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
Differential opeators on homogeneous spaces, Acta Math. 102 (l959),
239-299.
17
Fundamental solutions of invariant differential operators on symmetric 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 differential operators on symmetric spaces, Amer. J. Math. 86 (l964), 565-601.
109
The Radon transform on Euclidean spaces, compact two-point
homogeneous spaces and Grassmann manifolds, Acta Math.
113 (l965), 153-180
147
Radon-Fourier transforms on symmetric spaces and related group
representations, Bull. Amer. Math. Soc. 71 (l965), 757-763.
175
A duality in integral geometry on symmetric spaces, Proc.
U.S.-Japan Seminar in Differential 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 transform
on certain symmetric spaces, Math. Ann. 165 (1966), 297-308.
213
SIGURD̄UR HELGASON
v
vi
CONTENTS
(with A. Korányi) A Fatou-type theorem for harmonic functions on
symmetric spaces, Bull. Amer. Math. Soc. 74 (l968), 258-263.
225
(with K. Johnson) The bounded spherical functions on symmetric
spaces, Adv. in Math. 3 (l969), 586-593.
231
Applications of the Radon transform to representations of semisimple Lie groups, Proc. Nat. Acad. Sci. USA 63 (1969), 643-647.
239
A duality for symmetric spaces with applications to group representations, 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. Diff. Geometry 6 (l972), 411-419.
407
Paley-Wiener theorems and surjectivity of invariant differential
operators on symmetric spaces and Lie groups, Bull. Amer. Math.
Soc. 79 (l973), 129-132.
417
The surjectivity of invariant differential operators on symmetric
spaces I, Ann. of Math. 98 (1973), 451-479.
421
Eigenspaces of the Laplacian; integral representations and irreducibility, J. Funct. Anal. 17 (1974), 328-353.
451
A duality for symmetric spaces with applications to group representations II. Differential equations and eigenspace representations,
Adv. in Math. 22 (1976), 187-219.
477
A duality for symmetric spaces with applications to group representations III. Tangent space analysis, Adv. in Math. 36 (1980),
297-323.
511
(with F. Gonzalez) Invariant differential operators on Grassmann manifolds, Adv. in Math. 60 (1986), 81-91.
539
Value-distribution theory for holomorphic almost periodic functions, 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 differential operators on symmetric spaces,
Amer. J. Math. 114 (1992), 789-811.
571
vi
THE SELECTED WORKS
CONTENTS
vii
The Flat Horocycle Transform for a Symmetric Space, Adv. in
Math. 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 Multitemporal 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 Symmetric
Space, J. Lie Theory 17 (2007), 307-315.
703
Acknowledgments
713
SIGURD̄UR HELGASON
vii
Preface
The present volume contains a selection of 32 articles by Sigurður
Helgason, covering various aspects of geometric analysis on Riemannian
symmetric spaces. The papers are arranged chronologically and cover a
time span of more than 50 years, from 1956 to 2007.
Sigurður Helgason has had a profound impact on geometric analysis,
more particularly in the geometry of homogeneous spaces, harmonic analysis on symmetric spaces, the theory of the Radon transform, and the study
of invariant differential operators and their eigenfunctions. The collection
is preceded by an introduction by Helgason. Here some background and
perspective is given on the individual papers, and they are brought into a
historical perspective as seen by the author himself today. The introduction organizes the articles according to the following list of topics, which
describes the evolution of Helgason’s work over the period:
1.
2.
3.
4.
5.
6.
7.
8.
Invariant differential operators,
Geometric properties of solutions,
Double fibrations in integral geometry, Radon transforms,
Spherical functions and spherical transforms,
Duality for symmetric spaces,
Representation theory,
Fourier transform on Riemannian symmetric spaces,
Multipliers.
Many people know the work of Helgason primarily from his outstanding books. The writing of Sigurður Helgason has been a model of exposition since the publication in 1962 of Differential Geometry and Symmetric
Spaces, [B1], which quickly became a classic. It was not only used by students and researchers in Lie group theory, the first chapter was-and still
is-an excellent introduction to Riemannian geometry. This book, and its
successor from 1984, Groups and Geometric Analysis, [B6], was awarded
the Steele price for expository writing in 1988.
In addition to these two classical and prize-winning books, Helgason
has published several other books, notably The Radon Transform [B4] and
Geometric Analysis on Symmetric Spaces [B7]. A complete list of books
and articles up to 2008 is enclosed in this volume. The above citations refer
to that list.
SIGURD̄UR HELGASON
ix
x
PREFACE
The exposition in the original papers is of the same outstanding clarity
as that in the books. It is our hope that by collecting these papers, we
can create renewed attention on all those aspects of his work which are not
covered by the books.
We would like to thank Ed Dunne and Cristin Zannella at AMS for their
invaluable help to get this project started and going. We would also thank
the publishers of the original articles for their cooperation and permission
to include the articles in this volume.
Gestur Ólafsson and Henrik Schlichtkrull
x
THE SELECTED WORKS
Biographical note
Sigurður Helgason was born on September 30, 1927, in Akureyri, Iceland. He went to the Gymnasium in Akureyri 1939-1945. After a short
period at the University of Iceland in Reykjavík, he began studies at the
University of Copenhagen in 1946. Here he received the Gold Medal in 1951
for a paper summarized in [60] and the M.Sc. degree in 1952. He went to
Princeton as a graduate student in 1952 and received the ph.D. from there
in 1954. He was a Moore instructor at M.I.T. 1954-56, after which returned to Princeton 1956-57. He was at University of Chicago 1957-59, and
at Columbia University 1959-60. Since then he has been at M.I.T., where
he 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-Leffler
Institute, Stockholm. He has been awarded honorary doctor by University
of Iceland, University of Copenhagen and University of Uppsala. In 1988
he received the Steele prize of the A.M.S. for expository writing. He carries
the Major Knights Cross of the Icelandic Falcon since 1991.
