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Radon transformation on reductive symmetric spaces: support theorems J.J. Kuit

Thesis committee: Prof. dr. G.J. Heckman, Radboud Universiteit Nijmegen Prof. dr. S. Helgason, Massachusetts Institute of Technology Prof. dr. T. Kobayashi, University of Tokyo Prof. dr. E. M. Opdam, Universiteit van Amsterdam Prof. dr. H. Schlichtkrull, Københavns Universitet Cover illustration: The illustration on the cover visualizes a single horosphere in the Lie group G D SL.2; R/, considered here as the symmetric space .G G/=diag.G/. The group G is depicted as an open solid torus via the diffeomorphism described in [DK00, p. 13 – 16]. The surface inside the torus represents the image of the horosphere 1 x 1 0 W x; y 2 R 0 1 y 1 under this diffeomorphism.

ISBN: 978-90-393-5564-0

Radon transformation on reductive symmetric spaces: support theorems

Radontransformatie op reductieve symmetrische ruimten: dragerstellingen (met een samenvatting in het Nederlands)

Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de rector magnificus, prof. dr. G.J. van der Zwaan, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op maandag 16 mei 2011 des middags te 2.30 uur door

Job Jacob Kuit geboren op 13 januari 1983 te Ede

Promotor: Prof. dr. E.P. van den Ban

Voor mijn grootouders

Joost van den Vondel, De vernieuwde gulden winckel der kunstlievende Nederlanders, naar de eerste brief van Petrus; Paulus van Ravensteyn voor Dirck Pietersz. Pers, Amsterdam, 1622.

Table of Contents Introduction 1

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Radon transformation and support theorems 1.1 Integral transforms . . . . . . . . . . . . 1.2 Double fibrations . . . . . . . . . . . . . 1.3 Radon transforms . . . . . . . . . . . . . 1.4 Principal problems . . . . . . . . . . . . 1.5 Support theorems . . . . . . . . . . . . .

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Lie groups and symmetric spaces 2.1 Reductive Lie groups of the Harish-Chandra class 2.2 Parabolic subgroups . . . . . . . . . . . . . . . . 2.3 Reductive Symmetric spaces . . . . . . . . . . . 2.4 The class of ı -stable parabolic subgroups . . 2.5 Notation . . . . . . . . . . . . . . . . . . . . . .

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Radon transformation on a reductive symmetric space 3.1 Horospheres . . . . . . . . . . . . . . . . . . . . . 3.2 Double fibration . . . . . . . . . . . . . . . . . . . 3.3 Invariant measures . . . . . . . . . . . . . . . . . 3.4 Radon transforms . . . . . . . . . . . . . . . . . . 3.5 Spaces of functions and distributions . . . . . . . . 3.6 Extensions of the Radon transforms . . . . . . . . 3.7 Relations between Radon transforms . . . . . . . . 3.8 Some convex geometry . . . . . . . . . . . . . . . 3.9 Support of a transformed function or distribution .

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15 16 16 20 21 24

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27 28 29 32 35 35 42 49 51 53

i

Table of contents

4

5

Support theorem for the horospherical transform 4.1 The Euclidean Fourier transform and Paley-Wiener estimates . . 4.2 The unnormalized Fourier transform . . . . . . . . . . . . . . . 4.3 The -spherical Fourier transform FP 0 ; . . . . . . . . . . . . . 4.4 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Inversion formula . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 A support theorem for the horospherical transform for functions Support theorems 5.1 Spaces of distributions 5.2 Support theorems . . . 5.3 Injectivity . . . . . . . 5.4 Further remarks . . . .

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Appendix A: Transversality

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61 62 65 74 78 84 85

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91 92 96 100 105 107

Appendix B: Hyperbolic space 113 B.1 The structure of Hor.X / . . . . . . . . . . . . . . . . . . . . . . . 114 B.2 Incidence between subsets of X and horospheres . . . . . . . . . . 116 References

119

Indices 123 Index of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 General index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

ii

Samenvatting voor niet-wiskundigen

127

Dankwoord

131

Curriculum Vitae

133

Introduction Let G be a connected semisimple Lie group with finite center, let G D KAN be an Iwasawa decomposition of G and let M be the centralizer of A in K. A horosphere in the Riemannian symmetric space X D G=K of non-compact type is a submanifold of the form gN K, with g 2 G. The set of horospheres is in bijection with the manifold D G=MN via the map E W g MN 7! gN K: The horospherical transform on X is the Radon transform R mapping a compactly supported smooth function on X to the function Z R W g MN 7! .gn K/ N

on . In [Hel73, Lemma 8.1] S. Helgason proved the following support theorem for this transform. Let be a compactly supported smooth function and let V be a closed ball in X . Assume that R./ D 0 whenever E./ \ V D ;: Then .x/ D 0 for x … V:

Note that this theorem implies that the horospherical transform is injective on the space of compactly supported smooth functions. In this thesis, Helgason’s result is generalized to a support theorem for a class of Radon transforms (including the horospherical transforms) on a reductive symmetric space X D G=H with G a real reductive Lie group of the Harish-Chandra class

iii

Introduction

and H an essentially connected open subgroup of the fixed-point subgroup G of an involution on G. Let be a Cartan involution of G commuting with . For each ı -stable parabolic subgroup P with Langlands decomposition MP AP NP we consider the Radon transform RP mapping a function on X to the function on the homogeneous space P D G=.MP AP \ H /NP given by Z RP .g P / D .gn H / d n: NP

Here P denotes the coset e .MP AP \ H /NP containing the unit element e. This Radon transform, which is initially defined for compactly supported smooth functions, can be extended to a large class of distributions on X . If P0 is a minimal ı -stable parabolic subgroup of G contained in P , then AP is contained in AP0 . The Lie algebra aP0 of AP0 is -stable and decomposes as the direct sum of the C1 and 1 eigenspace for . The latter space is denoted by aq . The connected abelian Lie subgroup of G with Lie algebra aq is denoted by Aq . The maps K Aq ! X I .k; a/ 7! ka H and K Aq ! P I

.k; a/ 7! ka P

are surjective, just as in the Riemannian case. For a subset B of aq , we define X.B/ D K exp.B/ H

and P .B/ D K exp.B/ P :

The support of a transformed function or distribution is non-compact in general. In fact, if the support of a distribution is contained in X.B/ for some compact convex subset B of aq that is invariant under the action of the normalizer of aq in K \ H , then the support of the Radon transformed distribution RP is contained in P .B C P /, where P is the cone in aq spanned by the root vectors corresponding to roots that are positive with respect to P . The support theorem that we prove in this thesis is a partial converse to this statement for distributions in a suitable class of distributions, containing the compactly supported ones: Theorem 5.2.1 Let B be a convex compact subset of aq that is invariant under the action of the normalizer of aq in MP \ K \ H . If supp.RP / P .B C

iv

P /;

Introduction

then supp./ X.C /; where C is the maximal subset of B C normalizer of aq in K \ H .

P

that is invariant under the action of the

If K D H and P is a minimal parabolic subgroup of G, then C equals B. Our theorem reduces then to the support theorem of Helgason for the horospherical transform on a Riemannian symmetric space of non-compact type. Just as in the Riemannian case, the support theorem implies injectivity of the Radon transform. After a short introduction to the theory of Radon transformation in Chapter 1 and a short introduction to the theory of reductive Lie groups of the Harish-Chandra class, their parabolic subgroups and reductive symmetric spaces in Chapter 2, we introduce the transforms under consideration and establish some of their properties in Chapter 3. In Chapter 4 we consider the horospherical transform. A horospherical transform is a Radon transform RP0 as above, with P0 a minimal ı -stable parabolic subgroup. The horospherical transform is related to one component of the so called unnormalized Fourier transform on X corresponding to the parabolic subgroup P0 . As in the proof for [Hel73, Lemma 8.1], given the support of the horospherical transform of a function, it is possible to derive Paley-Wiener type estimates for this component of the unnormalized Fourier transform. The unnormalized Fourier transform is in turn related to the normalized Fourier transform corresponding to the opposite parabolic subgroup P 0 for K-finite functions on X . The Paley-Wiener estimates for the one component of the unnormalized Fourier transform yield Paley-Wiener estimates for one component of the normalized Fourier transform. A first support theorem (Proposition 4.6.2) is then obtained from these estimates by using the Fourier inversion formula of Van den Ban and Schlichtkrull ([BS99, Theorem 4.7]) and then by mimicking the proof for the classical PaleyWiener theorem for the Euclidean Fourier transform. However, this result gives information only on the intersection of the support of the function with a certain subset of X related to P0 . Using the equivariance of the horospherical transform, we then obtain the full support theorem (Theorem 4.6.4) for this transform. In Chapter 5 we generalize this (in Theorem 5.2.1) to a support theorem for the Radon transform RP on distributions for P an arbitrary ı -stable parabolic subgroup.

v

Chapter 1

Radon transformation and support theorems This chapter is meant as a short introduction to the theory of Radon transformation. After a few historical comments, we first introduce the concept of a double fibration in Section 1.2. For each double fibration satisfying a few technical conditions we then define two integral transforms, called Radon transforms. This is described in Section 1.3. In Section 1.4 the principal questions that are raised in the theory of Radon transformation are discussed. Finally in Section 1.5 we describe a class of theorems that describe the support of a function in terms of the support of its Radon transform: the support theorems. An example of a Radon transform is the integral transform obtained by mapping a suitable function on Rn to the function Rk on the manifold of k-dimensional affine subspaces in Rn , given by Z Rk ./ D .x/ dx:

Throughout this chapter we use this example as an illustration of the general theory.

