Contents

Preface by the editors ix

Biographical note xi

Introduction by S. Helgason xiii

Photos xxvii

Bibliography, up to 2008, of S. Helgason xxxvii

Multipliers of Banach algebras, Ann. of Math. 64 (l956), 240-254. 1

Di!erential opeators on homogeneous spaces, Acta Math. 102 (l959),239-299. 17

Fundamental solutions of invariant di!erential operators on sym-metric spaces, Bull. Amer. Math. Soc. 69 (l963), 778-781. 79

Duality and Radon transform for symmetric spaces, Amer. J.Math. 85 (l963), 667-692. 83

Fundamental solutions of invariant di!erential operators on sym-metric spaces, Amer. J. Math. 86 (l964), 565-601. 109

The Radon transform on Euclidean spaces, compact two-pointhomogeneous spaces and Grassmann manifolds, Acta Math.113 (l965), 153-180 147

Radon-Fourier transforms on symmetric spaces and related grouprepresentations, Bull. Amer. Math. Soc. 71 (l965), 757-763. 175

A duality in integral geometry on symmetric spaces, Proc.U.S.-Japan Seminar in Di!erential Geometry, Kyoto, Japan, l965,pp. 37-56. Nippon Hyronsha, Tokyo l966. 183

Totally geodesic spheres in compact symmetric spaces, Math.Ann. 165 (l966), 309-317. 203

An analogue of the Paley-Wiener theorem for the Fourier transformon certain symmetric spaces, Math. Ann. 165 (1966), 297-308. 213

SIGURDUR HELGASON v

vi CONTENTS

(with A. Korányi) A Fatou-type theorem for harmonic functions onsymmetric spaces, Bull. Amer. Math. Soc. 74 (l968), 258-263. 225

(with K. Johnson) The bounded spherical functions on symmetricspaces, Adv. in Math. 3 (l969), 586-593. 231

Applications of the Radon transform to representations of semi-simple Lie groups, Proc. Nat. Acad. Sci. USA 63 (1969), 643-647. 239

A duality for symmetric spaces with applications to group represen-tations, Adv. in Math. 5 (l970), 1-154. 245

Group representations and symmetric spaces, Proc. Internat.Congress Math. Nice 1970, Vol. 2. Gauthier-Villars, Paris 1971,313-320. 399

A formula for the radial part of the Laplace-Beltrami operator,J. Di!. Geometry 6 (l972), 411-419. 407

Paley-Wiener theorems and surjectivity of invariant di!erentialoperators on symmetric spaces and Lie groups, Bull. Amer. Math.Soc. 79 (l973), 129-132. 417

The surjectivity of invariant di!erential operators on symmetricspaces I, Ann. of Math. 98 (1973), 451-479. 421

Eigenspaces of the Laplacian; integral representations and irreduc-ibility, J. Funct. Anal. 17 (1974), 328-353. 451

A duality for symmetric spaces with applications to group repre-sentations II. Di!erential equations and eigenspace representations,Adv. in Math. 22 (1976), 187-219. 477

A duality for symmetric spaces with applications to group repre-sentations III. Tangent space analysis, Adv. in Math. 36 (1980),297-323. 511

(with F. Gonzalez) Invariant di!erential operators on Grass-mann manifolds, Adv. in Math. 60 (1986), 81-91. 539

Value-distribution theory for holomorphic almost periodic func-tions, The Harald Bohr Centenary (Copenhagen 1987). Mat.-Fys.Medd. Danske Vid. Selsk. 42 (1989), 93-102. 551

The totally geodesic Radon transform on constant curvature spaces,Integral geometry and tomography (Arcata, 1989). Contemp. Math.113 (1990), 141-149. 561

Some results on invariant di!erential operators on symmetric spaces,Amer. J. Math. 114 (1992), 789-811. 571

THE SELECTED WORKSvi

CONTENTS vii

The Flat Horocycle Transform for a Symmetric Space, Adv. inMath. 91 (1992), 232-251. 595

Huygens’ Principle for Wave Equations on Symmetric Spaces, J.Funct. Anal. 107 (1992) 279-288. 615

Radon transforms for double fibrations. Examples and viewpoints,Proc. of Symposium 75 years of Radon Transform, Vienna 1992.Int. Press, Boston 1994, 175-188. 625

Integral Geometry and Multitemporal Wave Equations, Comm.Pure and Appl. Math. 51 (1998) 1035-1071. 639

(with H. Schlichtkrull) The Paley-Wiener Space for the Multi-temporal Wave Equation, Comm. Pure and Appl. Math. 52 (1999)49-52. 677

The Abel, Fourier and Radon Transforms on Symmetric Spaces,Indag. Math. 16 (2005), 531-551. 681

The Inversion of the X-ray Transform on a Compact SymmetricSpace, J. Lie Theory 17 (2007), 307-315. 703

Acknowledgments 713

SIGURDUR HELGASON vii

Preface

The present volume contains a selection of 32 articles by Sigur"urHelgason, covering various aspects of geometric analysis on Riemanniansymmetric spaces. The papers are arranged chronologically and cover atime span of more than 50 years, from 1956 to 2007.

Sigur"ur Helgason has had a profound impact on geometric analysis,more particularly in the geometry of homogeneous spaces, harmonic analys-is on symmetric spaces, the theory of the Radon transform, and the studyof invariant di!erential operators and their eigenfunctions. The collectionis preceded by an introduction by Helgason. Here some background andperspective is given on the individual papers, and they are brought into ahistorical perspective as seen by the author himself today. The introducti-on organizes the articles according to the following list of topics, whichdescribes the evolution of Helgason’s work over the period:

1. Invariant di!erential operators,2. Geometric properties of solutions,3. Double fibrations in integral geometry, Radon transforms,4. Spherical functions and spherical transforms,5. Duality for symmetric spaces,6. Representation theory,7. Fourier transform on Riemannian symmetric spaces,8. Multipliers.

Many people know the work of Helgason primarily from his outstand-ing books. The writing of Sigur"ur Helgason has been a model of expositi-on since the publication in 1962 of Di!erential Geometry and SymmetricSpaces, [B1], which quickly became a classic. It was not only used by stu-dents and researchers in Lie group theory, the first chapter was-and stillis-an excellent introduction to Riemannian geometry. This book, and itssuccessor from 1984, Groups and Geometric Analysis, [B6], was awardedthe Steele price for expository writing in 1988.

In addition to these two classical and prize-winning books, Helgasonhas published several other books, notably The Radon Transform [B4] andGeometric Analysis on Symmetric Spaces [B7]. A complete list of booksand articles up to 2008 is enclosed in this volume. The above citations referto that list.

SIGURDUR HELGASON ix

x PREFACE

The exposition in the original papers is of the same outstanding clarityas that in the books. It is our hope that by collecting these papers, wecan create renewed attention on all those aspects of his work which are notcovered by the books.

We would like to thank Ed Dunne and Cristin Zannella at AMS for theirinvaluable help to get this project started and going. We would also thankthe publishers of the original articles for their cooperation and permissionto include the articles in this volume.