He and his wife live in Belmont, Massachusetts. They have a son and
a daughter, and 3 grandchildren.
SIGURD̄UR HELGASON
xi
Introduction
I am deeply grateful to the American Mathematical Society and to
Gestur Ólafsson and Henrik Schlichtkrull for their interest and their proposal to produce this volume with selections of my mathematical papers. The
editors and publisher kindly asked me to write some explanatory comments
on these papers.
My interest in mathematics I trace to my mathematics teacher in the
Gymnasium in Akureyri in North Iceland, Trausti Einarsson. Although an
astronomer, his deep respect for mathematics was highly infectious. The
geometry textbooks by the remarkable mathematician Ólafur Danielsson,
the pioneering founder of mathematics education in Iceland, were written
by 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 the
radius of a sphere inscribed in a regular dodecahedron of edge length 1.
Danielsson, who wrote several beautiful papers in geometry along with his
demanding Gymnasium teaching, was profoundly influenced by the rich
geometry tradition in Denmark, which started around 1870 with J. Petersen
and H. Zeuthen.
When I entered the Gymnasium in Akureyri around 1940, the Icelandic
population totaled about 125,000. Thus the Gymnasium program put
considerable emphasis on the teaching of several foreign languages. Our
French teacher, Thórarinn Björnsson, was particularly memorable because
of his outspoken admiration of French authors like Maupassant, Daudet
and Rolland, in addition to his fondness for the French language in which
he left us with a decent reading ability. Thus my later difficulties in understanding Élie Cartan’s papers were not of linguistic nature.
Later, at the University of Copenhagen, I benefited from splendid lectures by Harald Bohr, Børge Jessen and Werner Fenchel. The first two gave
substantial courses in complex function theory and real analysis, respectively. With Fenchel I had also a kind of reading course in projective geometry. 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 such
problems: Construct a triangle ABC from ha , ma and r. This was consistent with my expectation later to teach at a Gymnasium in Iceland.
SIGURD̄UR HELGASON
xiii
xiv
INTRODUCTION
One day I happened to notice a posting of the collection of prize questions which were posed by the university each year. The mathematics
problem on the list called for a generalization to analytic almost periodic 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 submitted and judged anonymously so I could not consult anybody. Thus
four months went by before I understood what the problem was about. My
paper solving the problem became my masters thesis. A summary is given
in paper [60] in this volume.
After completing my thesis in Princeton enjoying supportive and stimulating supervision by Bochner, my interest shifted to Lie group theory
with the intention of studying Harish–Chandra’s work. A related interest
was geometric analysis, having received from my friend Leifur Ásgeirsson
the page proofs of Fritz John’s famous book Plane Waves and Spherical
Means. Although the term ‘group’ does not appear in this book I sensed
quickly 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 the
hands of Sophus Lie, the study of differential equations led to the concept
of Lie groups which in turn went on to penetrate numerous other fields
of mathematics. Since Lie group theory had in the mid-fifties become so
well developed I became interested in a kind of converse to Lie’s program,
namely to investigate differential operators and differential equations invariant under known groups. Examples were readily at hand: the Laplace–
Beltrami operator on Riemannian or pseudo-Riemannian manifolds, the
Dirac operator, the wave equation and its variations.
In the account that follows I describe some of my own work connected
to analysis on homogeneous Spaces. This is accompanied by brief mention
of some prior and subsequent work which is closely related. These personal
viewpoints do not represent any attempt at a historical account, the feature
of relevance and of value being highly subjective. As Peter Lax aptly
observed: “For a mathematician the central problem is the one he happens
to be working on”.
Inspired by Harish–Chandra’s brilliant work on the representation
theory of semisimple Lie groups G, my taste for geometry led me to a related topic, namely a study of the symmetric spaces of É. Cartan. Here enter
several differential geometric features: geodesics, curvature, K-orbits, totally geodesic submanifolds and horocycles; for the compact dual spaces U/K
we have also closed geodesics, especially those of minimal length, equators,
antipodal manifolds, and totally geodesic spheres. Geometric analysis means, 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 have
also enjoyed the privilege of extended, stimulating visits at the Institute
for Advanced Study in Princeton and at the Mittag–Leffler Institute in
Djursholm, Sweden.
xiv
THE SELECTED WORKS
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, at
times when nothing but mathematics mattered, has been admirable. She
even typed my first two books in pre-TEX times, giving stylistic suggestions
along the way.
1. Invariant Differential Operators.
1
Articles: [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 the
algebra D(G/H) of G-invariant differential operators on a homogeneous
space G/H. For G/H reductive some preliminaries are done in [9]; for
G/H of maximum mobility, namely two-point homogeneous, the algebra
is easily shown to be generated by the Laplace–Beltrami operator. For
G/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 operators
in D(G/K) are not all induced from operators in the algebra Z(G) of biinvariant differential operators on G. While this was verified in [16] to
be true for all classical irreducible G/K, it fails for exactly four of the
exceptional G/K [63]. However, even here, each D is induced by a fraction
!2 where Z
!i ∈ D(G/K) is induced
Z1 /Z2 (Zi ∈ Z(G)), that is Z!1 D = Z
by Zi via the action of G on G/K. The algebra D(G/K) is the subject of
Chapter II in [B6] and is related to the Iwasawa decomposition
G = N AK ,
g = n +a +k,
g = n exp A(g)k ,
A(g) ∈ a .
Given D ∈ D(G/K) and its radial part under the action of N on
X = G/K, it is proved in [16] that the Abel transform (the Radon transform on K-invariant functions on X) is a simultaneous transmutation
operator for all D ∈ D(G/K), that is, “converts” them into constant
coefficient operators on A. Combining this with Harish–Chandra’s work
on the spherical transform and Hörmander’s divisibility theorem for tempered distribution on Rn , I showed in [16] that each D ∈ D(G/K) has a
fundamental solution. The later Paley–Wiener theorem furnishes a shorter proof but requires the surjectivity of a constant coefficient differential
operator on D! (Rn ).