1

1.

1.1

Radon transformation and support theorems

Integral transforms

In 1906 H.B.A. Bockwinkel published an article ([Boc06]) in the proceedings of the Koninklijke Akademie van Wetenschappen (the Royal Netherlands Academy of Sciences) on electromagnetic fields in a crystal, in which he wrote the following passage on the determination of the electromagnetic field in a crystal due to a periodic electromotive force (E.M.F.).

More explicitly, Bockwinkel considered the transform mapping a function on R3 to a function on the manifold of affine 2-dimensional planes by integrating over these planes as is stated in formula (2). Formula (1), which according to Bockwinkel was proved by H.A. Lorentz, is an inversion formula for this transform. In 1917 J. Radon considered in his article [Rad17] a generalization of this transform: a suitable function on Rn is mapped to the function R on the manifold of affine hyperplanes in Rn by defining Z R./ D

.x/ dx:

In his article Radon derived an inversion formula for this transform, generalizing the formula obtained by Lorentz.

2

1.2.

1.2

Double fibrations

Double fibrations

In [Hel66] Helgason used the concept of incidence, introduced by Chern in 1942 ([Che42]), to generalize the integral transform considered by Radon and several other transforms to a class of integral transforms named after Radon. In the theory of Radon transformation as introduced by Helgason one considers a double fibration of homogeneous spaces Z D G=.S \ TV/ VVV˘ VVVV

˘Xhhhh

G=S D X

hh s hhh h

VV+

(1.2.1) D G=T

where G is a Lie group, S and T are two closed subgroups of G and ˘X and ˘ are the canonical projections. The map from Z to X g .S \ T / 7! .g S; g T / is an embedding. Via this embedding points in Z are identified with points in X . A double fibration defines an incidence relation: x 2 X and 2 are said to be incident if .x; / 2 Z. Note that x and are incident if and only if 2 ˘ .˘X 1 .x//, or equivalently, x 2 ˘X .˘ 1 .//. If the set-valued maps X 3 x 7! ˘ .˘X 1 .x// and 3 7! ˘X .˘ 1 .// are both injective, then following Helgason, we say that S and T are transversal. Example 1.2.1. Let n 2 N. Let G D O.n/ n Rn be the semidirect product of the orthogonal group O.n/ and Rn , i.e. G is the group consisting of pairs .O; t / 2 O.n/ Rn equipped with the multiplication given by .O1 ; t1 /.O2 ; t2 / D .O1 O2 ; O2 1 t1 C t2 /

.O1;2 ; t1;2 / 2 G :

Let k 2 Z, with 0 < k < n. We define k to be the manifold of k-dimensional affine subspaces of Rn . The group G has a natural transitive action on Rn given by .O; t / x D O.x C t /

.O; t / 2 G; x 2 Rn :

In fact G is the group of isometries of Rn . The action of G on Rn induces in an obvious way a transitive action of G on k . We fix an element 0 2 k containing

3

1.


the origin 0 2 Rn and define S to be the stabilizer in G of the origin and T to be the stabilizer in G of 0 . Then S D O.n/ n f0g

and T D O.0 / O.0? / n 0 ;

where 0? denotes the orthocomplement of 0 . Note that O.0 / O.0? / equals the stabilizer of 0 in O.n/. Now Rn ' G=S and k ' G=T . Accordingly, the homogeneous space G=.S \ T / is diffeomorphic to the submanifold of Rn k given by Z D f.x; / 2 Rn k W x 2 g: The double fibration oo Z OOOOO OOO ooo o o O' wo o n k R

(1.2.2)

describes the incidence relations between points in Rn and k-dimensional affine subspaces in Rn : x 2 Rn and 2 k are incident if and only if x 2 . Since an affine subspace is determined by the points in it and a point in Rn is determined by the k-dimensional affine subspaces through it, the subgroups S and T are transversal.

1.3

Radon transforms

Following L. Schwartz, we denote spaces of compactly supported smooth functions and spaces of smooth functions by D and E , respectively. Spaces of distributions and spaces of compactly supported distributions we denote by D 0 and E 0 , respectively. To be able to define the Radon transforms for a double fibration (1.2.1), we need the following assumptions (A) There exist non-zero Radon measures dS \T s and dS \T t on S=.S \ T / and T =.S \ T / invariant under S and T , respectively. (B1) For every 2 D.X / the function T =.S \ T / ! C given by t .S \ T / 7! .t S / is absolutely integrable with respect to d.S \T / t .

4

1.3.

(B2) For every

Radon transforms

2 D. / the function S=.S \ T / ! C given by s .S \ T / 7! .s T /

is absolutely integrable with respect to d.S \T / s. The Radon transforms R and S for the double fibration (1.2.1) are defined to be the transforms mapping functions 2 D.X / and 2 D. / to the functions R W ! C and S

WX !C

given by Z R.g T / D

T =.S \T /

.gt S / dS \T t

and Z S .g S / D

.gs T / dS \T s;

S=.S \T /

respectively. If the condition (B) ST is a closed subset of G is satisfied, then (B1) and (B2) hold and R and S are continuous operators from D.X/ to E . / and from D. / to E .X /, respectively. (See [Hel99, Section II.1].) Note that the Radon transforms are G-equivariant. If moreover (C) There exist non-zero G-invariant Radon measures dx and d on X and , then S and R are dual to each other in the sense that for all 2 D.X / and 2 D./ Z Z R./ ./ d D .x/ S .x/ dx; (1.3.1)

X

assuming the measures are suitably normalized. This duality allows to extend the Radon transform to the space of compactly supported distributions: for 2 E 0 .X / and 2 E 0 . /, we define R 2 D 0 . / and S 2 D 0 .X / to be the distributions R W D. / 3

7! .S / and S W D.X / 3 7! .R/:

(1.3.2)

5

1.


Example 1.3.1. We continue in the setting of Example 1.2.1. The homogeneous space T =.S \ T / is diffeomorphic to 0 . The Lebesgue measure on 0 is T -invariant. Since S is compact, there exists an S-invariant measure on S=.S \ T /. Therefore (A) holds in this case. Furthermore, since S is compact and T is closed in G, the set S T is closed in G and hence (B) holds. The Lebesgue measure on Rn is G-invariant, just as the measure on k defined by Z Z f O.t C 0 / dt dO; D.k / 3 f 7! O.n/

0?

where dO is the Haar measure on O.n/ and dt the Lebesgue measure on 0? . We therefore finally conclude that (C) is satisfied in this case. The Radon transforms for the double fibration (1.2.2) are given by Z 2 D.Rn /; 2 k ; (1.3.3) Rk ./ D .x/ d x

where d x is the Lebesgue measure on . The group O.n/ acts on the compact submanifold f 2 k W x 2 g of k in a natural way. Let dx be the normalized invariant measure on this submanifold. Then Z (1.3.4) Sk .x/ D ./ dx 2 D.k /; x 2 Rn : 3x

1.4

Principal problems

Although many rather strong theorems have been proved for particular examples of Radon transforms, it seems to be impossible to obtain strong results in the general abstract setup described in the previous section. As Helgason mentions in [Hel99, Section 2.2] it may be better to regard this setup as a framework for examples rather than as a set of axioms for a general theory. When dealing with a certain example one often considers the following problems. Domain. When considering a double fibration (1.2.1) satisfying assumption (A), the first problem is to find a suitable space of functions or distributions on which the Radon transforms can be defined. If assumption (B) is satisfied then at least the integrals are absolutely convergent for compactly supported continuous functions, but in important examples the same holds for larger spaces of functions.

6

1.4.