Gestur Ólafsson and Henrik Schlichtkrull

THE SELECTED WORKSx

Biographical note

Sigur"ur Helgason was born on September 30, 1927, in Akureyri, Ice-land. He went to the Gymnasium in Akureyri 1939-1945. After a shortperiod at the University of Iceland in Reykjavík, he began studies at theUniversity of Copenhagen in 1946. Here he received the Gold Medal in 1951for a paper summarized in [60] and the M.Sc. degree in 1952. He went toPrinceton as a graduate student in 1952 and received the ph.D. from therein 1954. He was a Moore instructor at M.I.T. 1954-56, after which retur-ned to Princeton 1956-57. He was at University of Chicago 1957-59, andat Columbia University 1959-60. Since then he has been at M.I.T., wherehe was appointed full professor in 1965. The periods 1964-66, 1974-75,1983 (fall) and 1998 (spring) he spent at the Institute for Advanced Study,Princeton, and the periods 1970-71 and 1995 (fall) at the Mittag-Le#erInstitute, Stockholm. He has been awarded honorary doctor by Universityof Iceland, University of Copenhagen and University of Uppsala. In 1988he received the Steele prize of the A.M.S. for expository writing. He carriesthe Major Knights Cross of the Icelandic Falcon since 1991.

He and his wife live in Belmont, Massachusetts. They have a son anda daughter, and 3 grandchildren.

SIGURDUR HELGASON xi

Introduction

I am deeply grateful to the American Mathematical Society and toGestur Ólafsson and Henrik Schlichtkrull for their interest and their propo-sal to produce this volume with selections of my mathematical papers. Theeditors and publisher kindly asked me to write some explanatory commentson these papers.

My interest in mathematics I trace to my mathematics teacher in theGymnasium in Akureyri in North Iceland, Trausti Einarsson. Although anastronomer, his deep respect for mathematics was highly infectious. Thegeometry textbooks by the remarkable mathematician Ólafur Danielsson,the pioneering founder of mathematics education in Iceland, were writtenby a man with a real mission. The program was on a respectable level —for example the final exam, for students at age 17, included finding theradius of a sphere inscribed in a regular dodecahedron of edge length 1.Danielsson, who wrote several beautiful papers in geometry along with hisdemanding Gymnasium teaching, was profoundly influenced by the richgeometry tradition in Denmark, which started around 1870 with J. Petersenand H. Zeuthen.

When I entered the Gymnasium in Akureyri around 1940, the Icelandicpopulation totaled about 125,000. Thus the Gymnasium program putconsiderable emphasis on the teaching of several foreign languages. OurFrench teacher, Thórarinn Björnsson, was particularly memorable becauseof his outspoken admiration of French authors like Maupassant, Daudetand Rolland, in addition to his fondness for the French language in whichhe left us with a decent reading ability. Thus my later di$culties in und-erstanding Élie Cartan’s papers were not of linguistic nature.

Later, at the University of Copenhagen, I benefited from splendid lect-ures by Harald Bohr, Børge Jessen and Werner Fenchel. The first two gavesubstantial courses in complex function theory and real analysis, respecti-vely. With Fenchel I had also a kind of reading course in projective geome-try. At that time I obtained a copy of Julius Petersen’s remarkable book,“Methods and Theories for Solutions of Geometric Construction Problems”(1866). I worked on many of the 300 problems there. Example of suchproblems: Construct a triangle ABC from ha, ma and r. This was cons-istent with my expectation later to teach at a Gymnasium in Iceland.

SIGURDUR HELGASON xiii

xiv INTRODUCTION

One day I happened to notice a posting of the collection of prize qu-estions which were posed by the university each year. The mathematicsproblem on the list called for a generalization to analytic almost peri-odic functions of the value-distribution theory of Ahlfors and Nevanlinna.Everyone under 30 was free to try for a year. A solution was to be su-bmitted and judged anonymously so I could not consult anybody. Thusfour months went by before I understood what the problem was about. Mypaper solving the problem became my masters thesis. A summary is givenin paper [60] in this volume.

After completing my thesis in Princeton enjoying supportive and sti-mulating supervision by Bochner, my interest shifted to Lie group theorywith the intention of studying Harish–Chandra’s work. A related interestwas geometric analysis, having received from my friend Leifur Ásgeirssonthe page proofs of Fritz John’s famous book Plane Waves and SphericalMeans. Although the term ‘group’ does not appear in this book I sensedquickly that some of its themes were ripe for group-theoretic variations.

At an early stage it was a source of inspiration to me how at thehands of Sophus Lie, the study of di!erential equations led to the conceptof Lie groups which in turn went on to penetrate numerous other fieldsof mathematics. Since Lie group theory had in the mid-fifties become sowell developed I became interested in a kind of converse to Lie’s program,namely to investigate di!erential operators and di!erential equations in-variant under known groups. Examples were readily at hand: the Laplace–Beltrami operator on Riemannian or pseudo-Riemannian manifolds, theDirac operator, the wave equation and its variations.

In the account that follows I describe some of my own work connectedto analysis on homogeneous Spaces. This is accompanied by brief mentionof some prior and subsequent work which is closely related. These personalviewpoints do not represent any attempt at a historical account, the featureof relevance and of value being highly subjective. As Peter Lax aptlyobserved: “For a mathematician the central problem is the one he happensto be working on”.

Inspired by Harish–Chandra’s brilliant work on the representationtheory of semisimple Lie groups G, my taste for geometry led me to a rela-ted topic, namely a study of the symmetric spaces of É. Cartan. Here enterseveral di!erential geometric features: geodesics, curvature, K-orbits, tota-lly geodesic submanifolds and horocycles; for the compact dual spaces U/Kwe have also closed geodesics, especially those of minimal length, equators,antipodal manifolds, and totally geodesic spheres. Geometric analysis me-ans, in this context, analysis formulated in terms of these geometric objects.

I have had the fortune of teaching for 50 years at M.I.T. but havealso enjoyed the privilege of extended, stimulating visits at the Institutefor Advanced Study in Princeton and at the Mittag–Le#er Institute inDjursholm, Sweden.

THE SELECTED WORKSxiv

Introduction xv

In conclusion I want to emphasize my wife’s role in my life and work.Her support during these 50 years has been precious, and her tolerance, attimes when nothing but mathematics mattered, has been admirable. Sheeven typed my first two books in pre-TEX times, giving stylistic suggestionsalong the way.

1. Invariant Di!erential Operators.

Articles:1 [6], *[9], [13], *[16], *[23], *[32], *[35], *[37], *[43], [44], [46],[49], [55], *[59], *[63], [64], *[71], *[81], *[82].Books: [B2], [B6], [B7]The general project outlined above suggested the task of describing thealgebra D(G/H) of G-invariant di!erential operators on a homogeneousspace G/H. For G/H reductive some preliminaries are done in [9]; forG/H of maximum mobility, namely two-point homogeneous, the algebrais easily shown to be generated by the Laplace–Beltrami operator. ForG/K symmetric, Harish–Chandra’s results for G implied that D(G/K)is commutative and could in fact be described rather explicitly. However,contrary to a statement appearing frequently in the literature, the operatorsin D(G/K) are not all induced from operators in the algebra Z(G) of bi-invariant di!erential operators on G. While this was verified in [16] tobe true for all classical irreducible G/K, it fails for exactly four of theexceptional G/K [63]. However, even here, each D is induced by a fractionZ1/Z2 (Zi ! Z(G)), that is !Z1D = !Z2 where !Zi ! D(G/K) is inducedby Zi via the action of G on G/K. The algebra D(G/K) is the subject ofChapter II in [B6] and is related to the Iwasawa decomposition

G = NAK , g = n + a + k , g = n expA(g)k , A(g) ! a .

