Sci. Ecole Norm. Sup. 11 (1978), 167-210. 1984 Les equations de Hua d'un domaine borne symetrique de type tube. Invent. Math. 77(1984), 129-161. LAX, P., and PHILLIPS, R. S. 1967 Scattering Theory. Academic Press, New York, 1967. 616 BIBLIOGRAPHY 1978 An example of Huygens' principle. Comm. Pure Appl. Math 31 (1978), 415-423. 1979 Translation representations for the solution of the non-Euclidean wave equation. Comm. Pure Appl. Math. 32 (1979), 617-667. 1982 A local Paley-Wiener theorem for the Radon transform of L2 functions in a non- Euclidean setting. Comm. Pure Appl. Math. 35 (1982), 531-554. LEE, C.Y. 1974 Invariant polynomials of Weyl groups and applications to the centers of universal enveloping algebras. Can. J. Math. XXVI (1974), 583-592. LEPOWSKY, J. 1975 Conical vectors in induced modules. Trans. Amer. Math. Soc. 208 (1975), 219-272. 1976 Linear factorization of conical polynomials over certain nonassociative algebras. Trans. Amer. Math. Soc. 216 (1976), 237-248. LEWIS, J. B. 1978 Eigenfunctions on symmetric spaces with distribution-valued boundary forms. /. Func. Anal. 29, (1978), 287-307. LEVY-BRUHL, P. 1990 Resolubilite d'operateurs differentiels sur des espaces symetriques nilpotents. /. Funct. Anal. 89 (1990), 303-317. LIMIC, N., NIDERLE, J., and RACZKA, R. 1967 Eigenfunction expansions associated with the second-order invariant operator of hyperboloids and cones, III. /. Math. Phys. 8 (1967), 1079-1093. LINDAHL, L.-A. 1972 Fatou's theorem for symmetric spaces. Ark. Mat. 10 (1972), 33-47. LIONS, J. L. 1952 Supports dans la transformation de Laplace. J. Analyse Math. 2 (1952-53), 123-151. LIONS, J. L., and MAGENES, E. 1963 Problemes aux limites non homogenes (VII) Ann. Mat. Pura Appl. 4 63 (1963), 201-224. LOOMIS, L. H. 1953 "Abstact Harmonic Analysis. " Van Nostrand Reinhold, New York, 1953. LOHOUE. N. and RYCHENER, T. 1982 Die Resolvente von A auf symmetrischen Raumen von nichtkompakten typ. Comment. Math. Helv. 57 (1982), 445-468. LOWDENSLAGER, D. 1958 Potential theory in bounded symmetric homogeneous complex domains. Ann. of Math. 67 (1958), 467-484. LUDWIG, D. 1966 The Radon transform on Euclidean space. Comm. Pure Appl. Math. 23 (1966), 49- 81. MACKEY, GW. 1952 Induced representations of locally compact groups, I. Ann. of Math. 55 (1952), 101- 139. 1953 Induced representations of locally compact groups, II. Ann. of Math. 58 (1953), 193- 221. MADYCH, W. R., and SOLMON, D.C. 1988 A range theorem for the Radon transform. Proc. Amer. Math. Soc. 104 (1988), 79- 85. MALGRANGE, B. 1955 Existence et approximation des solutions des equations aux derivees partielles et des equations de convolution. Ann. Inst. Fourier Grenoble 6 (1955-56), 271-355. MANO, G 2006 Radon transform of functions supported on a homogeneous cone. Thesis, Kyoto University, 2006. BIBLIOGRAPHY 617 MAUTNER, F. I. 1951 Fourier analysis and symmetric spaces. Proc. Nat. Acad. Sci. USA 37 (1951), 529- 533. MAYER-LINDENBERG, F. 1981 Zur Dualitatstheorie symmetrischer paare. /. Reine Angew. Math. 321 (1981), 36- 52. MAZZEO, R.R. and VASY, A. 2005 Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type. J. Funct. Anal. 228 (2005), 311-368. MEANEY, C. 1986 The inverse Abel transform for SU(p,q). Ark. Mat. 24 (1986), 131-140. MENZALA, G.P. and SCHONBECK, T. 1984 Scattering frequencies for the wave equation with a potential term. /. Funct. Anal. 55 (1984), 297-322. MICHELSON, H. L. 1973 Fatou theorems for eigenfunctions of the invariant differential operators on symmetric spaces. Trans. Amer. Math. Soc. 177 (1973), 257-274. MIZONY, M. 1976 Algebres et noyaux de convolution sur le dual spherique d'un groupe de Lie semisimple, non-compact et de rang 1. Publ. Dep. Math. Lyon. 13 (1976), 1-14. MOHANTY, P., RAY, S.K, SARKAR, R.P. and SITARAM, A. 2004 The Helgason-Fourier transform for symmetric spaces. II. /. Lie Theory 14 (2004), 227-242. MOORE, C. C. 1964 Compactifications of symmetric spaces I, Amer. J. Math. 86 (1964), 201-218; II, ibid.358-378. MOSTOW, G.D. 1973 "Strong Rigidity of Locally Symmetric Spaces." Ann. of Math. Studies, Princeton Univ. Press, 1973. NATTERER, F. 1986 "The Mathematics of Computerized Tomography. " Wiley, New York, 1986. NELSON, E. 1959 Analytic vectors. Ann. of Math 70 (1959), 572-615. ODA, H. 2007 Generalization of Harish-Chandra's basic theorem for Riemannian symmetric spaces of non-compact type. Adv. Math. 208, no.2 (2007), 549-596. 