Principal problems

Range. Once a domain for the Radon transform has been determined, one often tries to determine the image under the transform of the whole domain or of certain subspaces. More generally, one tries to relate function spaces on X and to each other by means of the Radon transform. In particular it is usually interesting to find the kernel of the transform. Inversion formulae. If the transform is injective on the domain or on a certain subspace of the domain, then one of the main problems is to invert the transform. Invariant differential operators. For the analysis of G-invariant differential operators on the homogeneous space X D G=S, it may be useful to consider the b from the space D.X / of invariant differential question if there exists a map D 7! D operators on X to the space D. / of invariant differential operators on with the property that for every function in the domain of R and every D 2 D.X / b R.D/ D DR: As mentioned before, it is hard to obtain any results that hold in the general setting introduced in Section 1.2 and 1.3, but the formalism can help to prove results for specific Radon transforms. As an example we now derive an inversion formula for the k-plane transform introduced in Example 1.3.1, which is due to Helgason (see [Hel99, Theorem 6.2]). Example 1.4.1. We continue in the setting of Example 1.2.1 and Example 1.3.1. For the domain of the Radon transform Rk we choose the space S .Rn / of Schwartz functions on Rn and for the domain of Sk we choose E .k /. We denote the space of tempered distributions on Rn by S 0 .Rn /. To be able to formulate the inversion formula, we first need to introduce fractional powers of the laplacian. Let V be the subspace of S 0 .Rn / consisting of tempered distributions u with the property that the Fourier transform Fu of u is a locally integrable function. Let ˛ 0. We define the ˛-th power . /˛ of the differential operator to be the operator on V given by . /˛ u D F

1

k k2˛ Fu

.u 2 V /:

Note that if Fu is a locally integrable function, then k k2˛ Fu is locally integrable as well and therefore a distribution. In fact it is a tempered distribution and thus the inverse Fourier transform of k k2˛ Fu is well defined. Note further that if ˛ is a nonnegative integer, then . /˛ is the ordinary ˛-th power of .

7

1.


Theorem 1.4.2. Let 2 S .Rn /. Then Sk .Rk / is a tempered distribution of which the Fourier transform is a locally integrable function. Moreover, D

n k 2

.4/

k 2

k

n 2

. / 2 Sk .Rk /:

(1.4.1)

To prove the theorem we use some elementary distribution theory. For completeness for several well known results, we refer to the recently published book [DK10] by the authors former teachers, J.J. Duistermaat and J.A.C. Kolk. The plan of the proof is as follows. We analyze the function Sk .Rk / using the equivariance of the Radon transforms and the explicit formulas (1.3.3) and (1.3.4). We then determine the precise form of the Fourier transform of Sk .Rk / up to a multiplicative constant and show that up to the prefactor the inversion formula is correct. Finally, we perform some explicit calculations on one specific Schwartz function to determine the constant. We start with a lemma. Lemma 1.4.3. Let m > n and let 2 D 0 .Rn /. If is O.n/-invariant and homogeneous of degree m, then there exists a constant c 2 C, such that D ck km : Proof. Let p n and let ˚ W Rn ! Rp be a smooth submersion. We recall that the pull-back along ˚ E .Rp / ! E .Rn /I

7! ˚ D ı ˚

extends to a continuous linear mapping D 0 .Rp / ! D 0 .Rn /. Here smooth functions on Rn and Rp are identified with distributions via the Lebesgue measures on Rn and Rp , respectively. If ˚ is a diffeomorphism, then ˚ D .j˚

1

/ ı .˚

1

/ ;

where j˚ 1 is the absolute value of the Jacobian of ˚ 1 . Using the method of [DK10, Exercise 13.12] to describe the distributions on RnC1 n f0g that are invariant under Lorentz-transformations, one shows that the restriction of to Rn n f0g equals the pullback of a distribution on R>0 along the norm-function k k. Since is homogeneous of degree m, the distribution has to be homogeneous of the same degree, hence up to a multiplicative constant equals

8

1.4.

Principal problems

the function R>0 3 x 7! x m . This implies that outside the origin is equal to the function ck km for some c 2 C. Since this function is locally integrable on Rn , it follows that D ck km C for some distribution 2 E 0 .Rn / supported in the origin. Since has support contained in f0g, it follows that is of finite order, say d , and is of the form X D cˇ @ˇ ı: jˇ jd

(See e.g. [DK10, Theorem 8.10].) Here ı denotes the Dirac delta distribution supported in the origin and P the sum is taken over all multi-indices ˇ D .ˇ1 ; : : : ; ˇn / 2 Zn0 of length jˇj D jnD1 ˇj at most d . Both and ck km are homogeneous of degree m, hence the same has to hold for . Since @ˇ ı is homogeneous of degree n jˇj, it follows that D 0. We conclude that there exists a c 2 C such that D ck km . Proof of Theorem 1.4.2. Let be the tempered distribution D ı ı Sk ı Rk D Sk .Rk ı/; where ı denotes the Dirac delta distribution supported in the origin. Since both Sk and Rk are G-equivariant transforms, and ı is O.n/-invariant, we see that for every O 2 O.n/ O D ı ı Sk ı Rk ı .O

1

/ D ı ı .O

1

/ ı Sk ı Rk D O .ı/ ı Sk ı Rk D :

Therefore is O.n/-invariant. Moreover, if c > 0, then Z Z n 1 n .c 1 x/ d x d0 D c k c ./ D c .c / D c 30

./

for every 2 D.Rn /, hence is homogeneous of degree k there exists a constant c 2 C such that D ck kk

n

n

n. By Lemma 1.4.3

:

Note that is a tempered distribution. Now let 2 S .Rn /. For x 2 Rn we define Tx to be the translation y 7! y C x. Furthermore, we define W Rn ! Rn to be the reflection y 7! y in the origin. Since both Rk and Sk are G-equivariant, Sk .Rk /.x/ D Tx Sk .Rk / .0/ D ı ı Sk ı Rk .Tx / D .Tx /:

9

1.


Furthermore, since is O.n/-invariant, D and it follows that Sk .Rk /.x/ D .Tx / D .x/; where denotes the convolution product. Since is a tempered distribution, the convolution defines a tempered distribution. The Fourier transform of equals F. / D .F/F: (1.4.2) A tempered distribution is O.n/-invariant if and only if its Fourier transform is O.n/-invariant and it is homogeneous of degree k n if and only if its Fourier transform is homogeneous of degree k. (See e.g. [DK10, Exercise 14.30].) Since is O.n/-invariant and homogeneous of degree k n, it follows from Lemma 1.4.3 that there exists a constant c 0 2 C such that F D c 0 k k

k

:

(1.4.3)

Combining (1.4.2) and (1.4.3), we obtain that the Fourier transform of Sk .Rk / equals .F/F D c 0 kk k F, which is a locally integrable function. Furthermore, k

. / 2 Sk .Rk / D F

1

k kk F. / D c 0 F

1

k kk k k

k

F D c 0 :

It remains to determine the constant c 0 . To do this, we calculate both sides of the identity (1.4.4) .0 / D c 0 F 1 k k k .0 /; where 0 W Rn ! R is the function given by 0 .x/ D e

kxk2 2

.x 2 Rn /:

We first compute the left-hand side: Z .0 / D Sk .Rk 0 /.0/ D

Z 0 .x/ dx d :

30

Since the integrand is O.n/-invariant, the inner integral is independent of and equal to Z k Z kxk2 y2 k e 2 dx D e 2 dy D .2/ 2 Rk

10

R

1.5.

Support theorems

k

hence .0 / D .2/ 2 . The right-hand side of (1.4.4) equals Z Z n 0 k 1 0 2 c kxk F 0 .x/ dx D c .2/ kxk k e n n R R n k 2 n Z r2 2 2 2 D c0 r n k 1 e 2 dr D c 0 k : n n R>0 22 2 2 We conclude that

k

n 2

.4/ 2 c D 0

n k 2

kxk2 2

dx

:

An easy computation shows that ı Sn

1

D Sn

where is the differential operator on n .p! C ! ? / D

@2 .p! C ! ? / @p 2

1

1

ı ;

(1.4.5)

given by .

2 E .n

1 /; p

2 R; ! 2 S n

1

/:

The inversion formula (1) in [Boc06] for the Radon transform R2 on R3 is obtained from the inversion formula in Theorem 1.4.2 by applying (1.4.5). The difference in the prefactor is due to the fact that Bockwinkel used the (unnormalized) Euclidean measure on the sphere, while we use normalized measures on the submanifolds f 2 2 W 3 xg of 2 for x 2 R3 .