Given D ! D(G/K) and its radial part under the action of N onX = G/K, it is proved in [16] that the Abel transform (the Radon trans-form on K-invariant functions on X) is a simultaneous transmutationoperator for all D ! D(G/K), that is, “converts” them into constantcoe$cient operators on A. Combining this with Harish–Chandra’s workon the spherical transform and Hörmander’s divisibility theorem for tem-pered distribution on Rn, I showed in [16] that each D ! D(G/K) has afundamental solution. The later Paley–Wiener theorem furnishes a short-er proof but requires the surjectivity of a constant coe$cient di!erentialoperator on D!(Rn).

The next step was to prove the surjectivity of each D ! D(G/K) onthe space C"(G/K), i.e., the existence of a global solution to the equationDu = f for arbitrary smooth f . Already in some notes in 1964 I hadreduced this to a support theorem for the horocycle Radon transform on X.I used Harish–Chandra’s expansion for Eisenstein integrals but was stymiedby singularities on a#c whose location seemed unpredictable. But by looking

1Papers in this volume are indicated by an asterisk.

SIGURDUR HELGASON xv

xvi Introduction

at the simplest possible examples I discovered, but not until 1972, that thesesingularities could be canceled out by a certain intertwining procedure.This yielded the support theorem for the horocycle Radon transform andthereby the surjectivity

D C"(X) = C"(X)

follows in [37]. The local solvability of each Z ! Z(G) is also proved thereinvoking methods of Raïs and Harish-Chandra. General local solvabilitywas later proved by Duflo. The global solvability however fails (Cerèzo–Rouvière, Sem. École Polytech. (1973)) although it holds for the Casimiroperator (Cerèzo–Rouvière loc. cit., Rauch and Wigner, Ann. of Math.(1976)). Other surjectivity results are proved in [35], [37], [43]. The non-Riemannian symmetric case has been investigated by Duflo, van den Banand Schlichtkrull and others. The support theorem above was given anew proof by Gonzalez and Quinto. This approach has led Quinto andcollaborators to various interesting support theorems in much more generalcontext.

2. Geometric Properties of Solutions.

Articles: *[9], *[15], *[16], *[23], [25], *[32], *[40], [55], *[71], [78], *[81],*[82].

In his paper (C.R. Acad. Sci. Paris (1952)) Godement defined a harmonicfunction on G/K as a solution to Du = 0 for all D ! D(G/K) annihilat-ing the constants. He proved that they had a characterization in termsof mean values over orbits of K and their translates. In [9] I proveda similar extension of Ásgeirsson’s mean-value theorem for solutions ofLx(u(x, y)) = Ly(u(x, y)) on Rn " Rn (L-Laplacian) to the systemDxu = Dyu on G/K " G/K for D ! D(G/K). Inspired by work ofÁsgeirsson and Günther on Huygens’ principle I became interested in stu-dying the wave equation on symmetric spaces G/K and on Lorentzianmanifolds of constant curvature. In both cases the natural equation isLx + c = !2

!t2 , where c is a constant related to the scalar curvature. Papers[16], [45], [55], [58], [71], [78] and a chapter in [B7] deal with explicitsolution formulas for the natural wave equation on G/K and the occur-rence of Huygens’ principle. My work on these wave equations is relatedto extensive contributions by Branson, Ólafsson, Lax, Phillips, Schlicht-krull, Shahshahani, Solomatina and Ørsted. Papers [81] and [82] (the latterwith Schlichtkrull) deal with a hyperbolic system on G/K introduced bySemenov–Tian–Shansky, Izvestija (1976). The time variable has dimensi-on rank(X) and for rank(X) = 1 it reduces to the above wave equati-on. Developing further the work of Semenov–Tian–Shansky, Shahshahaniand Phillips, paper [81], gives an explicit solution of the Cauchy problemin terms of the Radon transform and in terms of the Fourier transform.Paper [82] with Schlichtkrull deals with an associated range problem.

THE SELECTED WORKSxvi

Introduction xvii

For a Lorentzian manifold X of constant curvature, paper [9] sets upan analog of Riesz potentials and used properties of these to show that afunction u on X is determined by a formula

u = c limr$0

rn%2Q(!)Mru ,

where (Mru)(x) is the “mean value” of u over a “Lorentzian sphere” ofradius r and center x, Q(!) being a polynomial in the Laplacian ! on X.This can be used to deduce a result of Schimming and Schlichtkrull (ActaMath. (1994)) that each factor ! + c in Q(!) satisfies Huygens’ principle.

It was proved by Furstenberg and Karpelevic that a bounded solutionof the Laplace equation Lu = 0 on the symmetric space G/K is necessarilyharmonic (in Godement’s sense above). This led them to a Poisson typeintegral formula for such functions. While the analog of the Schwarz theor-em for continuous boundary values is easy, in [23] Korányi and I provedthe corresponding Fatou theorem for such functions u(x); namely

limt$"

u(!(t)) exists

for almost all geodesics, starting at the origin o = eK. The result hasmany refinements, variations and generalizations due to Knapp, Korányi,Lindahl, Michelson, Schlichtkrull, Sjögren, Stein, Williamson and others.

3. Double Fibrations in Integral Geometry.Radon Transforms.

Articles: *[9], *[15], [17], *[18], *[19], *[20], *[28], [52], [53], *[62], *[69],*[74], [78], *[91], *[92].Books: [B4], [B6], [B7].

In Radon’s pioneering paper (1917) a function f on R2 is explicitly determ-ined by its line integrals. In projective geometry points and lines are puton equal footing. Observing that R2 and the set P2 of lines are bothpermuted transitively by the same group M(2), the isometry group of R2,the following generalization was proposed in [20]. Given a locally compactgroup and K and H two closed subgroups satisfying some natural conditi-ons we consider the coset spaces X = G/K, ! = G/H. Elements x = gK," = !H are (by Chern (1942)) said to be incident if the sets gK and !Hintersect as subsets of G. Given x ! X, " ! ! let

x = {" ! ! : x, " incident} ,

"" = {x ! X : x, " incident} .

Then we have two integral transforms

f ! Cc(X) # "f ! C(!) , # ! Cc(!) # # ! C(X)

SIGURDUR HELGASON xvii

xviii Introduction

given by"f(") =

#

""

f(x) dm(x) , #(x) =#

x

#(") dµ(")

with canonically defined measures dm and dµ. In group-theoretic termswith L = K $ H, dkL , dhL invariant measures on K/L and H/L we have

"f(!H) =#

H/L

f(!hK) dhL , #(gK) =#

K/L

#(gkH) dkL .

These geometrically dual transforms are also adjoint:#

X

f(x)#(x) dx =#

!

"f(")#(") d" ,

where dx and d" are invariant measures, dx = dgK , d" = dgH .The principal problems for these transforms f # "f , # # # would be

(i) Injectivity(ii) Inversion Formulas(iii) Range Questions(iv) Support Theorems. These are results of the following type:

Cc(X) = {f ! C(X) : "f ! Cc(!)} , but there are many variations.