The next step was to prove the surjectivity of each D ∈ D(G/K) on
the space C ∞ (G/K), i.e., the existence of a global solution to the equation
Du = f for arbitrary smooth f . Already in some notes in 1964 I had
reduced this to a support theorem for the horocycle Radon transform on X.
I used Harish–Chandra’s expansion for Eisenstein integrals but was stymied
by singularities on a∗c whose location seemed unpredictable. But by looking
1 Papers
in this volume are indicated by an asterisk.
SIGURD̄UR HELGASON
xv
xvi
Introduction
at the simplest possible examples I discovered, but not until 1972, that these
singularities could be canceled out by a certain intertwining procedure.
This yielded the support theorem for the horocycle Radon transform and
thereby the surjectivity
D C ∞ (X) = C ∞ (X)
follows in [37]. The local solvability of each Z ∈ Z(G) is also proved there
invoking methods of Raïs and Harish-Chandra. General local solvability
was later proved by Duflo. The global solvability however fails (Cerèzo–
Rouvière, Sem. École Polytech. (1973)) although it holds for the Casimir
operator (Cerèzo–Rouvière loc. cit., Rauch and Wigner, Ann. of Math.
(1976)). Other surjectivity results are proved in [35], [37], [43]. The nonRiemannian symmetric case has been investigated by Duflo, van den Ban
and Schlichtkrull and others. The support theorem above was given a
new proof by Gonzalez and Quinto. This approach has led Quinto and
collaborators to various interesting support theorems in much more general
context.
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 harmonic
function on G/K as a solution to Du = 0 for all D ∈ D(G/K) annihilating the constants. He proved that they had a characterization in terms
of mean values over orbits of K and their translates. In [9] I proved
a similar extension of Ásgeirsson’s mean-value theorem for solutions of
Lx (u(x, y)) = Ly (u(x, y)) on Rn × Rn (L-Laplacian) to the system
Dx u = Dy u on G/K × G/K for D ∈ D(G/K). Inspired by work of
Ásgeirsson and Günther on Huygens’ principle I became interested in studying the wave equation on symmetric spaces G/K and on Lorentzian
manifolds of constant curvature. In both cases the natural equation is
∂2
Lx + c = ∂t
2 , where c is a constant related to the scalar curvature. Papers
[16], [45], [55], [58], [71], [78] and a chapter in [B7] deal with explicit
solution formulas for the natural wave equation on G/K and the occurrence of Huygens’ principle. My work on these wave equations is related
to extensive contributions by Branson, Ólafsson, Lax, Phillips, Schlichtkrull, Shahshahani, Solomatina and Ørsted. Papers [81] and [82] (the latter
with Schlichtkrull) deal with a hyperbolic system on G/K introduced by
Semenov–Tian–Shansky, Izvestija (1976). The time variable has dimension rank(X) and for rank(X) = 1 it reduces to the above wave equation. Developing further the work of Semenov–Tian–Shansky, Shahshahani
and Phillips, paper [81], gives an explicit solution of the Cauchy problem
in terms of the Radon transform and in terms of the Fourier transform.
Paper [82] with Schlichtkrull deals with an associated range problem.
xvi
THE SELECTED WORKS
Introduction
xvii
For a Lorentzian manifold X of constant curvature, paper [9] sets up
an analog of Riesz potentials and used properties of these to show that a
function u on X is determined by a formula
u = c lim r n−2 Q(!)M r u ,
r→0
where (M u)(x) is the “mean value” of u over a “Lorentzian sphere” of
radius 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 (Acta
Math. (1994)) that each factor ! + c in Q(!) satisfies Huygens’ principle.
It was proved by Furstenberg and Karpelevic that a bounded solution
of the Laplace equation Lu = 0 on the symmetric space G/K is necessarily
harmonic (in Godement’s sense above). This led them to a Poisson type
integral formula for such functions. While the analog of the Schwarz theorem for continuous boundary values is easy, in [23] Korányi and I proved
the corresponding Fatou theorem for such functions u(x); namely
r
lim u(γ(t)) exists
t→∞
for almost all geodesics, starting at the origin o = eK. The result has
many 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 determined by its line integrals. In projective geometry points and lines are put
on equal footing. Observing that R2 and the set P2 of lines are both
permuted transitively by the same group M(2), the isometry group of R2 ,
the following generalization was proposed in [20]. Given a locally compact
group and K and H two closed subgroups satisfying some natural conditions we consider the coset spaces X = G/K, Ξ = G/H. Elements x = gK,
ξ = γH are (by Chern (1942)) said to be incident if the sets gK and γH
intersect as subsets of G. Given x ∈ X, ξ ∈ Ξ let
x̌ = {ξ ∈ Ξ : x, ξ incident} ,
ξ" = {x ∈ X : x, ξ incident} .
Then we have two integral transforms
f ∈ Cc (X) → f" ∈ C(Ξ) , ϕ ∈ Cc (Ξ) → ϕ̌ ∈ C(X)
SIGURD̄UR HELGASON
xvii
xviii
Introduction
given by
f"(ξ) =
#
f (x) dm(x) , ϕ̌(x) =
#
ϕ(ξ) dµ(ξ)
x̌
ξ"
with canonically defined measures dm and dµ. In group-theoretic terms
with L = K ∩ H, dkL , dhL invariant measures on K/L and H/L we have
#
#
"
f (γhK) dhL , ϕ̌(gK) =
ϕ(gkH) dkL .
f (γH) =
H/L
K/L
These geometrically dual transforms are also adjoint:
#
#
f (x) ϕ̌(x) dx = f"(ξ)ϕ(ξ) dξ ,
X
Ξ
where dx and dξ are invariant measures, dx = dgK , dξ = dgH .