1.5

Support theorems

As mentioned in Section 1.4, when considering a specific Radon transform, one of the principal problems is to determine its range and the image of certain subspaces of the domain. Related to these problems are the so called support theorems. Given the support of a function, it is usually not very difficult to give bounds on the support of the Radon transform of that function. A support theorem is the converse to such a result: it is a description of the support of a function in terms of the support of the Radon transformed function.

11

1.


Example 1.5.1. We continue in the setting of Example 1.2.1. See also Example 1.3.1 and 1.4.1. Let V be a compact convex subset of Rn and let 2 S .Rn /. It is clear that if supp./ V , then Rk ./ D 0 whenever 2 k satisfies \V D ;. The support theorem of Helgason ([Hel65, Theorem 2.1]) is the converse of this statement. Theorem 1.5.2. Let 2 S .Rn / and let V be a compact convex subset of Rn . Assume that Rk ./ D 0 whenever \ V D ;: Then supp./ V: We sketch the proof given by J.J.O.O. Wiegerinck in [Wie85]. Although the Radon transform Rk does not belong to the class of Radon transforms we consider in the later Chapters 3, 4 and 5, there are many similarities between Wiegerinck’s proof and the proof of the support theorem (Theorem 5.2.1) that we give in Chapter 4 and Chapter 5. Since every affine hyperplane that does not intersect with a given subset V of n R is a union of k-dimensional affine subspaces that do not intersect with V either, the following statement holds. Observation 1.5.3. Let 2 S .Rn / and let V be a closed convex subset in Rn . Assume that Rk ./ D 0 for every 2 k with \ V D ;: Then Rn

1 ./

D 0 for every 2 n

1

with \ V D ;:

Because of this observation we only need to consider the hyperplane transform Rn 1 . Furthermore, since a convex compact subset V of Rn equals the intersection of all closed balls containing V , it suffices to only consider the case in which V is a closed ball. Moreover, since Rn 1 intertwines the actions of the group of translations on Rn and n 1 respectively, without loss of generality we can assume that the ball is centered at the origin. It thus remains to prove the following (seemingly weaker) theorem. Theorem 1.5.4. Let 2 S .Rn / and let r > 0. Let Br be the closed ball in Rn centered at the origin with radius r. Assume that Rn

12

1 ./

D0

whenever \ Br D ;:

1.5.

Support theorems

Then supp./ Br : We denote the Fourier transforms on Rn and R by Fn and F1 , respectively. The proof for Theorem 1.5.4 is based on the following observation. Observation 1.5.5. Let 2 S .Rn /. Then for every r 2 R and every ! 2 S n 1 Z Z Z i rh!;xi Fn .r!/ D .x/e dx D .p! C y/ dy e i rp dp Rn R !? D F1 p 7! Rn 1 .p! C ! ? / .r/: (1.5.1) Sketch of the proof for Theorem 1.5.4. We sketch the proof given by Wiegerinck in [Wie85]. Let ! 2 S n 1 . Since Rn 1 is compactly supported, it follows from (1.5.1) that C 3 z 7! Fn .z!/ is holomorphic. Furthermore, since the support of R 3 p 7! Rn 1 .p! C ! ? / is contained in the interval Œ r; r, the directional derivatives of Fn in the origin satisfy ˇZ ˇ ˇ ˇ k ? k ˇ j@! Fn .0/j D ˇ Rn 1 .p! C ! /. ip/ dp ˇˇ cr k .k 2 Z0 /; R

where c equals the L1 .Rn /-norm of . Using the main lemma in [WK85] one can ˛ estimate for any multi-index ˛ the mixed derivative P @ F˛n in the origin in terms of the directional derivatives of Fn . We write ˛ c˛ for the Taylor series of Fn ./ in the origin. From the estimates it follows that there exists a constant a > 0 such that for every > 0 and every multi-index ˛ jc˛ j a

.r C /j˛j : j˛jŠ

From these estimates it follows that the Taylor series of Fn in the origin defines a holomorphic function F . Since for every ! 2 S n 1 the restriction of Fn to the complex line through ! is entire, it follows that Fn is real analytic on Rn and equal to F on Rn . Moreover, jF ./j ae .rC/

Pn

j D1

jj j

:

From this estimate, the Fourier inversion formula and the Plancherel-Pólya theorem ([Ron74, Theorem 3.4.2]) it then follows that supp./ is contained in Br .

13

Chapter 2

Lie groups and symmetric spaces This chapter is meant as a short introduction to the Lie theory that we need in later chapters. In Section 2.1 we define the so called reductive Lie groups of the HarishChandra class and in Section 2.2 we give a description of their parabolic subgroups. Then in Section 2.3 we give the definition of a reductive symmetric space. A reductive symmetric space is a homogeneous space G=H of a specific kind, where G is a reductive Lie group of the Harish-Chandra class. In Section 2.4, we consider a special class of parabolic subgroups of a reductive Lie group G of the HarishChandra class related to a reductive symmetric space G=H . Finally, in Section 2.5, we introduce some notation that we need in later chapters. The theory discussed in this chapter can be found in [Var77], [Kna02] and [BSD05].

15

2.

Lie groups and symmetric spaces

2.1

Reductive Lie groups of the Harish-Chandra class

A Lie group G is said to be a reductive Lie group of the Harish-Chandra class if (i) G is a real Lie group and its Lie algebra g is reductive. (ii) G has finitely many connected components. (iii) The image of G under the adjoint representation Ad W G ! GL.gC / is contained in the identity component of Aut.gC /. (iv) The center of the analytic subgroup with Lie algebra Œg; g is finite. Note that a connected semisimple Lie group is a reductive Lie group of the HarishChandra class if and only if it has finite center. Let G be a reductive Lie group of the Harish-Chandra class. A Cartan involution on G is an involution of G whose set of fixed points K D G is a maximal compact subgroup of G. It is known that all Cartan involutions are conjugate to a given Cartan involution via an element of Ad.G/. If is a Cartan involution of G, then we denote the corresponding involution on g also by . The latter involution is called a Cartan involution of g. The Lie algebra of g decomposes as a direct sum of vector spaces g D k ˚ p;

(2.1.1)

where k and p are the C1 and 1-eigenspace of respectively. Note that since Œp; p k, the subspace p is a Lie subalgebra of g if and only if p is abelian. The C1-eigenspace k on the other hand is a Lie subalgebra; it is the Lie algebra of K. Note furthermore that K acts on p via the adjoint representation and Œk; p p. The decomposition (2.1.1) is called the Cartan decomposition of g associated to . This decomposition has a global counterpart: the map K p ! GI

.k; Y / 7! k exp.Y /

is a diffeomorphism. The corresponding decomposition of G is called the Cartan decomposition of G associated to .

2.2

Parabolic subgroups

Let G be a reductive Lie group of the Harish-Chandra class and let be a Cartan involution of G.

16

2.2.

Parabolic subgroups

We fix a maximal abelian subspace a of p and write A for exp.a/. (It can be shown that every other maximal abelian subspace of p is conjugate to a via K.) The linear operators ad.Y / 2 End.g/, Y 2 a, are simultaneously diagonalizable. For ˛ 2 a we write g˛ D fY 2 g W ad.Z/Y D ˛.Z/Y for all Z 2 ag: The subset ˙.g; a/ of a , consisting of non-zero elements ˛ with g˛ ¤ f0g, is a (possibly non-reduced) root system and g decomposes as M gDm˚a˚ g˛ : ˛2˙.g;a/

Here m is the centralizer of a in k. Note that Œg˛ ; gˇ g˛Cˇ . We choose a set of positive roots ˙ C .g; a/ in the following manner. Let Z0 2 a be such that ˛.Z0 / ¤ 0 for every ˛ 2 ˙.g; a/. We choose ˙ C .g; a/ to be ˙ C .g; a/ D f˛ 2 ˙.g; a/ W ˛.Z0 / > 0g: We write nD

M

g˛ :

˛2˙ C .g;a/

The Lie algebra n is nilpotent. Since .g˛ / D g

˛

for ˛ 2 ˙.g; a/, it follows that

g D .n/ ˚ m ˚ a ˚ n: Let N D exp.n/ and let M D ZK .a/ be the centralizer of a in K. Then MAN is a subgroup of G. We define a minimal parabolic subgroup of G to be a subgroup of G that is conjugate to MAN . Furthermore, we define a parabolic subgroup of G to be a subgroup of G that contains a minimal parabolic subgroup. For a parabolic subgroup P of G, we define LP to be the subgroup P \ .P / of P and lP to be its Lie algebra. We further define aP D p \ Z.lP / and MP D ZK .aP / exp p \ ŒZg .aP /; Zg .aP / where ZK .aP / denotes the centralizer of aP in K and Zg .aP / denotes the centralizer of aP in g. Let AP D exp.aP /. The multiplication map MP AP ! LP I

.m; a/ 7! ma

is a diffeomorphism. We write mP for the Lie algebra of MP .