0RSTED, B. 1981 The conformal invariance of Huygens' principle. J. Differential Geom. 16 (1981), 1-9. OKAMOTO, K. 1971 Harmonic analysis on homogeneous vector bundles. Lecture Notes in Math., 266, 255-271, Springer-Verlag, New York. OLAFSSON, G, and QUINTO, E. (Eds.) 2006 The Radon Transform, inverse problems and Tomography. Proc. Symposia in Appl. Math. 63 AMS Short Cours Lee. Notes, 2006. OLAFSSON, G, and PASQUALE, A. 2004 Paley-Wiener theorems for the Theta-spherical transform; an overview. Acta Apppl. Math. 81, no. 1-3 (2004), 275-309. OLAFSSON, G., and SCHLICHTKRULL, H. 1992 Wave propagation on Riemannian symmetric spaces. J. Funct. Anal. 107 (1992), 270-278. 2008 Fourier series on compact symmetric spaces, (preprint) 2008 Representation theory, Radon transform and the heat equation on a Riemannian symmetric space, (preprint) 618 BIBLIOGRAPHY OLEVSKY, M. 1944 Some mean value theorems on spaces of constant curvature. Dokl. Akad. Nauk, USSR 45 (1944), 95-98. ORLOFF, J. 1985 Limit formulas and Riesz potentials for orbital integrals on symmetric spaces. Thesis, MIT, 1985. 1990a Invariant Radon transforms on a symmetric space. Contemp. Math. 113 (1990), 233 -242. 1990b Invariant Radon transforms on a symmetric space. Trans. Amer. Math. Soc. 318 (1990). 581-600. OSHIMA, T., SABURI, Y. and WAKAYAMA, M. A note on Ehrenpreis' fundamental principle on a symmetric space. Algebraic Analysis, vol.11. 681-697, Academic Press, Boston, 1988. OSHIMA, T., and SEKIGUCHI, J. 1980 Eigenspaces of invariant differential operators on an affine symmetric space. Invent. Math. 57(1980), 1-81. PALAMODOV, V., and DENISJUK, A. 1988 Inversion de la transformation de Radon d'apres des donnees incompletes. C.R. Acad. Sci. Paris, Math. 307 (1988), 181-183. PALEY, R., and WIENER, N. 1934 "Fourier Transforms in the Complex Domain." Amer. Math,. Soc, Providence , R.I. 1934. PARTHASARATHY, K. R., RANGA RAO, R., and VARADARAJAN, V.S. 1967 Representations of complex semisimple Lie groups and Lie algebras. Ann. of Math. 85 (1967), 383-429. PASQUALE, A. 2000 A Paley-Wiener theorem for the inverse spherical transform. Pacific J. Math. 193 (2000), 143-176. PENNEY, R. 1999 The Paley-Wiener theorem for the Hua system. J. Funct. Anal. 162 (1999), 323- 345. PESENSON, I. 2006 Deconvolution of band limited functions on non-compact symmetric spaces. Houston J. Math. 32 no. 1 (2006), 183-204. 2008 A Discrete Helgason-Fourier transform for Sobolev and Besov functions on noncompact symmetric spaces. Contemp. Math. (2008), (to appear). PHILLIPS, R. S., and SHAHSHAHANI, M. 1993 Scattering theory for symmetric spaces of noncompact type . Duke Math. J. 72 (1993), 1-29 POISSON, S.D. 1820 Nouveaux Memoires de l'Acad. des Sci. Vol III, 1820. QUINTO, E. T. 1981 Topological restrictions on double fibrations and Radon transforms. Proc. Amer. Math. Soc. 81 (1981), 570-574. 1982 Null spaces and ranges for the classical and spherical Radon transforms . J. Math. Anal. Appl. 90 (1982), 408-420. 1983 The invertibility of rotation invariant Radon transforms. /. Math. Anal. Appl. 91 (1983), 510-521; Erratum, ibid . 94 (1983), 602-603. 1987 Injectivity of rotation invariant Radon transforms on complex hyperplanes in Cn. Contemp. Math. 63 (1987), 245-260. 1993 Real analytic Radon transforms on rank one symmetric spaces. Proc. Amer. Math. Soc. 117(1993), 179-186. 1993 Pompeiu transforms on geodesic spheres on real analytic manifolds. Israel J. Math. 84 (1993), 353-363. BIBLIOGRAPHY 619 2006 Support theorems for the spherical Radon transform on manifolds. Internal Math. Research Not. (2006), 1-17. 2008 Helgason's support theorem and spherical Radon transforms. Contemp. Math, (to appear). RADER, C. 1988 Spherical functions on Cartan motion groups. Trans. Amer. Math. Soc. 310 (1988), 1-45. RADON, J. 1917 Uber die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Mannigfaltigkeiten. Ber. Verh. Sachs. Akad. Wiss. Leipzig, Math. Nat. Kl. 69 (1917), 262-277. RAIS, M. 1971 Solutions elementaires des operateurs differentiels bi-invariants sur un groupe de Lie nilpotent. C.R. Acad. Sci. Paris, 273 (1971), 495-498. 1975 Actions de certains groupes dans des espaces de fonctions C00. In "Non- commutative Harmonic Analysis ." Lecture Notes in Math. No. 466, pp. 147-150. Springer-Verlag, Berlin and New York, 1975. 1983 Groupes lineaires compacts et fonctions C00 covariantes. Bull. Sc. Math. 107 (1983), 93-111. RAUCH, J. and WIGNER, D. 1976 Global solvability of the Casimir operator. Ann. of Math. 103 (1976), 229-236. REVUZ, A. 1975 "Markov Chains." North Holl. Publ. Co. New York, 1975. RICHTER, F. 1986a "Differentialoperatoren auf Euclidischen k-Ebenraumen und Radon Transformationen. "Dissertation, Humboldt Universitat, Berlin, 1986. 