The principal problems for these transforms f → f", ϕ → ϕ̌ 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 (Ξ)} , but there are many variations.
In addition to considering the classical case X = Rn , Ξ = Pn , the
space of hyperplanes, where I determined the image of D(Rn ) under the
Radon transform, I obtained in [9] and [62] inversion formulas for the case
when X is a Riemannian manifold of constant curvature and Ξ the family of
totally geodesic submanifolds of a given dimension. This was later extensively studied by Semyanisty, Rubin, Palamodov, Rouvière, Kurusa, Berenstein and Casadio-Tarabusi. Substantial generalization to totally geodesic
submanifolds in a noncompact symmetric space X = G/K was done by Ishikawa (2003). Paper [18] contains an inversion formula for the case when
X is a compact symmetric space of rank 1 and Ξ the set of antipodal
submanifolds. The extensive calculation was later simplified (see Integral
Geometry and Radon Transforms, book to appear) using Rouvière’s method for the noncompact analog (Enseign. Math. (2001)). This implies the
injectivity of the X-ray transform on any compact symmetric space (the
case rank(X) > 1 being settled by Strichartz and Gindikin using Fourier
series on a torus). For X = G/K noncompact there is, in addition, a
support theorem for the X-ray transform [51].
In his paper (C.R. Acad. Sci. Paris 342 (2006), 1–6) Rouvière proved
an inversion formula for the X-ray transform on the noncompact symmetric
space X = G/K. His method is based on a skillful reduction to the hyperbolic plane H2 . The method does not work for the compact space U/K
because of the antipodal manifolds. While the X-ray transform on the
compact space U/K is injective (as mentioned) one can still ask:
xviii
THE SELECTED WORKS
Introduction
xix
(a) Is there a general inversion formula?
(b) If one only integrates over minimal geodesics (this transform
is called the Funk transform), is this transform injective?
(a) A global formula from [92] implies an inversion formula for the Funk
transform of f provided f is supported in a ball of radius π/4 (with U/K
of maximum sectional curvature 1; here U/K is assumed irreducible and
simply connected). The proof is based on results from [21] according to
which all the minimal geodesics are conjugate under U . In the absence of
the support condition the global formula mentioned also involves integration over the antipodal manifolds.
(b) In a recent preprint by Klein, Thorbergsson and Verhóczki, the Funk
transform is shown to be injective without the support condition.
The case of X = G/K and Ξ = G/M N , the space of horocycles, is
discussed in a later section.
One particularly simple instance of the dual setup is when Ξ is the
unit disk, X its boundary, both having the group G = SU(1, 1) acting
transitively by
az + b
|z| ≤ 1
|a|2 − |b|2 = 1 .
g:z→
b̄z + ā
Here the transform f → f" from C(X) to C(Ξ) turns out to be just the
usual Poisson integral and the inversion formula is just the classical Schwarz
theorem for the boundary values of a harmonic function. Other classical
theorems in Potential Theory furnish answers to the range question (iii).
While the range here consists of the harmonic functions there is a striking
analogy with the case X = R3 , Ξ = space of lines in X, where John (Duke
Math. J. (1938)) gave a precise description of the range as solutions of
Ásgeirsson’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 algebra Cc (K\G/K) of K-bi-invariant functions on G is commutative under
convolution. Given its standard topology, the continuous homomorphisms
of the algebra Cc (K\G/K) onto C are functionals
#
f → f (x)ϕ(x) dx
G
where the functions ϕ are the so-called zonal spherical functions on G.
Harish–Chandra gave a formula for these ϕ in terms of the Iwasawa decomposition of G whereby the set of these ϕ = ϕλ is parametrized by λ in
a∗c /W . On the other hand the space L1 (K\G/K) is a commutative Banach
SIGURD̄UR HELGASON
xix
xx
Introduction
algebra under convolution and its maximal ideal space corresponds to those
spherical functions ϕλ which are bounded. Johnson and I found in [26] that
the bounded spherical functions are those ϕλ (in the parametrization) for
which λ lies in a certain explicit tube in a∗c over a certain convex set in
a∗ . Flensted–Jensen has discovered an interesting relationship between the
spherical functions on G and its complexification GC (J. Funct. Anal.
(1978)). This leads, for example, to a somewhat simplified proof of the
boundedness criterion above.
The paper [22] is devoted to the problem of finding the range of the
space D(K\G/K) under the spherical transform. For this I used Harish–
Chandra’s expansion of the spherical function on the positive Weyl chamber a+ . The coefficients are certain meromorphic functions on a∗ and the
problem is to prove uniform compact support of the integral
#
a∗
ϕλ (exp H)F (λ) |c(λ)|−2 dλ .
This is proved in [22] for each term ψµ (λ, H) in the series for ϕλ by shifting
the contour a∗ into the complex space a∗c yet avoiding the singularities in
the coefficients ψµ (λ, H) as well as c(λ). Fortunately, there was a halfspace in a∗c free of singularities. It remained to justify the term-by-term
integration
# $
a∗
µ
ψµ (λ, H)F (λ) |c(λ)|
−2
dλ =
$#
µ a∗
ψµ (λ, H)F (λ) |c(λ)|−2 dλ
and this was done by Gangolli (Ann. of Math. (1971) 150–165) by the
device of multiplying the expansion of ϕλ by a suitable factor. A simplified
justification is done in my paper [28], p. 38 using the same device.
An exposition of Harish–Chandra’s theory with some simplifications
is given in [B6], incorporating also Rosenberg’s method for the inversion
formula (Proc. Amer. Math. Soc. (1977)). Another major simplification is
Anker’s proof (J. Funct. Anal. (1991)) of Harish-Chandra’s determination
of the image of the Schwartz space under the spherical transform.