17

2.


We write nP for the nilpotent radical of the Lie algebra of P , i.e. the maximal ideal nP with the property that there exists a k 2 N such that for all Y1 ; : : : Yk 2 nP ad.Y1 / ı ad.Y2 / ı ı ad.Yk / D 0: We define NP D exp.nP /. Then the multiplication map LP NP ! P I

.l; n/ 7! ln

is a diffeomorphism. This implies that the map MP AP NP ! P I

.m; a; n/ 7! man

is a diffeomorphism as well. The corresponding decomposition P D MP AP NP is called the Langlands decomposition of P . If Pm is a minimal parabolic subgroup, then its Langlands decomposition Pm D MPm APm NPm has the following properties. The Lie algebra aPm of APm is a maximal abelian subspace of p and MPm is the centralizer in K of aPm . Furthermore, there exists a unique choice of a set ˙ C .g; aPm I Pm / of positive roots for ˙.g; aPm / such that NPm D exp.nPm /, where nPm equals the direct sum of all root spaces g˛ with ˛ 2 ˙ C .g; aPm I Pm /. Related to each minimal parabolic subgroup Pm of G there exists a so called Iwasawa decomposition: the map K APm NPm ! GI

.k; a; n/ 7! kan

is a diffeomorphism. The corresponding decomposition of G is called an Iwasawa decomposition of G. Note that since KPm D G for every minimal parabolic subgroup Pm , we have KP D G for every parabolic subgroup P of G. It follows from the Iwasawa decomposition that the minimal parabolic subgroups are conjugate to each other via K. In the next lemma we list some elementary properties of parabolic subgroups and their Langlands decomposition. Lemma 2.2.1. Let P be a parabolic subgroup of G. (i) LP equals the centralizer of aP in G. (ii) A P if and only if aP a.

18

2.2.

Parabolic subgroups

(iii) Assume Pm D MANPm is a minimal parabolic subgroup contained in P . Then M nP D g˛ : C ˛2˙ ˇ .g;aIPm / ˛ˇ ¤0 aP

(iv) LP normalizes NP . (v) LP is a reductive Lie group of the Harish-Chandra class. Proof. For (i), see e.g. [Kna02, Proposition 7.82 and Proposition 7.83]. Assume A P . Since A is -stable, it follows that A is contained in LP D P \ .P /. By (i) the group A centralizes AP or equivalently a centralizes aP . Since a is a maximal abelian subspace of p and aP is contained in p, it follows that aP a. For the converse, assume aP a. Then a centralizes aP and is therefore contained in lP by (i). This proves (ii). For (iii), see e.g. [Kna02, Section VII.7]. Statement (iv) holds for P if and only if it holds for every K-conjugate of P . Without loss of generality we can therefore assume that a minimal parabolic subgroup Pm of the formˇ Pm D MANPm is contained in P . Let ˛ 2 ˙ C .g; aI Pm / be a root such that ˛ ˇaP ¤ 0. Since LP centralizes aP , it follows that for every l 2 LP , Y 2 a and Z 2 g˛ ŒY; Ad.l/Z D Ad.l/ŒY; Z D ˛.Y /Ad.l/Z: n f0g the space Therefore LP normalizes for each ˇ 2 aP

M

g˛ :

˛2˙ C .g;aIPm / ˛

ˇ ˇ

aP

Dˇ

This proves in particular that LP normalizes nP and thus that LP normalizes NP . For (v), see e.g. [Var77, Section II.6.3, Theorem 13]. Lemma 2.2.2. Let P and Q be two parabolic subgroups of G. Assume that P Q. Then LP LQ ; AQ AP and NQ NP :

19

2.


Proof. Since P Q, we have .P / .Q/. Hence LP LQ and lP lQ . The latter implies that the center Z.lQ / of lQ centralizes lP . Since Z.lQ / centralizes aP in particular, it is contained in lP by Lemma 2.2.1 (i). This implies that Z.lQ / Z.lP / and therefore aQ aP and AQ AP . The last claim follows from Lemma 2.2.1(iii). Q

Assume that P and Q are parabolic subgroups of G and P Q. We write NP Q Q for the intersection NP \ LQ . The Lie algebra of NP is denoted by nP . Note that the multiplication map Q NQ NP ! NP is a diffeomorphism.

2.3

Reductive Symmetric spaces

A symmetric space is a homogeneous space G=H for a Lie group G, where H is an open subgroup of the fixed point subgroup of some involution of G. Let G be a reductive Lie group of the Harish-Chandra class, let be an involution of G and let X be the symmetric space G=H , where H is an open subgroup of G . We denote the corresponding involution on g by as well. Lemma 2.3.1. There exists a Cartan involution that commutes with , i.e., ı D ı : For a proof, see for example [BSD05, Lemma 3.1, p.19]. The Lie algebra g has an eigenspace decomposition gDh˚q for . Here the first component is the C1 and the second the 1 eigenspace. Note that h is the Lie algebra of H . Since Œq; q h, the subspace q is a Lie subalgebra of g if and only if q is abelian. Since and commute, the Lie algebra g decomposes also as g D .k \ h/ ˚ .k \ q/ ˚ .p \ h/ ˚ .p \ q/: We fix a maximal abelian subspace aq of p\q. The subgroup H is called essentially connected if H D ZK\H .aq /H 0 ;

20

2.4.

The class of ı -stable parabolic subgroups

where H 0 is the identity component of H and ZK\H .aq / the centralizer of aq in K \ H . (See [Ban86, p. 24].) Every Cartan involution commuting with is conjugate to via an element of Ad.H / and every maximal abelian subspace of p \ q is conjugate to aq via an element of K \ H . This implies that the notion of H being essentially connected is independent of the choice for aq . If H is essentially connected, then X D G=H is called a reductive symmetric space of the HarishChandra class, or for short a reductive symmetric space. Examples of reductive symmetric spaces are spheres, Euclidean spaces, pseudoRiemannian hyperbolic spaces and the De Sitter and the anti De Sitter space. Also a Lie group G of the Harish-Chandra class may be viewed as a reductive symmetric space. In fact .G G/=diag.G/ ! GI

.g1 ; g2 / diag.G/ 7! g1 g2 1

is a diffeomorphism and diag.G/ is the fixed point subgroup of the involution .g1 ; g2 / 7! .g2 ; g1 / of G. Therefore .G G/=diag.G/ is a symmetric space. Since ZK .a/ meets every connected component of G (see [Kna02, 7.33]), the subgroup diag.G/ of G G is essentially connected, hence .G G/=diag.G/ is of the Harish-Chandra class. Another important class of symmetric spaces is the class of Riemannian symmetric spaces of non-compact type. These symmetric spaces are obtained by taking G to be a non-compact connected semi-simple Lie group and to be equal to a Cartan involution of G. The symmetric space is then of the form X D G=K. From now on we will always assume the following. (i) G is a reductive Lie group of the Harish-Chandra class. (ii) is an involution of G and H is an open subgroup of the fixed point subgroup G . (iii) is a Cartan involution commuting with and K D G . (iv) H is a essentially connected. (v) X is the reductive symmetric space G=H .

2.4


In this section we describe the ı -stable parabolic subgroups, i.e., the parabolic subgroups P with the property that ı .P / D P .

21

2.


We fix a maximal abelian subspace aq of p \ q and a -stable maximal abelian subspace a of p containing aq . Being -stable, a decomposes as a D .a \ h/ ˚ aq : We write A and Aq for the connected abelian subgroups of G with Lie algebras a and aq , respectively. We define ˇ ˙.g; aq / to be the set consisting of non-zero elements ˇ 2 aq such that ˇ D ˛ ˇaq for some root ˛ 2 ˙.g; a/. The set ˙.g; aq / is a root-system in aq . For ˛ 2 ˙.g; aq / we define g˛ D fY 2 g W ad.Z/Y D ˛.Z/Y for all Z 2 aq g: Let ˙ C .g; aq / be a choice of a positive system for ˙.g; aq /. Let M n0 D g˛ ˛2˙ C .g;aq /

and let N0 D exp.n0 /. Then ZG .aq / normalizes N0 and therefore P0 D ZG .aq /N0 is a subgroup of G. In fact P0 is a ı -stable parabolic subgroup that is minimal in the sense that if Q is a ı -stable parabolic subgroup contained in P0 , then Q D P0 . Every other minimal ı -stable parabolic subgroup of G is conjugate to P0 via an element of K. If P0 is a minimal ı -stable parabolic subgroup, then aP0 \ q is a maximal abelian subspace of p \ q and LP0 equals the centralizer in G of aP0 \ q. Furthermore, there exists a unique choice of a positive system ˙ C .g; aP0 \ qI P0 / for ˙.g; aP0 \ q/ such that nP0 equals the direct sum of root spaces g˛ with ˛ 2 ˙ C .g; aP0 \ qI P0 /. Lemma 2.4.1. Let P be a ı -stable parabolic subgroup of G. (i) LP equals the centralizer of aP \ q in G. (ii) Aq P if and only if aP \ q aq . (iii) Assume P0 is a minimal ı -stable parabolic subgroup contained in P and Aq P0 . Then M nP D g˛ : ˛2˙ C .g;aq IP0 / ˛

22

ˇ ˇ

aP \aq

¤0

2.4.