1986b On the k-dimensional Radon transform of rapidly decreasing functions. Lecture Notes in Math. Springer Berlin, 1986. 1990 On fundamental differential operators and the p-plane Radon transform. Ann. Global Anal. Geom. 8 (1990), 61-75. ROSSMANN, W. 1978 Analysis on real hyperbolic spaces. J. Fund. Anal. 30 (1978), 448-477. ROUVIERE, F. 1976a Sur la resolubilite locale des operateurs bi-invariants. Ann. Scuola Norm. Sup. Pisa 3 (1976), 231-244 1976b Solutions distributions de l'operateur de Casimir. C.R. Acad. Sci. Paris Ser. A-B 282 (1976), 853-856. 1978 Invariant differential equations on certain semisimple Lie groups. Trans. Amer. Math. Soc. 243 (1978), 97-114. 1983 Sur la transformation dAbel de groupes des Lie semisimples de rang un. Ann. Scuola Norm. Sup. Pisa 10 (1983), 263-290. 1986 Espaces symetriques et methode de Kashiwara-Vergne. Ann. Sc. Ec. Norm. Sup. 19 (1986), 553-581. 1990- 1991a Invariant analysis and contractions of symmetric spaces. I, II. Compositio Math. 73 (1990), 241-270; ibid. 80 (1991), 111-136. 1991b Une propriete de symetrie des espaces symetriques. C.R. Acad. Sci. Paris, 313 Seriel,(1991), 5-8. 1994 Fibres en droites sur un espace symetrique et analyse invariante. /. Fund. Anal 124 (1994), 263-291. 1994a Transformations de Radon. Lecture Notes, Universite de Nice, Nice, France, 1994. 2001 Inverting Radon transforms: the group-theoretic approach. Enseign. Math. 47 (2001), 205-252. 620 BIBLIOGRAPHY RUBIN, B. 2002 Helgason-Marchand inversion formulas for Radon transforms. Proc. Amer. Math. Soc. 130 (2002), 3017-3023. 2003 Notes on Radon transforms in integral geometry. Frant. Cole. Appl. Anal. 6, no. 1 (2003), 25-72. 2004a Radon transforms on affine Grassmannians. Trans. Amer. Math. Soc. 356 (2004), 5045-5070. 2004b Reconstruction of functions from their integrals over &-planes. Israel. J. Math. 141 (2004), 93-117. RUDIN, W. 1984 Eigenfunctions of the invariant Laplacian in B. /. D'Anal. Math. 43 (1984), 136- 148. SARKAR, R.P. and SENGUPTA, J. 2008 Beurling's theorem for Riemannian symmetric spaces II. Proc. Amer. Math. Soc. 136, no. 5 (2008), 1841-1853. SARKAR, R.P. and SITARAM, A. 2003 The Helgason Fourier transform for symmetric spaces. In "Perspectives in Geometry and Representation Theorey", Hindustan Book Agency, 2003, 467-473. SCHIFFMANN, G. 1971 Integrales d'entrelacement et fonctions de Whittaker. Bull. Soc. Math. France 99 (1971), 3-72. 1979 Travaux de Kostant sur la serie principale. In "Analyse Harmonique sur les Groupes de Lie, II," Lecture Notes in Math.No. 739, pp.460-510. Springer-Verlag, Berlin and New York, 1979. SCHIMMING, R. and SCHLICHTKRULL, H. 1994 Helmholtz operators on harmonic manifolds. Acta Math. 173 (1994), 235-258. SCHLICHTKRULL,.H. 1984a One-dimensional K-types in finite-dimensional representations of semisimple Lie groups. A generalization of Helgason's theorem. Math. Scand. 54 (1984), 279-294. 1984b "Hyperfunctions and Harmonic Analysis on Symmetric Spaces," Birkhauser, Boston, 1984. 1987 Eigenspaces of the Laplacian on hyperbolic spaces; composition series and integral transforms. J. Fund. Anal. 70 (1987), 194-219. SCHLICHTKRULL, H., and STETICER, H. 1987 Scalar irreducibility of eigenspaces on the tangent space of a reductive symmetric space. J. Funct. Anal. 74 (1987), 292-299. SCHMID, W. 1969 Die Randwerte holomorpher Funktionen auf Hermitesch symmetrischen Raumen. Invent. Math. 9 (1969), 61-80. 1971 On a conjecture of Langlands. Ann. of Math. 93 (1971), 1-42. 1992 Analytic and geometric realization of representations. In "New Developments in Lie Theory and their Applications." (J. Tirao and N. Wallach, eds.). Birkhauser, Boston, 1992. SCHWARTZ, G. and ZHU, C.-B. 1994 Invariant differential operators on symplectic Grassmann manifolds. Manncripta Math. 82(1994), 191-266. SCHWARTZ, L. 1966 "Theorie des Distributions." 2nd ed. Hermann, Paris, 1966. SEMENOV-TJAN-SHANSKI, M. A. 1976 Harmonic analysis on Riemannian symmetric spaces of negative curvature and scattering theory. Math. USSR, Izvestija 10 (1976), 535-563. SEMYANISTY, V. I. 1961 Homogeneous functions and some problems of integral geometry in spaces of constant curvature. Sov. Math. Dokl. 2 (1961), 59-62. BIBLIOGRAPHY 621 SENGUPTA, J. 2002 The uncertainty principle on Riemannian symmetric spaces of the noncompact type. Proc. Amer. Math. Soc. 130 (2002), 1009-1017. SERRE, J.P. 1954 Representations lineares et espaces Kahlerians des groupes de Lie compacts. Exp. 100, Seminaire Bourbaki, Inst. Henri Poincare, Paris, 1954. SHAHSHAHANI, M. 1983 Invariant hyperbolic systems on symmetric spaces. In "Differential Geometry." (R. Brooks et al. eds.) Birkhauser, Basel and Boston, pp. 203-233, 1983. 