In addition to considering the classical case X = Rn, ! = Pn, thespace of hyperplanes, where I determined the image of D(Rn) under theRadon transform, I obtained in [9] and [62] inversion formulas for the casewhen X is a Riemannian manifold of constant curvature and ! the family oftotally geodesic submanifolds of a given dimension. This was later extensi-vely studied by Semyanisty, Rubin, Palamodov, Rouvière, Kurusa, Beren-stein and Casadio-Tarabusi. Substantial generalization to totally geodesicsubmanifolds in a noncompact symmetric space X = G/K was done by Is-hikawa (2003). Paper [18] contains an inversion formula for the case whenX is a compact symmetric space of rank 1 and ! the set of antipodalsubmanifolds. The extensive calculation was later simplified (see IntegralGeometry and Radon Transforms, book to appear) using Rouvière’s met-hod for the noncompact analog (Enseign. Math. (2001)). This implies theinjectivity of the X-ray transform on any compact symmetric space (thecase rank(X) > 1 being settled by Strichartz and Gindikin using Fourierseries on a torus). For X = G/K noncompact there is, in addition, asupport theorem for the X-ray transform [51].

In his paper (C.R. Acad. Sci. Paris 342 (2006), 1–6) Rouvière provedan inversion formula for the X-ray transform on the noncompact symmetricspace X = G/K. His method is based on a skillful reduction to the hyper-bolic plane H2. The method does not work for the compact space U/Kbecause of the antipodal manifolds. While the X-ray transform on thecompact space U/K is injective (as mentioned) one can still ask:

THE SELECTED WORKSxviii

Introduction xix

(a) Is there a general inversion formula?(b) If one only integrates over minimal geodesics (this transformis called the Funk transform), is this transform injective?

(a) A global formula from [92] implies an inversion formula for the Funktransform of f provided f is supported in a ball of radius $/4 (with U/Kof maximum sectional curvature 1; here U/K is assumed irreducible andsimply connected). The proof is based on results from [21] according towhich all the minimal geodesics are conjugate under U . In the absence ofthe support condition the global formula mentioned also involves integrati-on over the antipodal manifolds.(b) In a recent preprint by Klein, Thorbergsson and Verhóczki, the Funktransform is shown to be injective without the support condition.

The case of X = G/K and ! = G/MN , the space of horocycles, isdiscussed in a later section.

One particularly simple instance of the dual setup is when ! is theunit disk, X its boundary, both having the group G = SU(1, 1) actingtransitively by

g : z # az + b

bz + a|z| % 1 |a|2 & |b|2 = 1 .

Here the transform f # "f from C(X) to C(!) turns out to be just theusual Poisson integral and the inversion formula is just the classical Schwarztheorem for the boundary values of a harmonic function. Other classicaltheorems in Potential Theory furnish answers to the range question (iii).While the range here consists of the harmonic functions there is a strikinganalogy with the case X = R3, ! = space of lines in X, where John (DukeMath. J. (1938)) gave a precise description of the range as solutions ofÁsgeirsson’s ultrahyperbolic equation in four variables.

4. Spherical Functions. Spherical Transforms.

Articles: *[22], *[26], *[28], *[37], [38], *[43], *[48], *[91]Books: [B6], [B7]

Let G be a connected noncompact semisimple Lie group with finite center,K a maximal compact subgroup. As proved by Gelfand, the convolution al-gebra Cc(K\G/K) of K-bi-invariant functions on G is commutative underconvolution. Given its standard topology, the continuous homomorphismsof the algebra Cc(K\G/K) onto C are functionals

f ##

G

f(x)#(x) dx

where the functions # are the so-called zonal spherical functions on G.Harish–Chandra gave a formula for these # in terms of the Iwasawa decom-position of G whereby the set of these # = ## is parametrized by % ina#c/W . On the other hand the space L1(K\G/K) is a commutative Banach

SIGURDUR HELGASON xix

xx Introduction

algebra under convolution and its maximal ideal space corresponds to thosespherical functions ## which are bounded. Johnson and I found in [26] thatthe bounded spherical functions are those ## (in the parametrization) forwhich % lies in a certain explicit tube in a#c over a certain convex set ina#. Flensted–Jensen has discovered an interesting relationship between thespherical functions on G and its complexification GC (J. Funct. Anal.(1978)). This leads, for example, to a somewhat simplified proof of theboundedness criterion above.

The paper [22] is devoted to the problem of finding the range of thespace D(K\G/K) under the spherical transform. For this I used Harish–Chandra’s expansion of the spherical function on the positive Weyl cham-ber a+. The coe$cients are certain meromorphic functions on a# and theproblem is to prove uniform compact support of the integral

#

a!

##(expH)F (%) |c(%)|%2 d% .

This is proved in [22] for each term &µ(%, H) in the series for ## by shiftingthe contour a# into the complex space a#c yet avoiding the singularities inthe coe$cients &µ(%, H) as well as c(%). Fortunately, there was a half-space in a#c free of singularities. It remained to justify the term-by-termintegration

#

a!

$

µ

&µ(%, H)F (%) |c(%)|%2 d% =$

µ

#

a!

&µ(%, H)F (%) |c(%)|%2 d%

and this was done by Gangolli (Ann. of Math. (1971) 150–165) by thedevice of multiplying the expansion of ## by a suitable factor. A simplifiedjustification is done in my paper [28], p. 38 using the same device.

An exposition of Harish–Chandra’s theory with some simplificationsis given in [B6], incorporating also Rosenberg’s method for the inversionformula (Proc. Amer. Math. Soc. (1977)). Another major simplification isAnker’s proof (J. Funct. Anal. (1991)) of Harish-Chandra’s determinationof the image of the Schwartz space under the spherical transform.

The Abel transform mentioned earlier is considered again in [91] andsome new operational properties established, for example that its adjo-int A# is a bijection of C"

W (A) onto C"(K\G/K), the subscript denotingWeyl group invariance.

Harish–Chandra’s Eisenstein integrals (generalized spherical functions)defined for each ' ! "K are studied in [28] and [43]. Here I proved that theysatisfy functional equations, one for each s ! W , and that they can beexpressed in terms of the zonal spherical function ##. This plays a role inthe proof of the Paley–Wiener theorem for the Fourier transform.

THE SELECTED WORKSxx

Introduction xxi

5. Duality for Symmetric Spaces.

Articles: *[15], [17], *[18], *[19], *[20], *[27], *[28], *[43], *[48], *[74],Book: [B7]

The duality referred to here is between the symmetric space X = G/Kand the space ! = G/MN of horocycles in X. Here G has the Iwasawadecomposition G = NAK and M is the centralizer of A in K. In theframework of an earlier section, we have the double fibration

X = G/K ! = G/MN

G/M

!!! "

""

and the transforms f # "f , # # # are

"f(") =#

"

f(x) dm(x) (the horocycle transform)

#(x) =#

"&x

#(") dµ(") (the dual transform).

The injectivity of f # "f on L1(X) can be deduced from the boundednesscriterion of ## mentioned earlier. For G a complex classical group Gelfand,Naimark and Graev had shown how a function F ! C"

c (G) can be explicitlygiven in terms of its integrals over the conjugacy classes in G. Combinedwith the decomposition G = NAK this proves an inversion formula forf # "f for complex classical G. For general G the inversion and Plancherelformula for f # "f were given in [15], [17], [19]. The kernel of the map# # # and the surjectivity C"(!)' = C"(X) are established in [B7].

The main theme in [20], [28] is the pattern of analogies between G/Kand G/MN . The algebra D(G/MN) is commutative as is the algebraD(G/K) and the transforms f # "f , # # # intertwine the operators inD(G/K) and D(G/MN). The analogs for G/MN of spherical functionson G/K are the conical distributions on G/MN which by definition are(i) MN -invariant.(ii) Eigendistributions of each D ! D(G/MN).