The Abel transform mentioned earlier is considered again in [91] and
some new operational properties established, for example that its adjo∞
(A) onto C ∞ (K\G/K), the subscript denoting
int A∗ is a bijection of CW
Weyl group invariance.
Harish–Chandra’s Eisenstein integrals (generalized spherical functions)
" are studied in [28] and [43]. Here I proved that they
defined for each δ ∈ K
satisfy functional equations, one for each s ∈ W , and that they can be
expressed in terms of the zonal spherical function ϕλ . This plays a role in
the proof of the Paley–Wiener theorem for the Fourier transform.
xx
THE SELECTED WORKS
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/K
and the space Ξ = G/M N of horocycles in X. Here G has the Iwasawa
decomposition G = N AK and M is the centralizer of A in K. In the
framework of an earlier section, we have the double fibration
G/M
!
!
"
!
"
"
X = G/K
Ξ = G/M N
"
and the transforms f → f , ϕ → ϕ̌ are
f"(ξ) =
ϕ̌(x) =
#
f (x) dm(x) (the horocycle transform)
ξ
#
ϕ(ξ) dµ(ξ) (the dual transform).
ξ&x
The injectivity of f → f" on L1 (X) can be deduced from the boundedness
criterion of ϕλ mentioned earlier. For G a complex classical group Gelfand,
Naimark and Graev had shown how a function F ∈ Cc∞ (G) can be explicitly
given in terms of its integrals over the conjugacy classes in G. Combined
with the decomposition G = N AK this proves an inversion formula for
f → f" for complex classical G. For general G the inversion and Plancherel
formula for f → f" were given in [15], [17], [19]. The kernel of the map
ϕ → ϕ̌ and the surjectivity C ∞ (Ξ)∨ = C ∞ (X) are established in [B7].
The main theme in [20], [28] is the pattern of analogies between G/K
and G/M N . The algebra D(G/M N ) is commutative as is the algebra
D(G/K) and the transforms f → f", ϕ → ϕ̌ intertwine the operators in
D(G/K) and D(G/M N ). The analogs for G/M N of spherical functions
on G/K are the conical distributions on G/M N which by definition are
(i) M N -invariant.
(ii) Eigendistributions of each D ∈ D(G/M N ).
While the set of spherical functions is parametrized by a∗c /W (corresponding to K\G/K ∼ A/W ) I proved in [28] and [43] that the set
of conical distributions is parametrized by a∗c × W , corresponding to the
decomposition Ξ = M N · (A × W ) · ξ0 via the Bruhat decomposition of G.
The injectivity criterion for the Poisson transform Pλ mentioned below is
of central importance for this problem. The parametrization required the
exclusion of certain singular eigenvalues. For X of rank 1 this restriction
was removed by Men Cheng Hu (MIT thesis of 1973; Bull. Am. Math.
SIGURD̄UR HELGASON
xxi
xxii
Introduction
Soc. (1975)). In particular, Hu found for λ = 0 an additional conical
distribution which was omitted in Lemma 4.13, Ch. III in [28].
A representation of G is spherical if there is a fixed vector under K
and conical if there is a fixed vector under M N . It is proved in [17] and
[28] that an irreducible finite-dimensional representation is spherical if and
only if it is conical. This leads to an explicit characterization of their
highest weights. It also leads to simultaneous imbeddings of X and Ξ into
the same finite-dimensional vector space. The horocycles then appear as
certain plane sections with X ([B7], Ch. II, §4).
With π an irreducible finite-dimensional spherical representation of G
!
a K fixed vector under
on V let vK be a π(K) fixed vector and vK
the contragredient representation π̌. Then π is equivalent to the natural
representation of G on the space of functions
!
)
g → (π(g −1 )v, vK
v∈V
and in this model the spherical and conical vectors are respectively given
by
!
ϕK (gK) = (π(g −1 )vK , vK
),
ψM N (gK) = e−Λ(A(g)) ,
Λ being the highest restricted weight of π and A(g) as on page 2. In a
lecture in Iceland, 2007, David Vogan interpreted the identity above of the
spherical and conical representation in terms of a “deformation” of K into
M N . Also Adam Korányi has shown that the M N -fixed vector is a suitable
limit of an orbit of the K-fixed vector.
The continuous decompositions of L2 (X) and L2 (Ξ) into irreducibles
correspond under the maps f → f" and ϕ → ϕ̌. The map ϕ → ϕ̌ induces for
each λ ∈ a∗c a “Poisson transform” Pλ which maps functions (and functionals) on B = K/M into a joint eigenspace Eλ (X) of the operators in
D(G/K). In [27], [28] and [43] it is stated and proved that Pλ is injecti−1
ve if and only if Γ+
*= 0 where Γ+
X (λ)
X , the Gamma function of X, is
the denominator in the Gindikin–Karpelevic formula for Harish–Chandra’s
c(λ) function. For X of rank one this was done in [28] by analyzing the
Poisson transform at ∞. For X of higher rank this was done in [43] by studying its behavior at the origin, using methods from Kostant’s work on the
spherical principal series. The surjectivity of Pλ for hyperbolic spaces X
(with analytic functionals on B) and the surjectivity of Pλ for general X
on the K-finite level was proved in [40] and [43]. The surjectivity without
the K-finiteness assumption was proved by Kashiwara, Kowata, Minemura,
Okamoto, Oshima and Tanaka (Ann. of Math. (1978)). Professor Okamoto
has kindly explained how this remarkable collaboration came about.
“ I still remember vividly how we reached our proof of your great
conjecture. We tried very hard in vain to prove your conjecture in the
higher rank case, using the concept of analytical functionals. To understand
xxii
THE SELECTED WORKS
Introduction
xxiii
your conjecture in a different way, we decided to apply directly the original
idea of Prof. Sato about hyperfunctions to the new approach.