1989 Poincare inequality, uncertainty principle and scattering theory on symmetric spaces. Amer. J. Math. Ill (1989), 197-224. SHAHSHAHANI, M., and SITARAM, A. 1987 The Pompeiu problem in exterior domains in symmetric spaces. Contemp. Math. 63 (1987), 267-277. SHERMAN, T. 1975 Fourier analysis on the sphere. Trans. Amer. Math. Soc. 209 (1975), 1-31. 1977 Fourier analysis on compact symmetric spaces. Bull. Amer. Math. Soc. 83 (1977), 378-380. 1990 The Helgason Fourier transform for compact Riemannian symmetric spaces of rank one. Acta Math. 164 (1990), 73 -144. SHIMENO, N. 1990 Eigenspaces of invariant differential operators on a homogeneous line bundle on a Riemannian symmetric space. /. Fac. Sci. Univ. Tokyo Sect. I A Math. 37, no. 1 (1990), 201-234. 1996 Boundary value problems for the Shilov boundary of a bounded symmetric domain of tube type. (English summary). J. Funct. Anal. 140, no.l (1996), 124-141. 2001 An analog of Hardy's theorem for the Harish-Chandra transform. Hiroshima Math. J. 31 (2001), 383-390. SHIMURA, G. 1984 On differential operators attached to certain representations of classical groupa. Invwent. Math. 11 (1984), 463-488. 1990 Invariant differential operators on Hermitian symmetric spaces. Ann. Math. 132 (1990), 232-272. SITARAM, A. 1980 An analog of the Wiener Tauberian theorem for the spherical transform on semisimple Lie groups. Fac. J. Math. 89 (1980), 439-445. 1988 On an analog of the Wiener Tauberian theorem for symmetric spaces of the noncompact type. Fac. J. Math. 133 (1988), 197-208. SITARAM, A. and SUNDARI, M. 1997 An analog of Hardy's theorem for very rapidly decreasing functions on semisimple Lie groups. Pacific J. Math. Ill (1997), 187-200. SJOGREN, P. 1981 Characterizations of Poisson integrals on symmetric spaces. Math. Scand. 49 (1981), 229-249. 1984 A Fatou theorem for eigenfunctions of the Laplace-Beltrami operator in a symmetric space. Duke Math. J. 51 (1984), 47-56. 1988 Asymtotic behaviour of generalized Poisson integrals in rank one symmetric spaces and trees. Ann. ScuolaNorm. Sup. Pisa CI. Sci. 15 (1988), 98-113. SOLMON, D. C. 1976 The X-ray transform. /. Math. Anal. Appl. 56 (1976), 61-83. 1987 Asymtotic formulas for the dual Radon transform. Math. Zeitschr. 195 (1987), 321- 343. SOLOMATINA, L. E. 1986 Translation representation and Huygens' principle for an invariant wave equation in a Riemannian symmetric space. Soviet Math. Izv. 30 (1986), 108-111. 622 BIBLIOGRAPHY 1988 Fundamental solutions of invariant differential equations on symmetric spaces. Function Spaces and Eqs. of Mathematical Physics. 51-61 87 Voronezh, Gos. Univ. Voronezh, 1988. SPEH,B.,andVOGAN,D. 1980 Reducibility of generalized principal series representations. Acta Math. 145 (1980), 227-299. STANTON, R. J. 1976 On mean convergence of Fourier series on compact Lie groups. Trans. Amer. Math. Soc. 218 (1976), 61-87. STANTON, R. J. and TOMAS, P. A. 1978 Expansions for spherical functions on noncompact symmetric spaces. Acta Math. 140 (1978), 251-276. 1979 Pointwise inversion of the spherical transform on LP (l 1989 Harmonic analysis as spectral theory of Laplacians. /. Fund. Anal. 87 (1989), 51- 148. 1991 LP harmonic analysis and Radon transform on the Heisenberg group. /. Fund. Anal. 96(1991), 350-406. STROHMAIER, A. 2002 Hardy's theorem for the Helgason Fourier transform on noncompact rank one symmetric spaces. Colloq. Math. 94 (2002), 263-280. 2005 Analytic continuation of resolvent kernels on noncomapct symmetric spaces. (English summary). Math. Z. 250, no. 2 (2005), 411-425. SUGIURA, M. 1990 "Unitary Representations and Harmonic Analysis." 2nd ed. North Holland and Kodansha, Amsterdam and Tokyo, 1990. SULANKE, R. 1966 Croftonsche Formeln in Kleinschen Raumen. Math. Nachr. 