While the set of spherical functions is parametrized by a#c/W (cor-responding to K\G/K ' A/W ) I proved in [28] and [43] that the setof conical distributions is parametrized by a#c " W , corresponding to thedecomposition ! = MN · (A"W ) · "0 via the Bruhat decomposition of G.The injectivity criterion for the Poisson transform P# mentioned below isof central importance for this problem. The parametrization required theexclusion of certain singular eigenvalues. For X of rank 1 this restrictionwas removed by Men Cheng Hu (MIT thesis of 1973; Bull. Am. Math.

SIGURDUR HELGASON xxi

xxii Introduction

Soc. (1975)). In particular, Hu found for % = 0 an additional conicaldistribution which was omitted in Lemma 4.13, Ch. III in [28].

A representation of G is spherical if there is a fixed vector under Kand conical if there is a fixed vector under MN . It is proved in [17] and[28] that an irreducible finite-dimensional representation is spherical if andonly if it is conical. This leads to an explicit characterization of theirhighest weights. It also leads to simultaneous imbeddings of X and ! intothe same finite-dimensional vector space. The horocycles then appear ascertain plane sections with X ([B7], Ch. II, §4).

With $ an irreducible finite-dimensional spherical representation of Gon V let vK be a $(K) fixed vector and v!K a K fixed vector underthe contragredient representation $. Then $ is equivalent to the naturalrepresentation of G on the space of functions

g # ($(g%1)v, v!K) v ! V

and in this model the spherical and conical vectors are respectively givenby

#K(gK) = ($(g%1)vK , v!K) ,

&MN (gK) = e%"(A(g)) ,

" being the highest restricted weight of $ and A(g) as on page 2. In alecture in Iceland, 2007, David Vogan interpreted the identity above of thespherical and conical representation in terms of a “deformation” of K intoMN . Also Adam Korányi has shown that the MN -fixed vector is a suitablelimit of an orbit of the K-fixed vector.

The continuous decompositions of L2(X) and L2(!) into irreduciblescorrespond under the maps f # "f and # # #. The map # # # induces foreach % ! a#c a “Poisson transform” P# which maps functions (and functi-onals) on B = K/M into a joint eigenspace E#(X) of the operators inD(G/K). In [27], [28] and [43] it is stated and proved that P# is injecti-ve if and only if #+

X(%)%1 *= 0 where #+X , the Gamma function of X, is

the denominator in the Gindikin–Karpelevic formula for Harish–Chandra’sc(%) function. For X of rank one this was done in [28] by analyzing thePoisson transform at +. For X of higher rank this was done in [43] by stu-dying its behavior at the origin, using methods from Kostant’s work on thespherical principal series. The surjectivity of P# for hyperbolic spaces X(with analytic functionals on B) and the surjectivity of P# for general Xon the K-finite level was proved in [40] and [43]. The surjectivity withoutthe K-finiteness assumption was proved by Kashiwara, Kowata, Minemura,Okamoto, Oshima and Tanaka (Ann. of Math. (1978)). Professor Okamotohas kindly explained how this remarkable collaboration came about.

“ I still remember vividly how we reached our proof of your greatconjecture. We tried very hard in vain to prove your conjecture in thehigher rank case, using the concept of analytical functionals. To understand

THE SELECTED WORKSxxii

Introduction xxiii

your conjecture in a di!erent way, we decided to apply directly the originalidea of Prof. Sato about hyperfunctions to the new approach.

I took advantage of the fact that Prof. Sato is an uncle of Minemuraso that Minemura asked his uncle (Prof. Sato) to spare some time on Sat-urdays or Sundays for discussions about the possibility of applying directlythe original idea of Sato’s hyperfunctions to prove your conjecture. Wetraveled to Kyoto University every weekend. During this time Kashiwara(Kyoto University) and Oshima (Tokyo University) came to join in ourdiscussions.

To carry out our idea in the higher rank case, we realized that it isimportant to extend the notion of the ordinary di!erential equation withregular singularity to the notion of “the system of partial di!erential equati-ons with regular singularity”.

The splendid aspect of your conjecture is to have given birth to anew branch of Mathematics: Kashiwara–Oshima: Systems of di!erentialequations with regular singularities and their boundary value problems,Ann. of Math. (1977).”

An engaging account of this work is contained in a book by Schlichtkrull(Birkhäuser (1984)).

Schmid has indicated how the rank 1 method (which relied on hyper-geometric function estimates) might extend to the general rank.

The Poisson transform of distributions on B was investigated by Lewisand by Oshima–Sekiguchi. The latter authors also treat the case of a$nesymmetric G/H. Generalization to vector bundles was done by An Yang,(Thesis MIT, (1994)) and by Martin Olbrich (Dissertation, Berlin (1994)).

The flat analog of the double fibration above, replacing horocycles bytheir tangent planes, is investigated in [48] and [69]. The results are ana-logous but the expected tools like spherical function theory become de-generate so the results are obtained by looking at the flat case as a limitingcase of the curved case.

The complexification of the double fibration above, replacing eachgroup L by Lc has been actively studied in recent years (Gindikin, Krötz,Ólafsson, Stanton), and the extension of the joint eigenfunctions of D(G/K)to a tube around G/K in Gc/Kc plays a central role.

6. Representation Theory.

Articles: [17], *[19], *[20], *[27], *[28], [29], *[30], [31], *[43], [47], *[48],[49]Book: [B7]

Given a coset space L/H with D(L/H) commutative the group L actsnaturally on each joint eigenspace of the operators in D(L/H). I have calledthese representations eigenspace representations. If U/K is a symmetricspace of the compact type the eigenspace representations are precisely theirreducible spherical representations $ of U . Spherical means that there is

SIGURDUR HELGASON xxiii

xxiv Introduction

a fixed vector under $(K). If U is simply connected the highest weightsof these $ are given by the characterization mentioned before. This wasfurther generalized by Schlichtkrull (1984) and by Johnson (1987).

In the papers listed above the irreducibility question for the eigenspacerepresentations is considered for the cases Rn = M(n)/O(n), the noncom-pact symmetric space G/K, the dual space G/MN and G/N for G complex.For G/K the joint eigenspaces are parametrized by a#c/W (since each jointeigenspace contains a unique spherical function); the irreducibility turnsout to be equivalent to 1/#X(%) *= 0 where #X(%) is the denominator ofc(%)c(&%).

Using the structure of D(G/MN) the eigenspace representations forG/MN turn out to be just the members of the spherical principal seriesof G; the intertwining operators, realizing the equivalence between twoequivalent representations from this series, consist of convolutions withconical distributions ([27], Theorem 4.6). Another parallel determinationof these operators was done by Schi!mann and by Kunze, Knapp and Stein.The irreducibility of these representations is equivalent to 1/#X(%) *= 0;this gives another proof and interpretation of the algebraic irreducibilitycriterion of Wallach, Parthasarathy–Rao–Varadarajan and Kostant thoughstated in quite di!erent terms. For G complex, D(G/N) is still commutati-ve, and the eigenspace representations form the full principal series.

For a non-Riemannian symmetric space G/H the eigenspace represen-tations have been studied by Huang (Ann. of Math. (2001)). For thehyperbolic spaces over various fields this had been done very completely bySchlichtkrull (1987). For G nilpotent or solvable they have been e!ectivelystudied by Hole, Jacobsen and Stetkaer (Math. Scand. (1975, 1983, 1981)).