Proof. For (i), see [Ban88, Lemma 2.2]. Assume Aq P . Since Aq is -stable, it is contained in LP . This implies that the center Z.lP / of lP is contained in the centralizer Zg .aq / of aq in g. Therefore aP \ q D Z.lP / \ p \ q Zg .aq / \ p \ q: Since aq is a maximal abelian subspace of p \ q, the latter equals aq . For the converse, assume that aP \ q aq . Since aq is abelian, it centralizes aP \ q. By (i) the subalgebra aq is contained in lP . Therefore Aq D exp.aq / is contained in LP , which in turn is contained in P . This proves (ii). Since P is ı -stable, the Lie algebra nP , being the nilpotent radical of the Lie algebra of P , is ı -stable as well. This implies that g˛ is not contained in nP if ˛ 2 ˙.g; a/ vanishes on aP \q. The result now follows from Lemma 2.2.1(iii). We write P.a/ for the collection of parabolic subgroups P of G with aP D a, and P .aq / for the collection of ı -stable parabolic subgroups P of G with aP \ aq D aq . Note that P.a/ and P .aq / consist of the minimal parabolic subgroups containing A and the minimal ı -stable parabolic subgroups containing Aq , respectively. Lemma 2.4.2. Let Pm 2 P.a/ and let P0 be a minimal ı -stable parabolic subgroup containing Pm . Then P0 nP D nPm \ h m

and NPPm0 D NPm \ H:

Proof. We first note that P0 nPm \ h nPm \ .nPm / D nPm \ .nP0 ˚ nP /: m P0 As .nP0 / D nP0 and .nP / .lP0 / D lP0 , it follows that m P0 nPm \ h nP : m P0 For the converse inclusion, it suffices to show that nP h. For this we note that m P0 nP is a sum of root spaces g˛ with ˛ 2 ˙.g; a/. Assume that ˛ 2 ˙.g; a/ is such m

P0 that g˛ occurs in this sum. We will show that then g˛ nPm \ h. Since nP is m contained in lP0 , it centralizes aq . It follows that the restriction of ˛ to aq vanishes, hence ˛ D ˛ and g˛ is -stable. Therefore g˛ decomposes as a direct sum of vector spaces g˛ D gC ˛ ˚ g˛ ;

23

2.


where g˙ ˛ is the ˙1-eigenspace of . It suffices to show that g˛ D f0g. For this, let Y 2 g˛ . Then Y Y is contained in p \ q and centralizes aq . By maximality of aq it follows that Y belongs to this space. Since the intersection of aq and nP ˚ nP is zero, Y must be zero. This establishes the first identity, and we turn to the second. Since exp W nPm ! NPm is a diffeomorphism, it follows that NPm \ H NPm \ G D exp.nPm \ h/. This implies that NPm \ H D exp.nPm \ h/. The second identity now follows from the first. If P is a ı -stable parabolic subgroup containing Aq , then LP D .MP \ K/Aq .LP \ H / Furthermore, if Y; Y0 2 aq and exp.Y / 2 .MP \ K/ exp.Y0 /.LP \ H /, then Y D Ad.k/Y0 for some k in the normalizer NMP \K\H .aq / of aq in MP \ K \ H . This decomposition if LP is often called the polar decomposition of LP . Note that in particular G is a ı -stable parabolic subgroup and the polar decomposition of G is G D KAq H: To conclude this section, we describe a generalization of the Iwasawa decomposition. Let P0 be a minimal ı -stable parabolic subgroup. Then the double coset space P0 n G=H is finite. Furthermore, the sets P0 wH with w 2 NK .aq / are open subsets of G. In fact, these are all the open P H -orbits in G. The union of these open orbits is dense in G. Finally the map NP0 Aq .MP0 \ K/ MP0 \K\H H ! P0 H is a diffeomorphism.

2.5

Notation

C We recall that ˙.g; aq / is a possibly unreduced root system. We write ˙˙ .g; aq I P / C for the set of positive roots ˛ 2 ˙ .g; aq I P / for which the ˙1-eigenspace of ı in the root space g˛ is non-trivial. We denote by W the Weyl group of the root system ˙.g; aq /. Note that there is a natural isomorphism

W ' NK .aq /=ZK .aq /: If S is a subgroup of G, then we define WS to be the subgroup consisting of elements that can be realized as Ad.s/jaq for s 2 NK\S .aq /. We write W for a set

24

2.5.

Notation

of representatives in NK .aq / of W =WK\H . The Weyl group of ˙.lP ; aq / equals WMP \K . We write WMP for a set of representatives in NMP \K .aq / for WMP \K =WMP \K\H : We write B for an Ad.G/-invariant, -invariant, symmetric, non-degenerate bilinear form on g such that (i)

B.; / is a positive definite inner product on g,

(ii) k and p are orthogonal with respect to B.; /, (iii) the semisimple part of g and the center of g are orthogonal with respect to B.; / (iv) the restriction of B to the semisimple part of g equals the Killing form. For a root ˛ 2 ˙.g; aq /, we define H˛ 2 aq to be the element given by ˛.Y / D B.H˛ ; Y /

.Y 2 aq /:

(2.5.1)

For g 2 G and P a parabolic subgroup of G we denote the parabolic subgroup by P g . If P is a ı -stable parabolic subgroup containing Aq , then by aC q .P / we denote the set consisting of elements H 2 aq such that ˛.H / > 0 for all ˛ 2 ˙.g; aq I P /. Furthermore, for R 2 R, we define g

1P g

aq .P; R/ D f 2 aq;C W Re .H˛ / < R for all ˛ 2 ˙ C .g; aq I P /g: Finally if V is a Fréchet space and .; V / a continuous representation of G on V , then we denote the space of smooth vectors for by V 1 .

25

Chapter 3

Radon transformation on a reductive symmetric space In Sections 3.1 – 3.4 we first introduce for each pair of ı -stable parabolic subgroups P and Q, with P Q, a class of closed submanifolds (related to P ) of a certain homogeneous space Q (related to Q) and describe the Radon transforms that are obtained by integrating over these submanifolds. Then in Section 3.6 some basic estimates are derived. It follows from these estimates that the Radon transforms, which are initially defined on the space of compactly supported smooth functions, extend to larger spaces of functions and distributions defined in Section 3.5. In Section 3.7 we describe certain relations between Radon transforms related to different parabolic subgroups. Finally, in Section 3.9, we derive some properties of the support of the Radon transform of a function or distribution, in terms of the support of that function or distribution. For this we need a few results from convex analysis that we describe in Section 3.8.

27

3.

3.1

Radon transformation on a reductive symmetric space

Horospheres

For a ı -stable parabolic subgroup P , we define P D G=.LP \ H /NP : Since .LP \ H /NP is closed in P , hence in G, it follows that P is a smooth homogeneous space for G. Note that G is a ı -stable parabolic subgroup of G and G D X: The cosets e H and e .LP \ H /NP are denoted by x0 and P , respectively. Let P0 be a minimal ı -stable parabolic subgroup of G. A horosphere in X is an orbit of a subgroup of G conjugate to NP0 in X of maximal dimension, i.e., a submanifold of X (see Proposition 3.2.2) of the form g1 NP0 g2 x0 with dimension equal to the dimension of NP0 . The set of all horospheres in X is denoted by Hor.X/. According to [Ros79, Theorem 13] and [Mat79] the group G equals the union of subsets P0 kH , where k runs over a finite subset of K. This implies that Hor.X / is the union of finitely many G-orbits. The dimension of these orbits need not be constant. It follows from the same theorem in [Ros79] that the set of orbits of maximal dimension is in bijection with W via the map w 7! G .NP0 w x0 / D G .w 1 NP0w x0 /. Here the superscript w denotes conjugation with w 1 . The stabilizer in G of P0 equals .LP0 \ H /NP0 . (See Proposition A.2.) Therefore the G-orbits in Hor.X / of maximal dimension, i.e., the sets G P0w for w 2 W, are parametrized by the homogeneous spaces P0w . The double fibration G=.LP0 \ H /

s ss ss s s ss y s s ˘X

X

MMM ˘ MMMP0 MMM MMM &

(3.1.1)

P0

describes the incidence relation between points in X and horospheres in X which are parametrized by P0 : a point x is contained in a horosphere parametrized by 2 P0 if and only if x 2 ˘X ˘P1 ./ : 0

If H D K, then X is a Riemannian symmetric space. In that case, P0 is a minimal parabolic subgroup. If G D KAN is an Iwasawa decomposition for G,

28

3.2.