32 (1966), 217-241. TAKAHASHI, R. 1963 Sur les representations untaires des groupes de Lorentz generalises. Bull. Soc. Math. France 91 (1963), 289-433. TAKEUCHI, M. 1973 Polynomial representations associated with symmetric bounded domains. Osaka J. Math. 10 (1973), 441-473. TANNER, E.A. and WILSON, R. (Eds.) 1994 "Noncompact Lie Groups and Some of Their Applications. Kluwer Acad. Publishers, 1994. TAYLOR, M.E. 1986 "Noncommutative Harmonic Analysis" Math. Surveys and Monogr. Amer. Math. Soc, Providence RI, 1986. TEDONE, O. 1898 Sull'integrazione dell'equazione d2\lcX2 - I d2 I/dx{2 = 0. Ann. Mat. 1 (1898), 1-24. TERRAS, A. 1985,1988 "Harmonic Analysis on Symmetric Spaces and Applications, I, II. Springer-Verlag, Berlin and New York, 1985, 1988. THANGAVELU, S. 2002 Hardy's Theorem for the Helgason-Fourier Transform on Noncompact Rank One Symmetric Spaces (Eng). Colloq. Math. 94 (2002), 263-280. 2004a An Introduction to the Uncertainty Principle: Hardy's Theorem on Lie Groups, Birkhauser, Boston, 2004. 2004b On theorems of Hardy, Gelfand-Shilov and Beurling for semisimple groups. Pub. Res. Inst. Math. Sci. Kyoto 40 (2004), 311-344. THOMAS, E.G.F. 1984 An infinitesimal characterization of Gelfand pairs. "Proc. Conf. Modern Anal. And Probability", Yale Univ., 1984. TITCHMARSH, E, C. 1939 "The Theory of Functions," 2nd ed. Oxford Univ. Press, London and New York, 1939. TITS, J. 1955 Sur certaines classes d'espace homogenes de groupes de Lie. Acad. Roy. Belg. CI. Sci. Mem, Coll. 29 (1955), No. 3. TORASSO, P. 1977 Le theoreme de Paley-Wiener pour I'espace des fonctions indefinement differentiables et a support compact sur un espace symetrique de type non compact. /. Fund. Anal. 26 (1977), 201-213. 624 BIBLIOGRAPHY TREVES, F. 1966 "Linear Partial Differential Equations with Constant Coefficients." Gordon and Breach, New York, 1966. TRIMECHE, K. 1991 Operateurs de permutations et analyse harmonique associes a des operateurs aux derivees partielles. /. Math. Pure Appl. 9 (1991), 1-73. TROMBI, P., and VARADARAJAN, V. S. 1971 Spherical transforms on semisimple Lie groups. Ann. of Math. 94 (1971), 246-303. VARADARAJAN, V.S 1977 "Harmonic Analysis on Real Reductive Groups." Lecture Notes in Math. No. 576. Springer-Verlag, Berlin and New York, 1977. VILENKIN, N. 1968 "Special Functions and the Theory of Group Representations, " Translation of Math. Monogr. Vol 22. Amer. Math. Soc. Providence, R.I. 1968. VILENKIN, N., and KLIMYK, A.U. 1991-'93 "Representations of Lie Groups and Special Functions. "Vols. I, II, III. Kluwer, Dordrecht, 1991, 1993. VOGAN, D. 1981 "Representations of Real Reductive Groups." Birkhauser, Basel and Boston, 1981. VOGAN, D. and WALLACH, N. 1990 Intertwining operators for real reductive groups. Advan. Math. 82 (1990), 203-243. VRETARE,.L. 1976 Elementary spherical functions on symmetric spaces. Math. Scand. 39 (1976), 343- 358. 1977 On a recurrence formula for elementary spherical functions on symmetric space and its applications. Math. Scand. 41 (1977), 99-112. WALLACH, N. 1973 "Harmonic Analysis in Homogeneous Spaces." Dekker, New York, 1973. 1975 On Harish Chandra's generalized c-functions. Amer. J. Math. 97(1975), 386-403. 1983 Asymtotic expansions of generalized matrix entries of representations of real reductive groups. Lecture Notes in Math. No. 1024, pp. 287-369. Springer-Verlag, Berlin and New York, 1983. 1988, 1992 "Real Reductive Groups I, II. " Academic Press, New York, 1988, 1992. 1990 The powers of the resolvent on a locally symmetric space. Bull. Soc. Math. Belg. 62 (1990), 777-790. 1995 Invariant differential operators on a reductive Lie algebra and Weyl group representations. J. Amer. Math. Soc. 6 (1993), 779-816. WARNER, G. 1972 "Harmonic Analysis on Semisimple Lie Groups," Vols I, II. Springer-Verlag, Berlin and New York, 1972. WAWRZYNCZYK, A. 1985 Spectral analysis and mean periodic functions on rank-one symmetric spaces. Bol. Soc. Mat. Mex. 30 (1985), 15-29. 1984 "Group Representations and Special Functions." Reidel, Dordrecht, 1984. WEIL, A. 1940 "L integration dans les Groupes Topologiques et ses Applications." Hermann, Paris, 1940. WHITTAKER, E.T. and WATSON, G.N. "A Course of Modern Analysis", Cambr. Univ. Press, 1927. WIEGERINCK, J. J. O. O. 1985 A support theorem for the Radon transform on Rr. Nederl. Akad. Wetensch. Proc. A 88 1985. WIGNER, D. 1977 Bi-invariant operators on nilpotent Lie groups. Invent. Math. 41 (1977), 259-264. BIBLIOGRAPHY 625 WILLIAMS, F.L. 1985 Formula for the Casimir operator in Iwasawa coordinates. Tokyo J. Math. 8 (1985), 99-105. WILLIAMS, GD. 1978 The principal series of a p-adic group. Quarterly J. Math. 29 (1978), 31-56. WOLF, J. A. 1964 Self-adjoint function spaces on Riemannian symmetric manifolds. Trans. Amer. Math. Soc. 113 (1964), 299-315 2006 Spherical functions on Euclidean space. /. Funct. Anal. 239 (2006), 127-136. 