7. Fourier Transform on G/K.

Articles: *[19], *[20], [24], *[28], *[35], [36], *[37], *[43], [54], [57], [79],*[81], *[82] *[91]

The spherical function ## on G is K-bi-invariant, that is K-invariant onX = G/K and the theory of the spherical transform !f(%)=

%X f(x)##(x) dx

only concerns functions f on X which are K-invariant. The analog of ##for R2 is the Bessel function J0(%|x|) which is a radial eigenfunction ofthe Laplacian L on R2 and the Bessel transform on R+ becomes the ana-log of the spherical transform. In [19] I proposed a definition of a Fouriertransform on X = G/K, applying to all functions on X but reducing to thespherical transform for f K-invariant in the same way as the Fourier trans-form on R2 when used on radial functions reduces to the Bessel transform.Given the Iwasawa decomposition G = NAK g = n expA(g)k, A(g) ! a;we define the vector-valued inner product A(x, b) on X " B by

A(gK, kM) = A(k%1g) .

THE SELECTED WORKSxxiv

Introduction xxv

Putting e#,b(x) = e(i#+$)(A(x,b)) we define the Fourier transform of a functi-on f on X by

!f(%, b) =#

X

f(x)e%#,b(x) dx

for all (%, b) ! a#c " B for which the integral converges. The definition issuggested by the geometric analogy between horocycles and hyperplanes;A(x, b) is a “vector-valued distance” from o to the horocycle through xwith normal b. A representation-theoretic interpretation was pointed outby Sitaram (Pacific J. Math. (1988)) and Ólafsson–Schlichtkrull (Tributeto Mackey Volume, AMS (2008)). All the principal theorems from classicalFourier transform theory have their analogs for this Fourier transform.Thus [19] and [28] contain the Plancherel theorem (with range) and theinversion formula which can be written

'o =#

a!(B

e#,b dµ(%, b) ,

in analogy with the Euclidean version

'o =#

R+(Sn"1

e#,%%n%1 d% d( , e#,%(x) = ei#(x,%) .

The Paley–Wiener theorem (description of D(X))) is proved in [19] and[28]. This theorem was used in [43] to prove the necessary and su$cientcondition for the bijectivity of the Poisson transform for K-finite functionson B. In turn this was used to prove the irreducibility criterion for theeigenspace representations mentioned earlier. Another proof of the Paley–Wiener theorem was given by Torasso (1977). The paper [91] contains astrong form of the Riemann–Lebesgue lemma, proved jointly with Senguptaand Sitaram. These two and several others, Mohanty, Ray, Cowling, Bagci,Sarkar, Sundari, Rawat, have made great progress in topics such as theanalog of the Wiener–Tauberian theorem and Hardy’s theorem concerningsimultaneous exponential decay of a function and its Fourier transform.Okamoto and especially Eguchi have proved deep theorems for the Fouriertransform on the Schwartz spaces S(X) and their Lp analogs (Eguchi, J.Funct. Anal. (1979)).

The analog of the above Fourier transform for the dual compact sym-metric space U/K was developed by Sherman (Acta Math. (1990)), and forvector bundles over X by Camporesi (Pacific J. Math (1997)). A Paley–Wiener type theorem for the compact case has been obtained (2000) byGonzalez (on U) and more generally (2008) by Ólafsson and Schlichtkrull(on U/K).

In more recent years Fourier analysis theory for the non-Riemanniansymmetric spaces G/H has been vigorously developed by van den Ban,Carmona, Delorme, van Dijk, Faraut, Flensted–Jensen, Matsuki, Molcha-nov, Ólafsson, Oshima, Rossmann, Schlichtkrull and many others.

SIGURDUR HELGASON xxv

xxvi Introduction

8. Multipliers.

[1], [2], #[3], [4], [5]

Multipliers appear naturally in classical Fourier analysis on groups, wherethey correspond to operators on function spaces commuting with translati-ons. In the papers listed the setup is a semisimple Banach algebra A; asubalgebra A0, the derived algebra, is introduced as the algebra of elementsx ! A for which the multiplication y # xy is continuous from the topologyof the spectral norm ,y,sp = ,"y," to the original topology. Severalproperties of A0 are proved in [3], for example the determination of therepresentative algebra "A0 as the intersection of certain L1 spaces.

For various function algebras on a compact group G the derived al-gebra A0 is explicitly determined. For G the circle group the resultsspecialize to classical results of Littlewood, Sidon and Zygmund in Fourierseries theory; for G the Bohr compactification of R they solve the multiplierproblem for almost periodic functions on R. My Ph.D. thesis consisted of[2] and [3].

For A = L1(G), where G is a compact group (abelian or not) A0 turnsout to be L2(G) and in fact,

,f,2 %-

2 supg

{,f . g,1 : ,g,sp % 1} .

For G abelian it follows from a combination of the papers Edwards andRoss (Helgason’s number and lacunarity constants, Bull. Austr. Math.Soc. (1973)) and Sawa (Studia Math. (1985)) that

-2 can be replaced

by 2/-$ and that this is the best possible constant. For G nonabelian or

G/K symmetric it is not known whether the best constant is independentof G.

THE SELECTED WORKSxxvi

Photos

xxviii PHOTOS

THE SELECTED WORKSxxviii

Akureyri Gymnasium. (Photo courtesy of F. Rouviere.)

With family, Artie (wife), children Thor and Annie.

PHOTOS xxix

SIGURDUR HELGASON xxix

With student J. Dadok and Adviser S. Bochner.

Conference Roskilde Denmark (for Helgason's 65th birthday).

xxx PHOTOS

THE SELECTED WORKSxxx

At the Lie Memorial Conference, Oslo, 1992. (Photo courtesy of Artie Helgason.)

With Birgitte Burkovics, J. Radon's daughter.

With H. Schlichtkrull and M. Flersted-Jensen.

PHOTOS xxxi

SIGURDUR HELGASON xxxi

Receiving Major Knights Cross from President Vigdis Finnbogadottir, 1991.

From R. Sulanke's retirement. With Th. Friederich, U. Simon, R. Sulanke, A. Onishchik.

xxxii PHOTOS

THE SELECTED WORKSxxxii

At Dr.h.c. in Uppsala, 1996, with D. Hejhal and C. Kieselman. (Photo courtesy of Artie Helgason.)

With Gestur Olafsson, Tokyo, 2006. (Photo courtsey of B. Rubin.)

PHOTOS xxxiii

SIGURDUR HELGASON xxxiii

With A. Koranyi, Conference in Icleand, 2007. (Photo courtesy of F. Rouviere.)

Workshop in Tambara, Japan, 2006.

Bibliography, up to 2008, of S. Helgason

Books

[B1] Di!erential Geometry and Symmetric Spaces, Academic Press, New York,1962. Russian translation, MIR, Moscow, 1964. Reprint by AmericanMathematical Society, 2001.

[B2] Analysis on Lie Groups and Homogeneous Spaces, Conference Board ofMath. Science Lectures, Dartmouth 1971. Amer. Math. Soc. CBMSMonographs 1972. Corrected reprint 1977.

[B3] Di!erential Geometry, Lie Groups and Symmetric Spaces, Academic Press,New York, 1978. Russian translation, Factorial Press, 2005. Correctedreprint by American Mathematical Society in 2001.

[B4] The Radon Transform, Birkhäuser, Boston 1980, Russian translation, MIR,Moscow, 1983. 2nd edition, 1999.

[B5] Topics in Harmonic Analysis, Birkhäuser, Boston, 1981.