I took advantage of the fact that Prof. Sato is an uncle of Minemura
so that Minemura asked his uncle (Prof. Sato) to spare some time on Saturdays or Sundays for discussions about the possibility of applying directly
the original idea of Sato’s hyperfunctions to prove your conjecture. We
traveled to Kyoto University every weekend. During this time Kashiwara
(Kyoto University) and Oshima (Tokyo University) came to join in our
discussions.
To carry out our idea in the higher rank case, we realized that it is
important to extend the notion of the ordinary differential equation with
regular singularity to the notion of “the system of partial differential equations with regular singularity”.
The splendid aspect of your conjecture is to have given birth to a
new branch of Mathematics: Kashiwara–Oshima: Systems of differential
equations 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 hypergeometric function estimates) might extend to the general rank.
The Poisson transform of distributions on B was investigated by Lewis
and by Oshima–Sekiguchi. The latter authors also treat the case of affine
symmetric 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 by
their tangent planes, is investigated in [48] and [69]. The results are analogous but the expected tools like spherical function theory become degenerate so the results are obtained by looking at the flat case as a limiting
case of the curved case.
The complexification of the double fibration above, replacing each
group 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 /K c 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 acts
naturally on each joint eigenspace of the operators in D(L/H). I have called
these representations eigenspace representations. If U/K is a symmetric
space of the compact type the eigenspace representations are precisely the
irreducible spherical representations π of U . Spherical means that there is
SIGURD̄UR HELGASON
xxiii
xxiv
Introduction
a fixed vector under π(K). If U is simply connected the highest weights
of these π are given by the characterization mentioned before. This was
further generalized by Schlichtkrull (1984) and by Johnson (1987).
In the papers listed above the irreducibility question for the eigenspace
representations is considered for the cases Rn = M(n)/O(n), the noncompact symmetric space G/K, the dual space G/M N and G/N for G complex.
For G/K the joint eigenspaces are parametrized by a∗c /W (since each joint
eigenspace contains a unique spherical function); the irreducibility turns
out to be equivalent to 1/ΓX (λ) *= 0 where ΓX (λ) is the denominator of
c(λ)c(−λ).
Using the structure of D(G/M N ) the eigenspace representations for
G/M N turn out to be just the members of the spherical principal series
of G; the intertwining operators, realizing the equivalence between two
equivalent representations from this series, consist of convolutions with
conical distributions ([27], Theorem 4.6). Another parallel determination
of these operators was done by Schiffmann and by Kunze, Knapp and Stein.
The irreducibility of these representations is equivalent to 1/ΓX (λ) *= 0;
this gives another proof and interpretation of the algebraic irreducibility
criterion of Wallach, Parthasarathy–Rao–Varadarajan and Kostant though
stated in quite different terms. For G complex, D(G/N ) is still commutative, and the eigenspace representations form the full principal series.
For a non-Riemannian symmetric space G/H the eigenspace representations have been studied by Huang (Ann. of Math. (2001)). For the
hyperbolic spaces over various fields this had been done very completely by
Schlichtkrull (1987). For G nilpotent or solvable they have been effectively
studied 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 ϕλ on G is K-bi-invariant, that is% K-invariant on
X = G/K and the theory of the spherical transform f!(λ) = X f (x)ϕλ (x) dx
only concerns functions f on X which are K-invariant. The analog of ϕλ
for R2 is the Bessel function J0 (λ|x|) which is a radial eigenfunction of
the Laplacian L on R2 and the Bessel transform on R+ becomes the analog of the spherical transform. In [19] I proposed a definition of a Fourier
transform on X = G/K, applying to all functions on X but reducing to the
spherical transform for f K-invariant in the same way as the Fourier transform on R2 when used on radial functions reduces to the Bessel transform.
Given the Iwasawa decomposition G = N AK g = n exp A(g)k, A(g) ∈ a;
we define the vector-valued inner product A(x, b) on X × B by
A(gK, kM ) = A(k−1 g) .
xxiv
THE SELECTED WORKS
Introduction
xxv
Putting eλ,b (x) = e(iλ+ρ)(A(x,b)) we define the Fourier transform of a function f on X by
#
!
f (λ, b) = f (x)e−λ,b (x) dx
X
for all (λ, b) ∈ a∗c × B for which the integral converges. The definition is
suggested by the geometric analogy between horocycles and hyperplanes;
A(x, b) is a “vector-valued distance” from o to the horocycle through x
with normal b. A representation-theoretic interpretation was pointed out
by Sitaram (Pacific J. Math. (1988)) and Ólafsson–Schlichtkrull (Tribute
to Mackey Volume, AMS (2008)). All the principal theorems from classical
Fourier transform theory have their analogs for this Fourier transform.
Thus [19] and [28] contain the Plancherel theorem (with range) and the
inversion formula which can be written
#
eλ,b dµ(λ, b) ,
δo =
a∗ ×B
in analogy with the Euclidean version
#
δo =
eλ,ω λn−1 dλ dω ,
eλ,ω (x) = eiλ(x,ω) .
R+ ×Sn−1
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 sufficient
condition for the bijectivity of the Poisson transform for K-finite functions
on B. In turn this was used to prove the irreducibility criterion for the
eigenspace representations mentioned earlier. Another proof of the Paley–
Wiener theorem was given by Torasso (1977). The paper [91] contains a
strong form of the Riemann–Lebesgue lemma, proved jointly with Sengupta
and Sitaram. These two and several others, Mohanty, Ray, Cowling, Bagci,
Sarkar, Sundari, Rawat, have made great progress in topics such as the
analog of the Wiener–Tauberian theorem and Hardy’s theorem concerning
simultaneous exponential decay of a function and its Fourier transform.