Double fibration

then by a suitable conjugation we may arrange that P0 D MAN , where M is the centralizer of a in K. Then (3.1.1) reduces to ˘Xjjjjj

j u jjj

G=K D X

G=M SS SSS˘ SSSS S) D MN

The Radon transform corresponding to this double fibration is the horospherical transform described in the introduction.

3.2

Double fibration

In this and the following sections we will assume that P and Q are ı -stable parabolic subgroups such that A P Q. Then AQ AP A. Since LP is a closed subgroup of LQ , whereas Q ' LQ NQ , with LQ normalizing NQ , it follows that .LP \H /NQ is a closed subgroup of G. We recall that P D G=.LP \ H /NP and Q D G=.LQ \ H /NQ and consider the following generalization of (3.1.1). G=.LP \ H /NQ

p ppp p p ppp p x pp ˘Q

Q

NNN NN˘NP NNN NNN &

(3.2.1)

P

In view of the following proposition, this is a double fibration of the form (1.2.1). Proposition 3.2.1. Let P; Q be ı -stable parabolic subgroups with P Q. Then .LQ \ H /NQ \ .LP \ H /NP D .LP \ H /NQ : Proof. As LP LQ and NQ NP , it follows that the set on the right-hand side of the equality is contained in the intersection on the left-hand side. We turn to the converse inclusion. Assume that g belongs to the intersection on the left-hand side. Q Q Using that NP D NP NQ , that NP LQ and that LP LQ we see that g D ln for certain elements n 2 NQ and Q

l 2 .LQ \ H / \ .LP \ H /NP : Since P is ı -stable, .NP / D NP , so that NP has trivial intersection with H . Therefore, l 2 LP \ H .

29

3.


The double fibration (3.2.1) describes the incidence relations between points in Q and subsets of Q of the form gNP Q with g 2 G. Note that for Q D G and P D P0 a minimal ı -stable parabolic subgroup, (3.2.1) reduces to (3.1.1). Q For 2 P we define EP ./ to be the subset of Q given by Q EP ./ D ˘Q ˘P1 .fg/ : Moreover, we agree to write EP ./ for EPG ./. Note that for g 2 G, Q

Q

Q

EP .g P / D g EP .P / D gNP Q D gNP Q : Proposition 3.2.2. Let g 2 G. Then Q

(i) EP .g P / is a closed submanifold of Q . Q

Q

(ii) The map n 7! gn Q is a diffeomorphism from NP onto EP .g P /. Proof. Without loss of generality we may assume that g D e. The multiplication map defines a surjective submersion K Q ! G. As K \ Q D K \ MQ , this submersion induces a diffeomorphism K K\MQ Q ! G which is equivariant for the right action of the closed subgroup .LQ \ H /NQ of Q. Now Q ' LQ NQ and LQ normalizes NQ . Therefore, the above diffeomorphism factors through a diffeomorphism K K\MQ LQ =LQ \ H ! G=.LQ \ H /NQ . It follows that the map k; l .LQ \ H / 7! kl Q (3.2.2) K .MQ \K/ LQ =.LQ \ H / ! Q I is a diffeomorphism. Hence, l 7! l P is a diffeomorphism from LQ =.LQ \ H / onto the closed submanifold LQ Q of Q . Let now P0 be a minimal ı -stable parabolic subgroup contained in P . From [Ban86, Lemma 3.4] applied to LQ and the minimal ı -stable parabolic subgroup P0 \ LQ of LQ , it follows that the multiplication map Q

NP0 LP0 .LP0 \H / .LQ \ H / ! LQ

(3.2.3)

is a diffeomorphism onto the open subset .P0 \ LQ /.LQ \ H / of LQ . Therefore Q

NP0 LP0 =.LP0 \ H / ! LQ Q

30

3.2.

Double fibration

is a diffeomorphism onto the open subset .P0 \LQ /Q of the submanifold LQ Q of Q . As the multiplication map Q

Q

NPP0 NP ! NP0 Q

Q

is a diffeomorphism, the map NP 3 n 7! n Q is a diffeomorphism onto NP Q and this set is a submanifold of LQ Q and therefore a submanifold of Q . Q Furthermore, it follows that NP Q is closed in .P0 \ LQ / Q equipped with the Q subspace topology. It remains to be shown that NP Q is closed in Q . Q Let .nj /j 2N , .aj /j 2N and .mj /j 2N be sequences in NP0 , Aq and MP0 , respectively, such that .nj aj mj Q /j 2N converges to a point in the boundary of .P0 \ LQ / Q . Then, in view of [Ban86, Lemma 3.4], the set faj W j 2 Ng is not Q relatively compact in Aq . It follows that if .nj Q /j 2N is a sequence in NP Q converging to in Q , then cannot be an element of the boundary of .P0 \ LQ / Q , Q hence 2 .P0 \ LQ / Q . Since NP Q is closed in .P0 \ LQ / Q , we conclude Q Q that 2 NP Q . This proves that NP Q is closed in Q . Corollary 3.2.3. The set .LQ \ H /NQ .LP \ H /NP equals .LQ \ H /NP . The latter is a closed submanifold of G. Proof. Since LP LQ and NQ NP , .LQ \ H /NQ .LP \ H /NP D .LQ \ H /NQ NP .LP \ H / D .LQ \ H /NP .LP \ H / D .LQ \ H /.LP \ H /NP D .LQ \ H /NP : Q

The set NP .LQ \ H / equals the pre-image of EP .P / under the projection G ! Q Q . Since EP .P / is a closed submanifold according to Proposition 3.2.2 and G is a fiber bundle over Q , it follows that NP .LQ \ H / is a closed submanifold of G. The same holds for its image under the map g 7! g 1 , which is .LQ \ H /NP . As a consequence of Corollary 3.2.3, the double fibration (3.2.1) satisfies condition (B) of Section 1.3. According to Corollary A.4 of the appendix, the groups .LP \H /NP and .LQ \ H /NQ are transversal. We recall from Section 1 that this means in particular that the map Q 7! EP ./ is injective.

31

3.

3.3


Invariant measures

Properties (A) and (C) of Section 1.3 are satisfied for the double fibration 3.2.1 if certain invariant measures exist. In this section we will prove the existence of such measures. Lemma 3.3.1. Let r be a reductive Lie algebra with Cartan involution . Every -stable Lie subalgebra of r is reductive. Proof. Let s be a -stable Lie subalgebra of r. The restriction of the positive definite inner product h; i D B.; / to s s is a positive definite inner product on s. Assume that i is an ideal of s, then s D i ˚ i? as vector spaces. Here the orthocomplement is taken with respect to h; i. Let X 2 i? and Y 2 s. Then for every Z 2 i we have hŒX; Y ; Zi D D

B.ŒX; Y ; Z/ D

B.X; ŒY; Z/

B.X; ŒY; Z/ D hX; ŒY; Zi D 0:

For the last equality it was used that ŒY; Z 2 i. This shows that ŒX; Y 2 i? , and we conclude that i? is an ideal of s complementary to i. It follows that s is reductive. We retain the notation of the previous sections. In particular, P and Q are ı -stable parabolic subgroups containing A and such that P Q. Proposition 3.3.2. The group .LP \ H /NQ is unimodular. Proof. Let be the modular function of .LP \ H /NQ . Note that .LP \ H /NQ D .LP \ H \ K/.LP \ H /0 NQ ; where .LP \H /0 denotes the identity component of .LP \H /. Since .LP \H \K/ is compact, .kg/ D .g/ for all k 2 LP \ H \ K and g 2 .LP \ H /0 NQ . If n 2 NQ , then ad.log.n// is upper triangular with respect to the usual basis of .lP \ h/ ˚ nQ , hence ˇ ˇ .n/ D j det.Ad.n/ˇ.lP \h/˚nQ /j D exp tr ad.log.n//ˇ.lP \h/˚nQ D 1: The group .LP \ H /0 normalizes both lP \ h and nQ . Let l 2 .LP \ H /0 , then it follows that ˇ ˇ ˇ ˇ ˇ ˇ .l/ D ˇ det Ad.l/ˇlP \h ˇ ˇ det Ad.l/ˇnQ ˇ:

32

3.3.