2007 "Harmonic Analysis on Commutative Spaces" Amer. Math. Soc, 2007. YANG, A. 1998 Poisson transforms on vector bundles. Trans. Amer. Math. Soc. 350, no. 3 (1998), 857-887. ZHANG, G. 2007 Radon transform on real, complex and quaternionic Grassmannians. Duke Math. J. 138(2007), 137-160. ZHELOBENKO, D. P. 1974 Harmonic analysis on complex semisimple Lie groups. Proc. Int. Congr. Math. Vancouver, 1974, Vol. II. ZHU, CHEN-BO. 1992 Invariant distributions of classical groups. Duke Math. J. 65 (1992), 85-119. ZORICH, A.V. 1991 Inversion of horospherical integral transform on Lorentz group and on some other real semisimple Lie groups. RIMS, Kyoto, 1991, 1-37. SYMBOLS FREQUENTLY USED Ad: adjoint representation of a Lie group, 5 ad: adjoint representation of a Lie algebra, 5 A(r): spherical area, 420, 484 A(g): component in g = nexpA(G)k) 86, 99 A(B): space of analytic functions on JB, 530 A'{B)\ space of analytic functionals (hyperfunctions) on B, 530 a, ac, a*, a*: abelian subspaces and their duals, 61 a': 68 a*{6): subset of a*, 237 a+, a+: Weyl chambers in a and a*, 61, 202 la\ transpose, 29 Ay. vector in ac, corresponding to A, 61 A(x,b): composite distance, 99 AQ\ projection from p to a, 289 A: Abel transform, 381 A*: dual Abel transform, 382 A-*'- space of analytic vectors, 416 r Br(p), B (p): open ball with radius r, center p, 3 B: Killing form, 61 *B: set of bounded spherical functions, 341 pR: ball in 3, 364 BC(G): space of bounded continuous functions on G, 336 CI: closure, 3 conj: complex conjugate, 93 Cn: complex n-space, 298 Cn: special set, 6 C(X): space of continuous functions of X, 3 CC(X): space of continuous functions of compact support, 3 C\(G): space of iif-bi-invariant functions in CC(G), 80 CK(X): space of continuous functions with support in K, 3 Co(X): space of continuous functions vanishing at oo, 3 C°°(X)1C^°(X): set of differentiable functions, set of differentiable func tions of compact support, 4 c(A): Harish-Chandra's c-function, 90 cs(X): partial c-function, 141 Cs: generalized c-functions, 234 C+,~C,+C,C~: closures of Weyl chambers and their duals, 129 r, T: isomorphisms of differential operators, 74 rS)y- intertwining operator, 240 Tx(A): Gamma function for X, 284 627 628 SYMBOLS FREQUENTLY USED di'. partial derivative, 3 6: density, 213 V(X): C?(X), 4 V(X): set of distributions on X, 4 r V x\ eigenspace, 76 VK(X): set of / AMN(D), AK(D), AN(D): radial parts of L>, 75, 70 A(£jc,J)c): set of roots, 128 ds(A),es(A): factors in cs(A), 142 E{M): set of all differential operators on M, 36 £(X): C°°(X), 4 £'{X): space of distributions of compact support, 4 £*{X),£[(X): space of K-invariant elements in £(X),£'(X), 207, 381 £\,£(\),£^),£*,£x',£x16' eigenspaces, 76, 229, 282, 531 £^(G): space of if-bi-invariant members of £(G), 381 Ekm- eigenspace of Laplacian, 11 e\}b'- plane wave eigenfunction, 99 F(a,b;c;z): hypergeometric function, 328 / -> /: map from CC(G) to CC(G/H), 26, 155 f6: if-commuting function, 266 T\ spherical transform, 220 T{X)\ function space, 376 g^T^D^. images of g e £{M)^T € £>'(M), operator D under WA: Hilbert space inside £A(X),£A(p), 284, 309, 552 H6(a*): special holomorphic functions on a*, 275 H: Hilbert transform, 6, 390 rjx' if-fixed vector in £A(£), 243 H(g): component in g = k expif(g)n, 99 Im: imaginary part, 261 1(E): space of invariant polynomials on E1, 229 I7: Riesz potential, 6 I(X): group of isometries of X, 50 J2(G): if-bi-invariant Schwartz space, 220 I'x s: intertwining integral, 244 7A,s- normalized intertwining operator, 245 J: inversion, 162 JS(X): polynomial matrix, 287 Jn(z): Bessel function, 285 /CV(a*),/C(a* x B)w' exponential type, slow growth, 271 K,KM- unitary dual and subset, 227, 370 /CA: Hilbert space inside P^(S), 548 X8'. character of 5, 13 £: algebra in Cartan decomposition, 77 L1(X): space of integrable functions on X, 85 LP(X): space of / with |/|* G LX(X), 433 L — Lx'. Laplace-Beltrami operator on X, 5 L(g) = Lg: left translation by g, 5 1(6): dimension, 228 A: operator on P, 7, on H, 93, weight lattice, 240 i: orthocomplement of m in £, 71 A0: operator on Ho, 390 Mp: the tangent space to a manifold M at p, 3 m*: element, 64 Mr: mean-value operator, 77, 484 M(n): group of isometries of Rn, 1 m: centralizer of a in I, 61 Wl: set of continuous homomorphisms, 339 ffll(B): space of measures on JB, 439 J\f: kernel of dual transform, 13, 367 n: part of Iwasawa decomposition, 61 0(n),0(p,q): orthogonal groups, 1, 352 71 1 ftn: area of S ' , 9 Pn: set of hyperplanes in jRn, 8 Pi: space of homogeneous polynomials of degree /, 16 