Okamoto and especially Eguchi have proved deep theorems for the Fourier
transform 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 symmetric space U/K was developed by Sherman (Acta Math. (1990)), and for
vector bundles over X by Camporesi (Pacific J. Math (1997)). A Paley–
Wiener type theorem for the compact case has been obtained (2000) by
Gonzalez (on U ) and more generally (2008) by Ólafsson and Schlichtkrull
(on U/K).
In more recent years Fourier analysis theory for the non-Riemannian
symmetric spaces G/H has been vigorously developed by van den Ban,
Carmona, Delorme, van Dijk, Faraut, Flensted–Jensen, Matsuki, Molchanov, Ólafsson, Oshima, Rossmann, Schlichtkrull and many others.
SIGURD̄UR HELGASON
xxv
xxvi
Introduction
8. Multipliers.
[1], [2], ∗ [3], [4], [5]
Multipliers appear naturally in classical Fourier analysis on groups, where
they correspond to operators on function spaces commuting with translations. In the papers listed the setup is a semisimple Banach algebra A; a
subalgebra A0 , the derived algebra, is introduced as the algebra of elements
x ∈ A for which the multiplication y → xy is continuous from the topology
y ,∞ to the original topology. Several
of the spectral norm ,y,sp = ,"
properties of A0 are proved in [3], for example the determination of the
"0 as the intersection of certain L1 spaces.
representative algebra A
For various function algebras on a compact group G the derived algebra A0 is explicitly determined. For G the circle group the results
specialize to classical results of Littlewood, Sidon and Zygmund in Fourier
series theory; for G the Bohr compactification of R they solve the multiplier
problem 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 turns
out to be L2 (G) and in fact,
√
,f ,2 ≤ 2 sup {,f ∗ g,1 : ,g,sp ≤ 1} .
g
For G abelian it follows from a combination of the papers Edwards and
Ross (Helgason’s number and lacunarity constants, Bull.
Austr. Math.
√
Soc. √
(1973)) and Sawa (Studia Math. (1985)) that 2 can be replaced
by 2/ π and that this is the best possible constant. For G nonabelian or
G/K symmetric it is not known whether the best constant is independent
of G.
xxvi
THE SELECTED WORKS
Photos
xxviii
PHOTOS
Akureyri Gymnasium. (Photo courtesy of F. Rouviere.)
With family, Artie (wife), children Thor and Annie.
xxviii
THE SELECTED WORKS
PHOTOS
xxix
With student J. Dadok and Adviser S. Bochner.
Conference Roskilde Denmark (for Helgason's 65th birthday).
SIGURD̄UR HELGASON
xxix
xxx
PHOTOS
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.
xxx
THE SELECTED WORKS
PHOTOS
xxxi
Receiving Major Knights Cross from President
Vigdis Finnbogadottir, 1991.
From R. Sulanke's retirement. With Th. Friederich, U. Simon, R. Sulanke,
A. Onishchik.
SIGURD̄UR HELGASON
xxxi
xxxii
PHOTOS
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.)
xxxii
THE SELECTED WORKS
PHOTOS
xxxiii
With A. Koranyi, Conference in Icleand, 2007. (Photo courtesy of
F. Rouviere.)
Workshop in Tambara, Japan, 2006.
SIGURD̄UR HELGASON
xxxiii
Bibliography, up to 2008, of S. Helgason
Books
[B1] Differential Geometry and Symmetric Spaces, Academic Press, New York,
1962. Russian translation, MIR, Moscow, 1964. Reprint by American
Mathematical Society, 2001.
[B2] Analysis on Lie Groups and Homogeneous Spaces, Conference Board of
Math. Science Lectures, Dartmouth 1971. Amer. Math. Soc. CBMS
Monographs 1972. Corrected reprint 1977.
[B3] Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press,
New York, 1978. Russian translation, Factorial Press, 2005. Corrected
reprint 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 by
American Mathematical Society in 2000.
[B7] Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs. Amer. Math. Soc. 1994. 2nd edition, 2008.
[B8] (with Guðmundur Arnlaugsson) Stærðfræðingurinn Ólafur Daníelsson.
Saga Brautryðjanda. Háskólaútgáfan, 1996.
SIGURD̄UR 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 differential equations on Lie groups, Thirteenth Scand.
Congr. Helsinki, l957, 110-115.
Math.
[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] Differential operators on homogeneous spaces, Acta Math. 102 (l959), 239299.
[10] Some remarks on the exponential mapping of an affine 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 differential operators on symmetric spaces, 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 differential operators on symmetric spaces, Amer. J. Math. 86 (l964), 565-601.
[17] A duality in integral geometry; some generalizations of the Radon transform, Bull. Amer. Math. Soc. 70 (l964), 435-446.
xxxvi
xxxvi
THE SELECTED WORKS
Bibliography
xxxvii
*[18] The Radon transform on Euclidean spaces, compact two-point homogeneous
spaces and Grassmann manifolds, Acta Math. 113 (l965), 153-180.
*[19] Radon-Fourier transforms on symmetric spaces and related group representations, Bull. Amer. Math. Soc. 71 (l965), 757-763.
*[20] A duality in integral geometry on symmetric spaces, Proc. U.S.-Japan
Seminar in Differential Geometry, Kyoto, Japan, l965, pp. 37-56. Nippon
Hyronsha, 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 certain
symmetric spaces, Math. Ann. 165 (1966), 297-308.
*[23] (with A. Korányi) A Fatou- type theorem for harmonic functions on symmetric 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 differential 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 Lie
groups, 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. Congress
Math. Nice 1970, Vol. 2. Gauthier-Villars, Paris 1971, 313-320.
[31] Conical distributions and group representations, Symmetric Spaces, Short
courses 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. Diff.
Geometry 6 (l972), 411-419.
[33] Harmonic analysis in the non-Euclidean disk, Proc. Internat. Conf. on
Harmonic Analysis, Univ. of Maryland 1971. Lecture Notes in Mathematics 266, 151-156.