[B6] Groups and Geometric Analysis, Academic Press, New York and Orlando,1984. Russian translation, MIR, Moscow, 1987. Corrected reprint byAmerican Mathematical Society in 2000.

[B7] Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Mo-nographs. Amer. Math. Soc. 1994. 2nd edition, 2008.

[B8] (with Gu"mundur Arnlaugsson) Stær"fræ"ingurinn Ólafur Daníelsson.Saga Brautry"janda. Háskólaútgáfan, 1996.

SIGURDUR HELGASON xxxv

Papers

Asterisks indicate the paper is included in this volume

[1] The derived algebra of a Banach algebra, Proc. Nat. Acad. Sci. USA 40(l954), 994-995.

[2] Some problems in the theory of almost periodic functions, Math. Scand. 3(l955), 49-67.

*[3] Multipliers of Banach algebras, Ann. of Math. 64 (l956), 240-254.

[4] A characterization of the intersection of L1 spaces, Math. Scand. 4 (l956),5-8.

[5] Topologies of group algebras and a theorem of Littlewood, Trans. Amer.Math. Soc. 86 (l957), 269-283.

[6] Partial di!erential equations on Lie groups, Thirteenth Scand. Math.Congr. Helsinki, l957, 110-115.

[7] Lacunary Fourier series on noncommutative groups, Proc. Amer. Math.Soc. 9 (l958), 782-790.

[8] On Riemannian curvature of homogeneous spaces, Proc. Amer. Math.Soc. 9 (l958), 831-838.

*[9] Di!erential operators on homogeneous spaces, Acta Math. 102 (l959), 239-299.

[10] Some remarks on the exponential mapping of an a#ne connection, Math.Scand. 9 (l961), l29-l46.

[11] Some results on invariant theory, Bull. Amer. Math. Soc. 68 (l962),367-371.

[12] Invariants and fundamental functions, Acta Math. 109 (l963), 241-258.

*[13] Fundamental solutions of invariant di!erential operators on symmetric spa-ces, Bull. Amer. Math. Soc. 68 (l963), 778-781.

[14] Duality and Radon transform for symmetric spaces, Bull. Amer. Math.Soc. 69 (l963), 782-788.

*[15] Duality and Radon transform for symmetric spaces, Amer. J. Math. 85(l963), 667-692.

*[16] Fundamental solutions of invariant di!erential operators on symmetric spa-ces, Amer. J. Math. 86 (l964), 565-601.

[17] A duality in integral geometry; some generalizations of the Radon trans-form, Bull. Amer. Math. Soc. 70 (l964), 435-446.

xxxvi

THE SELECTED WORKSxxxvi

Bibliography xxxvii

*[18] The Radon transform on Euclidean spaces, compact two-point homogeneousspaces and Grassmann manifolds, Acta Math. 113 (l965), 153-180.

*[19] Radon-Fourier transforms on symmetric spaces and related group represent-ations, Bull. Amer. Math. Soc. 71 (l965), 757-763.

*[20] A duality in integral geometry on symmetric spaces, Proc. U.S.-JapanSeminar in Di!erential Geometry, Kyoto, Japan, l965, pp. 37-56. NipponHyronsha, Tokyo l966.

*[21] Totally geodesic spheres in compact symmetric spaces, Math. Ann. 165(l966), 309-317.

*[22] An analog of the Paley-Wiener theorem for the Fourier transform on certainsymmetric spaces, Math. Ann. 165 (1966), 297-308.

*[23] (with A. Korányi) A Fatou- type theorem for harmonic functions on sym-metric spaces, Bull. Amer. Math. Soc. 74 (l968), 258-263.

[24] Lie groups and symmetric spaces, Battelle Rencontres 1967, 1-71. W. A.Benjamin, Inc., New York l968.

[25] Some geometric properties of di!erential operators, Scientia Islandica,Anniversary Volume 1968, 41-43.

*[26] (with K. Johnson) The bounded spherical functions on symmetric spaces,Adv. in Math. 3 (l969), 586-593.

*[27] Applications of the Radon transform to representations of semisimple Liegroups, Proc. Nat. Acad. Sci. USA 63 (1969), 643-647.

*[28] A duality for symmetric spaces with applications to group representations,Adv. in Math. 5 (l970), 1-154.

[29] Representations of semisimple Lie groups, pp. 65-118, CIME lectures,Montecatini 1970, Edition Cremonese, Rome, 1971.

*[30] Group representations and symmetric spaces, Proc. Internat. CongressMath. Nice 1970, Vol. 2. Gauthier-Villars, Paris 1971, 313-320.

[31] Conical distributions and group representations, Symmetric Spaces, Shortcourses presented at Washington Univerity, Marcel Dekker, Inc. New York,l972, 141-156.

*[32] A formula for the radial part of the Laplace-Beltrami operator, J. Di!.Geometry 6 (l972), 411-419.

[33] Harmonic analysis in the non-Euclidean disk, Proc. Internat. Conf. onHarmonic Analysis, Univ. of Maryland 1971. Lecture Notes in Mat-hematics 266, 151-156.

SIGURDUR HELGASON xxxvii

xxxviii Bibliography

[34] Geometric reduction of di!erential operators, Symposia Mathematica 10(1972), 107-115.

*[35] Paley-Wiener theorems and surjectivity of invariant di!erential operatorson symmetric spaces and Lie groups, Bull. Amer. Math. Soc. 79 (l973),129-132.

[36] Functions on symmetric spaces, Proc. Sympos. Pure Math. 26, Amer.Math. Soc., Providence, R. I., 1973, 101-146.

*[37] The surjectivity of invariant di!erential operators on symmetric spaces,Ann. of Math. 98 (1973), 451-480.

[38] Spherical functions, spherical representations and correspondence theoremsfor the spherical Fourier transform, Colloque sur les fonctions sphériques etla théorie des groupes, Nancy, 5-9, 1971. Russian translation, Mathematika17 (1973).

[39] Elie Cartan’s Symmetriske Rum, Danish Math. Soc. Centennial Publicati-on, 1973, pp. 93-101.

*[40] Eigenspaces of the Laplacian; integral representations and irreducibility, J.Funct. Anal. 17 (1974), 328-353.

[41] The eigenfunctions of the Laplacian on a two-point homogeneous space;integral representations and irreducibility, Proc. Sympos. Pure Math. 27,357-360. Amer. Math. Soc., Providence 1975.

[42] Remarque sur R.G. McLenaghan: On the validity of Huygens principle forsecond order partial di!erential equations with four independent variables.I. Ann. Inst. Poincaré Sect. A 21 (1974), 347.

*[43] A duality for symmetric spaces with applications to group representationsII. Di!erential equations and eigenspace representations, Adv. in Math.22 (1976), 187-219.

[44] Solvability of invariant di!erential operators on homogeneous manifolds,C.I.M.E. seminar lecture (Varenna 1975), 281-310. Cremonese, Roma 1975.

[45] Solvability questions for invariant di!erential operators, Proc. Fifth In-ternat. Colloq. on Group Theoretical Methods in Physics (Montreal 1976),pp. 517-527. Academic Press 1977.

[46] Invariant di!erential equations on homogeneous manifolds, Bull. Amer.Math. Soc. 83 (1977), 751-774.

[47] Some results on eigenfunctions on symmetric spaces and eigenspace repre-sentations, Math. Scand. 41 (l977), 79-89.