P\,P\: Poisson transform, 300, 100 P6(X): inverse of Qs(\), 236 93: set of positive definite spherical functions, 340 7r(A): product of roots, 91, 154 630 SYMBOLS FREQUENTLY USED p: part of a Cart an decomposition, 61 Q5(\): polynomial matrix, 232 71: ring of functions on A+, 234 R: modified Radon transform, 220 Rn: real n-space, 1 JR+: set of reals > 0, 3 Re: real part, 90 Rg or R(g): right translation by #, 5 Res: residue, 6 p, po> P*: half sum of roots, 61, 323 S*: element, 64 Sn: n-sphere, 7 Sr (p): sphere of radius r and center p, 3 £(#, b): horocycle determined by x and 6, 99 : ^A,S^A,S conical distributions, 135, 142 ^A,<5- generalized Bessel function, 289 Z, Z+: the integers, the nonnegative integers, 3 Z{G): center of D{G), 322 Z(G/K): image of Z(G) in D(G/K), 322 ~: Fourier transform, spherical transform, lift of functions, distributions, 4, 77, 155, 198 A: Radon transform, incidence, 1, 31 V: Dual Radon transform, incidence, 1, 31 *, x: convolutions, adjoint operation, pullback, star operator, Fourier trans form, 6, 9, 26, 80, 82, 96, 137, 200, 557 ®: direct sum, 527 0: tensor product, 12, 108, 112 {,): inner product, 29 fcj,£^: space of if-invariants in E, 86, 90 •: operator, 8, 97 •p: operator on G(p, n), 41 —: closure, 3, restriction, 116 o: interior, 3 _L: annihilator, 16 INDEX A distributions, parametrization of, Abel transform, 381 106 Adjoint representation, 5 function, 105, for SL(n,R), 183 Analytical functionals, 529, 530 representation, 106, model for, 121 Analytic vector, 416 vector, 106 Annihilator, 17 Contraction, 422 Antipodal mapping, 162 Contragredient representation, 118 Convolutions, 6, 9, 26, 96, 98, 137 Cusp form, 3 B Banach algebra topology, 339 D Base, 541 d-plane transform, 45 Bessel function, 285, 289, 292 , 478 ^-spherical transform, 274, 279, 293 Bessel transform Darboux equation, 185 generalized, 293 Differential operator Borel imbedding, 444 image of, 4 Boundary component, 68 invariant, 35 Bounded growth, 418 radial part, 70, 74 Bounded, 29 Dirichlet problem, 422, 460 Bruhat decomposition, 63 Distributions, 4 spaces of, 12 C of compact support, 4 Cartan involution, 61 Double fibration, 30, 32, 388 Cartan subalgebras Dual transform, 1, 9, 85 conjugacy of, 59, 97, 480 inversion of, 20 Casimir operator, 534 Duality Cauchy problem, 410, 469, 497 topology compatible with, 29 Centralizer, 60 for a symmetric space, 62 Character of a representation, 13 Compatibility with projection, 542 E Composite distance, 64, 99 Eigenfunctions Conformal diffeomorphism, 486 of slow growth, 531 Conical exponentially bounded, 531 distribution, 105, 186 Eigenspace, 11 distribution, exceptional, 171 representation, 75 633 634 INDEX for distribution spaces, 540 zonal spherical function, 292 for vector bundles, 541 Geodesic of function spaces, 540 in a horocycle space, 65 Eigenvalue, 11 symmetry, 62 Eisenstein integral, 228, 327 in bounded symmetric domains, 444 Energy Grassmannian, 39 conservation of, 487 Green's kernel, 533 equipartition of, 488 Green's function, 533 kinetic, 487 potential, 487 H Euclidean imbeddings, 122 Haar measure, 26 Evaluation mapping, 29 Harish-Chandra c-function, 90 Exponential type Harish-Chandra imbedding, 443 of a holomorphic function, 261 Harmonic uniform, 261 function, 100, 421 Extreme polynomials, 16 point, 338 Heisenberg group, 190 weight vector, 558, 569 Hilbert transform, 6, 390 weight, 569 Holomorphic, representation, 565 F function of Fatou theorem, 430, 432, 438 exponential type, 261 Fiber, 541 function of Fixed point property, 426 uniform exponential type, 261 Flat in a symmetric space, 65 Homogeneous spaces in duality, 31 Fourier transform, 4, 9, 82 Horocycle, 60 Euclidean, 92 as plane section, 122, 127 on a symmetric space, 197, 199, 202, interior of, 182 315 normal to, 64, 99 self duality under, 197, 208 parallel, 65 Frechet space, 4, 30 plane, 387 Functional on the boundary, 528 transform, 2, 85 Fundamental solution, 402 Hua equations, 461 Funk-Hecke theorem, 18 Huygens principle, 468, 471, 473, 474, Furstenberg compatification, 439 477, 482, 485, 504, 538 converse of, 536 G Hyperbolic space, 184, 378, 398 Gamma function of a Hyperfunction, 530 symmetric space, 284 Hypergeometric functions, 328 Gelfand pair, 340 Hyperplane, 5 Gelfand transform, 340 Generalized Bessel function, 289, 292 I reduction to Incident, 31, 53 INDEX 635 Indivisible roots, 90 Lift Inductive limit, 4 of a function, 155 Inner product, 3 distribution, 155 Intertwining operator, 554, 556 Light cone, 482 