SIGURD̄UR HELGASON
xxxvii
xxxviii
Bibliography
[34] Geometric reduction of differential operators, Symposia Mathematica 10
(1972), 107-115.
*[35] Paley-Wiener theorems and surjectivity of invariant differential operators
on 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 differential operators on symmetric spaces,
Ann. of Math. 98 (1973), 451-480.
[38] Spherical functions, spherical representations and correspondence theorems
for the spherical Fourier transform, Colloque sur les fonctions sphériques et
la théorie des groupes, Nancy, 5-9, 1971. Russian translation, Mathematika
17 (1973).
[39] Elie Cartan’s Symmetriske Rum, Danish Math. Soc. Centennial Publication, 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 for
second order partial differential equations with four independent variables.
I. Ann. Inst. Poincaré Sect. A 21 (1974), 347.
*[43] A duality for symmetric spaces with applications to group representations
II. Differential equations and eigenspace representations, Adv. in Math.
22 (1976), 187-219.
[44] Solvability of invariant differential operators on homogeneous manifolds,
C.I.M.E. seminar lecture (Varenna 1975), 281-310. Cremonese, Roma 1975.
[45] Solvability questions for invariant differential operators, Proc. Fifth Internat. Colloq. on Group Theoretical Methods in Physics (Montreal 1976),
pp. 517-527. Academic Press 1977.
[46] Invariant differential equations on homogeneous manifolds, Bull. Amer.
Math. Soc. 83 (1977), 751-774.
[47] Some results on eigenfunctions on symmetric spaces and eigenspace representations, Math. Scand. 41 (l977), 79-89.
xxxviii
THE SELECTED WORKS
Bibliography
xxxix
*[48] A duality for symmetric spaces with applications to group representations
III. Tangent space analysis, Adv. in Math. 36 (1980), 297-323.
[49] Invariant differential operators and eigenspace representations, In Representations of Lie groups (G. Luke, ed.), London Math. Soc. Lecture Note
Series 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. Differential Geometry 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 Harmonic
Analysis (Turin/Milan, 1982). Ist. Naz. Alta Mat. Francesco Severi, Rome
1983, 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. Differential Geometry and its Applications, (Nove Mesto Morave
Czechoslovakia 1983). Publ. Univ. Karlova Praha 1984, 59-75.
[57] The Fourier transform on symmetric spaces, The mathematical heritage of
Élie Cartan, Astérisque 1985, Numero Hors Série, 151-164.
[58] Some results on Radon transforms, Huygens Principle and X-ray transforms, Integral geometry (Brunswick, 1984). Contemp. Math. 63 (1987),
151-177.
*[59] (with F. Gonzalez) Invariant differential operators on Grassmann manifolds, Adv. in Math. 60 (1986), 81-91.
*[60] Value-distribution theory for holomorphic almost periodic functions, The
Harald 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.
SIGURD̄UR HELGASON
xxxix
xl
Bibliography
*[62] The totally geodesic Radon transform on constant curvature spaces, Integral
geometry and tomography (Arcata, 1989). Contemp. Math. 113 (1990),
141-149.
*[63] Some results on invariant differential operators on symmetric spaces, Amer.
J. Math. 114 (1992), 789-811.
[64] Invariant differential 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. Intelligencer, 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. AMS
Summer Inst. on Differential Geometry (Los Angeles 1990), Proc. Sympos.
Pure Math. 54 (1993), 317-323.
[68] Tveir Töframenn Rúmfræðinnar, 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, Afbrigði 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 Quantum
Systems, Int. Wigner Symp. (Goslar 1991). World Sci. Publ. 1993, 149157.
[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. of
Symposium 75 years of Radon Transform, Vienna 1992. Int. Press, Boston
1994, 175-188.
[75] The Fourier Transform on Symmetric Spaces and Applications, Proc. Conf.
Differential Geometry and its Applications (Opava 1992), Silesian Univ.
Opava 1993, 23-28.
[76] Sophus Lie, the Mathematician, Proc. of The Sophus Lie Memorial Conference (Oslo 1992), Scand. Univ. Press, Oslo 1994, 3-21.
xl
THE SELECTED WORKS
Bibliography
xli
[77] Harish-Chandra’s c-function; a mathematical jewel. In Noncompact Lie
Groups and Some of Their Applications. NATO Adv. Sci. Inst. Ser. C
Math. 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 in
Math. 1684 (1998), 99-121
[79] Orbital Integrals, Symmetric Fourier Analysis and Eigenspace Representations, 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 and
Appl. Math. 51 (1998) 1035-1071.
*[82] (with H. Schlichtkrull) The Paley-Wiener Space for the Multitemporal Wave
Equation, Comm. Pure and Appl. Math. 52 (1999) 49-52.
[83] Harish-Chandra and his Mathematical Legacy. Some Personal Recollections, Current Sci. 74 (1998), 921-924.
[84] Harish-Chandra’s c-function; a mathematical jewel. Reprint of [77], The
mathematical legacy of Harish-Chandra, Proc. Sympos. Pure Math. 68
(2000), 273-283.
[85] Harish-Chandra Memorial Talk, ibid. 43-45.
[86] Hinn óevklíðski heimur, saga, rúmfræði og greining, Verpill, 2000.
[87] The Fourier Transform on Symmetric Spaces. Applications to Classical
and Multitemporal Wave Equations, Amer. Math. Soc. Proc. Symp. Pure
Math. (to appear).
[88] Rúmfræði og Raunveruleikinn, Icelandic Math Society Bulletin, 2000, 1-12.
[89] Non-Euclidean Analysis. In (Eds. Prékopa and Molnár) Non-Euclidean
Geometries, 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 Hyperbolic 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.
SIGURD̄UR HELGASON
xli