THE SELECTED WORKSxxxviii

Bibliography xxxix

*[48] A duality for symmetric spaces with applications to group representationsIII. Tangent space analysis, Adv. in Math. 36 (1980), 297-323.

[49] Invariant di!erential operators and eigenspace representations, In Repre-sentations of Lie groups (G. Luke, ed.), London Math. Soc. Lecture NoteSeries 34, Cambridge Univ. Press 1979, 236-286.

[50] Support of Radon transforms, Adv. in Math. 38 (l980), 91-100.

[51] The X-ray transform on a symmetric space, Proc. Conf. Di!erential Geome-try and Global Analysis (Berlin 1979). Lecture Notes in Math. 838 (1981),145-148.

[52] Ranges of Radon Transforms, In Computed tomography (Cincinnati, 1982),Proc. Symp. Appl. Math. 27, 63-70, Amer. Math. Soc. 1982.

[53] The Range of the Radon Transform on a Symmetric Space, Proc. Conf.Representations of Reductive Lie Groups (Park City, 1982). Birkhäuser,1983, 145-151.

[54] Topics in Geometric Fourier Analysis. In Topics in Modern HarmonicAnalysis (Turin/Milan, 1982). Ist. Naz. Alta Mat. Francesco Severi, Rome1983, 301-351.

[55] Wave equations on homogeneous spaces, Lie Group Representations III(College Park, 1983). Lecture Notes in Math. 1077 (1984), 254-287.

[56] Operational properties of the Radon transform with applications, Proc.Conf. Di!erential Geometry and its Applications, (Nove Mesto MoraveCzechoslovakia 1983). Publ. Univ. Karlova Praha 1984, 59-75.

[57] The Fourier transform on symmetric spaces, The mathematical heritage ofÉlie Cartan, Astérisque 1985, Numero Hors Série, 151-164.

[58] Some results on Radon transforms, Huygens Principle and X-ray trans-forms, Integral geometry (Brunswick, 1984). Contemp. Math. 63 (1987),151-177.

*[59] (with F. Gonzalez) Invariant di!erential operators on Grassmann mani-folds, Adv. in Math. 60 (1986), 81-91.

*[60] Value-distribution theory for holomorphic almost periodic functions, TheHarald Bohr Centenary (Copenhagen 1987). Mat.-Fys. Medd. Danske Vid.Selsk. 42 (1989), 93-102.

[61] Response to 1988 Steele Prize Award, Notices Amer. Math. Soc. 35 (1988),965-967.

SIGURDUR HELGASON xxxix

xl Bibliography

*[62] The totally geodesic Radon transform on constant curvature spaces, Integralgeometry and tomography (Arcata, 1989). Contemp. Math. 113 (1990),141-149.

*[63] Some results on invariant di!erential operators on symmetric spaces, Amer.J. Math. 114 (1992), 789-811.

[64] Invariant di!erential operators and Weyl group invariants. In (Ed. W.Barker and P. Sally) Harmonic Analysis on Reductive Groups, 193-200,Birkhäuser, Boston 1991.

[65] A centennial: Wilhelm Killing and the exceptional groups, Math. Intelli-gencer, 12 (1990) 54-57.

[66] Geometric and Analytic Features of the Radon Transform, Mitt. Math.Ges. Hamburg, 12 (1991), 977-988 (1992).

[67] Support Theorems in Integral Geometry and their Applications. Proc. AMSSummer Inst. on Di!erential Geometry (Los Angeles 1990), Proc. Sympos.Pure Math. 54 (1993), 317-323.

[68] Tveir Töframenn Rúmfræ"innar, Poncelet og Jacobi. Icelandic Math. Soc.Bulletin 1991, 6-17.

*[69] The Flat Horocycle Transform for a Symmetric Space, Adv. in Math. 91(1992), 232-251.

[70] Radon Vörpunin, Afbrig"i og Notkun, Leifur Asgeirsson Memorial Volume.Háskólaútgáfan, Reykjavik 1998.

*[71] Huygens’ Principle for Wave Equations on Symmetric Spaces, J. Funct.Anal. 107 (1992) 279-288.

[72] The Wave Equation on Symmetric Spaces. In Classical and QuantumSystems, Int. Wigner Symp. (Goslar 1991). World Sci. Publ. 1993, 149-157.

[73] Sophus Lie and the role of Lie groups in mathematics, Report of the 14.Nordic LMFK-congress Reykjavík, 1990.

*[74] Radon transforms for double fibrations. Examples and viewpoints, Proc. ofSymposium 75 years of Radon Transform, Vienna 1992. Int. Press, Boston1994, 175-188.

[75] The Fourier Transform on Symmetric Spaces and Applications, Proc. Conf.Di!erential Geometry and its Applications (Opava 1992), Silesian Univ.Opava 1993, 23-28.

[76] Sophus Lie, the Mathematician, Proc. of The Sophus Lie Memorial Conf-erence (Oslo 1992), Scand. Univ. Press, Oslo 1994, 3-21.

THE SELECTED WORKSxl

Bibliography xli

[77] Harish-Chandra’s c-function; a mathematical jewel. In Noncompact LieGroups and Some of Their Applications. NATO Adv. Sci. Inst. Ser. CMath. Phys. Sci., 429 (San Antonio 1993). Kluwer Acad. Publ. 1994,55-67.

[78] Radon Transforms and Wave Equations, C.I.M.E. 1996. Lecture Notes inMath. 1684 (1998), 99-121

[79] Orbital Integrals, Symmetric Fourier Analysis and Eigenspace Representati-ons, Proc. Sympos. Pure Math. 61 (1997), 167-189.

[80] Paul Günther’s work on Hadamard’s conjecture, Zeitschr. Anal. Anw. 16(1997), 5-6.

*[81] Integral Geometry and Multitemporal Wave Equations, Comm. Pure andAppl. Math. 51 (1998) 1035-1071.

*[82] (with H. Schlichtkrull) The Paley-Wiener Space for the Multitemporal WaveEquation, Comm. Pure and Appl. Math. 52 (1999) 49-52.

[83] Harish-Chandra and his Mathematical Legacy. Some Personal Recollecti-ons, Current Sci. 74 (1998), 921-924.

[84] Harish-Chandra’s c-function; a mathematical jewel. Reprint of [77], Themathematical legacy of Harish-Chandra, Proc. Sympos. Pure Math. 68(2000), 273-283.

[85] Harish-Chandra Memorial Talk, ibid. 43-45.

[86] Hinn óevklí"ski heimur, saga, rúmfræ"i og greining, Verpill, 2000.

[87] The Fourier Transform on Symmetric Spaces. Applications to Classicaland Multitemporal Wave Equations, Amer. Math. Soc. Proc. Symp. PureMath. (to appear).

[88] Rúmfræ"i og Raunveruleikinn, Icelandic Math Society Bulletin, 2000, 1-12.

[89] Non-Euclidean Analysis. In (Eds. Prékopa and Molnár) Non-EuclideanGeometries, Proceedings of the Janos Bolyai Mem. Conf., Budapest, July,2002. Springer, 2006. p. 367-384.

[90] Remarks on B. Rubin, Radon, Cosine and Sine Transforms on Real Hyper-bolic Space, Adv. in Math. 192 (2005), 225.

*[91] The Abel, Fourier and Radon Transforms on Symmetric Spaces, Indag.Math. 16 (2005), 531-551.

*[92] The Inversion of the X-ray Transform on a Compact Symmetric Space, J.Lie Theory 17 (2007), 307-315.

SIGURDUR HELGASON xli