Intertwining, 10 Line bundle, 541 Invariant differential operator, 35, 55 Local trivialization, 541 Invariant Lorentzian manifold, 482 operator, 25, 55 distribution, 25 M Inversion problem, 7 Maximal flat, 65 Inversion Maximal theorem, 473 constant curvature spaces for, 51 Maximum principle, 422 ^-spherical transform for, 279 Mean value, 77, 413 for N, 159 Mean value operator, 77 Fourier transform for, 201 expansion for, 78 Grassmann manifolds for, 42, 45 commutativity, 80, 415 horocycle planes for, 391 Measure, 3 horocycle transform for, 89 Moment condition, 11 hyperplane transform for, 5 Multiplicity, 61 spherical transform for, 221, 342 Isotropic vector, 298 N Iwasawa decomposition, 60 Normal to a horocycle, 64 J Normalizer, 61 J-polynomials, 286 O Jacobi functions, 352 Open mapping theorem, 29 transform, 352 Orbital integral, 482 Joint eigenspace, 75, 76 P K Paley-Wiener theorem K-finite joint eigenfunctions, 527 for the Fourier transform on X, 260 K-type, 14 for the K-types, 275 Kelvin transformation, 192 for the Radon transform on X, 365 Kernel, 13, 394 Parallel horocycles, 65 Killing form, 61 Peter-Weyl expansion, 14 Klein-Gordon equation, 472 Pizetti formula, 193 Poisson L integral, 2, 49 Lagrangian subspace, 53 kernel and the dual transform, 100 Laplacian, 5 kernel, 49, 456 Lebesgue differentiation theorem, 432 transform and the dual Legendre polynomial, 18, 52 transform, 102, 300 636 INDEX Polar coordinate representations, 62 conical, 105 Polydisk in a contragredient, 119 bounded symmetric domain, 451 eigenspace, 75, 540, 541 Pompeiu problem, 2 holomorphic, 565 Positive definite, 335 irreducible, 550 Principal series, 564 spherical, 106 Projection map, 541 scalar irreducible, 571 Pseudo-Riemannian manifold, 482 weights of, 127 Restricted roots, 61 Q for bounded symmetric domain, 445 Q-matrices, 232 Restricted weight vector, 127 Restricted weight, 127 R Retrograde cone, 482 Radial, part of a differential operator, Riesz potential, 6 70, 74 Roots, Radon inversion formula, 5 indivisible, 90 Radon transform, 1, 7, 9, 85 restricted, 61 double fibration for, 30 strongly orthogonal, 442 horocycle planes, for, 388 unmultipliable, 129 horocycles, for, 2, 85 injectivity, for, 85 S range problem for, 7 Scalar irreducible, 571 inversion formula for 89 Scattering theory, 90 Plancherel formula for, 89 Schwartz kernel theorem, 597 Range theorems, 11 Schwartz spaces, 4, 8, 214, 384 d-plane transform, 46 Schwarz' theorem, 2 ^-spherical transform, 275 Semi-norms, 4 Fourier transform, 261, 271, 275, Semireflexive, 30 281 Shilov boundary 453 horocycle plane transform, 394 Go-homogeneous, 454 hyperplane transform, 11 Ko-homogeneous, 454 Poisson transform, 529, 530, 531 Poisson kernel for, 456 Rank-one reduction, 137 Simple, 151, 165, 242, 300 Rapidly decreasing function, 4 Singular, 301 Reduced expression, 138 Singular support, 536 Reductive homogeneous space, 70 Slow growth, 91 Reflection of a symmetric space, 67 Solvability, 401, 403, 534, 535 Regular Solvable groups, 426 geodesic, 65 Space of p-planes 39 vector, 153, 301 through the origin, 41 Representation Sphere, area of, 9 adjoint, 5 Spherical character of, 13 function, 86, 340 INDEX 637 function, generalized, 228 Radon transform, 1, 5, 85 reduction to zonal Spherical transform, 90, 274 spherical function, 228 Twisted Radon transform, 44 functional equation for, 240, 278 X-ray transform, 2 principle series, 549 Transmutation operator, 402 compact models for, 550 Transpose map, 29 irreducibility, 550 Transversal manifold, 70 representation, 106 Transversality, 33 model for, 121 TschebyschefT polynomial, 52 transform, 90, 335, 340 Tube type, 462, 465 transform of type J, 275 Twisted transform, 44 vector, 106 zonal spherical function, 292 U Strong topology, 29 Ultrahyperbolic operator, 50 SU(2,l)-reduction, 257 Ultraspherical polynomial, 18 Sub-Laplacian, 191 Support problem, 7 Support theorem, 12, 182, 185, 392 Vector bundle associated to a representation, 542 complex, 541 Tempered distribution, 5 Theta series, 3 W Totally geodesic submanifold, 68 Wave Transform equation, 468 Abel, 381 propagator, 479 Bessel, 293 Weak topology, 29 d-plane transform, 45 Weak* topology, 29 Double fibration, 32, 57 Weight, 127 Dual transform, 1 Weight vector, 123, 127 Fourier, 4, 82, 199, 315 Weyl chamber, 61, 430, 439 Gelfand transform, 340 Weyl group, 61 Hilbert, 6, 390 Whittaker vector, 194 Horocycle transform, 2, 85 Horocycle plane transform, 388 Kelvin, 192 X-ray transform, 2 Poisson transform, 102