Invariant measures

Since .LP \ H /0 is reductive, the modular function of this group is trivial, hence the first factor in the product is equal to 1. Similarly, the modular function Q of Q evaluated in l equals ˇ ˇ ˇ ˇ ˇ ˇ Q .l/ D ˇ det Ad.l/ˇlQ ˇ ˇ det Ad.l/ˇnQ ˇ: Again the first factor in the product equals 1, this time by reductivity of L0Q . We conclude that .l/ D Q .l/: If m 2 MQ , a 2 AQ and n 2 NQ , then it is known that Q .man/ D a2Q , where Q is the element of aQ given by ˇ 1 Q .Y / D tr.ad.Y /ˇnQ /; 2

.Y 2 aQ /:

(3.3.1)

Let m 2 MQ and a 2 AQ be such that l D ma. Since both MQ and AQ are stable, it follows that a 2 AQ \ H . Now Q is ı -stable, hence ı .Q / D Q and we see that Q D 0 on aQ \ h, so that a2Q D 1. Therefore, .l/ D Q .l/ D 1 and we conclude that .LP \ H /NQ is unimodular. Corollary 3.3.3. (i) There exists a non-zero G-invariant Radon measure on G=.LP \ H /NQ . In particular there exists a non-zero G-invariant Radon measure on P . (ii) There exists a non-zero .LQ \ H /NQ -invariant Radon measure on .LQ \ H /NQ =.LP \ H /NQ : (iii) There exists a non-zero .LP \ H /NP -invariant Radon measure on .LP \ H /NP =.LP \ H /NQ : Proof. By Proposition 3.3.2, also applied with P D Q, all occurring groups are unimodular. As a consequence of Corollary 3.3.3, the double fibration (3.2.1) satisfies properties (A) and (C) of Section 1.3.

33

3.

Radon transformation on a reductive symmetric space Q

The groups .LP \ H /NQ , NP and .LP \ H /NP are unimodular (see Proposition 3.3.2) and the multiplication map Q

.LP \ H /NQ NP ! .LP \ H /NP is a diffeomorphism. Therefore, Z .ln .LP \ H /NQ / d.LP \H /NQ .ln/ D

.LP \H /NP =.LP \H /NQ

Z Q

NP

.n .LP \ H /NQ / d n

for every 2 L1 ..LP \ H /NP =.LP \ H /NQ / if the measures are suitably normalized. Similarly, Z .ln .LP \ H /NQ / d.LP \H /NQ .ln/ D .LQ \H /NQ =.LP \H /NQ Z .l .LP \ H /NQ / d l .LQ \H /=.LP \H /

for every 2 L1 ..LQ \ H /NQ =.LP \ H /NQ / if the measures are suitably normalized. If N is a simply connected subgroup of G with nilpotent Lie algebra n, then the Haar measure on N is related to the Lebesgue measure on n by Z Z .n/ d n D c .exp.Y // d Y 2 L1 .N / N

n

for some constant c > 0. Here d Y denotes the unit Lebesgue measure of n relative to inner product h; i. We will choose the normalization of measures on the groups Q Aq and NP always such that c D 1. If P , Q and S are parabolic subgroups with P Q S, then because of this choice for the normalization of the measures Z Z Z 2 L1 .NPS / : .n/ d n D .nn0 / d n0 d n (3.3.2) NPS

Q

NP

S NQ

Furthermore, we normalize the Haar measure on any compact group such that it is a probability measure.

34

3.4.

3.4

Radon transforms

Radon transforms

The Radon transforms RP and SP for the double fibration (3.2.1) are given by Z Q RP .g P / D .gn Q / d n . 2 D.Q /; g 2 G/ Q

NP

(3.4.1) Q

SP

Z .g Q / D

.LQ \H /=.LP \H /

.gh P / dLP \H h .

2 D.P /; g 2 G/: (3.4.2)

G We write RP for RG P and SP for SP . These Radon transforms are given by Z RP .g P / D .gn x0 / d n . 2 D.X /; g 2 G/ NP Z SP .g x0 / D .gh P / dLP \H h . 2 D.P /; g 2 G/: H=.LP \H /

If P D P0 is a minimal ı -stable parabolic subgroup, then RP0 is called the horospherical transform associated to P0 Remark 3.4.1. In [Krö09, Section 2] it is claimed that the set Hor.X / of horospheres in X can be given the structure of a connected real analytic manifold. However, the real analytic atlas for Hor.X / given there is not an atlas in the proper sense. In fact, each of the finitely many G-orbits in Hor.X / serves as the domain for a chart, but it is easily seen that not all of these orbits need to have maximal dimension. In Section 5.4 of Appendix B we discuss an example of this phenomena. In [Krö09, Remark 3.3] it is furthermore claimed that the horospherical transforms RP w with w 2 W induce a transform from the space of real analytic vectors for the left regular representation of G on L1 .X / to the space of real analytic functions on Hor.X /. Even if Hor.X / could be equipped with a canonical structure of a real analytic manifold, then it is not clear to us why this should be true.

3.5

Spaces of functions and distributions

From (3.2.2) and the fact that LP =.LP \ H / ' MP =.MP \ H / AP =.AP \ H /;

35

3.


it follows that there is a unique map AP W P ! aP \ aq such that AP .kman P / D q .log a/

.k 2 K; m 2 MP ; a 2 AP ; n 2 NP /;

where q denotes the orthogonal projection g ! q. Note that the map AP is real analytic. We define the function JP W P ! R by JP D e

2P ıAP

where P is defined as in (3.3.1) with P in place of Q. Note that JG equals the constant function 1 on X. Let L1 .P ; JP / D f W P ! C W JP 2 L1 .P /g: Endowed with the norm Z 7!

P

j./jJP ./ d ;

(3.5.1)

L1 .P ; JP / is a Banach space. Lemma 3.5.1. For every compact subset C of G, there exists a constant c > 0 such that for every g 2 C and 2 P c

1

JP .g / JP ./ cJP .g /:

(3.5.2)

Proof. Let Pm 2 P.a/ be a minimal parabolic subgroup contained in P . Let k 2 K, a 2 A and n 2 NPm . Note that there exist unique aMP 2 A \ MP , aAP 2 AP such that a D aMP aAP . Furthermore, there exist unique nP 2 NP and nP 2 NPPm D NPm \ MP such that n D nP nP . Hence AP .kan P / D AP .kaMP aAP nP nP P / D AP k.aMP nP /aAP nP P D q .aAP /: We conclude that AP .g P / D aP \q ı AKANPm .g/; where aP \q is the orthogonal projection (with respect to B.; /) onto aP \ q and AKANPm is the function G ! a given by g 2 K exp AKANPm .g/ NPm .g 2 G/: (3.5.3)

36

3.5.


Let k1 ; k10 ; k2 2 K, a1 ; a2 2 A and n2 2 NPm . Then, by [Kos73, Theorem 4.1], AKANPm .k1 a1 k10 k2 a2 n2 / 2AKANPm .a1 K/ C log a2 D ch.W .a/ log a1 / C log a2 ; where W .a/ denotes the Weyl-group NK .a/=ZK .a/. Therefore, if g1 ; g2 2 G, AKANPm .g1 g2 / D ch W .a/ AKAK .g1 / C AKANPm .g2 /: Here AKAK is the map from G to the closed positive Weyl chamber aC .Pm / given by g 2 K exp AKAK .g/ K .g 2 G/: For g 2 G and 2 P we now find AP .g / D Y .g/ C AP ./; with Y.g/ 2 aP \aq ch W .a/ AKAK .g/ . The estimate (3.5.2) now follows from the observation that Y .g/ is a locally bounded function of g. It follows from Lemma 3.5.1 that the space L1 .P ; JP / is invariant under left translation by elements of g. Accordingly, we define the representation of G in this space by Œ.g/./ WD .g

1

/;

. 2 L1 .P ; JP /; g 2 G/:

(3.5.4)

Proposition 3.5.2. The representation is a continuous Banach-representation. Proof. Put V WD L1 .P ; JP /, and write k kV for the norm given in (3.5.1). For a compact subset C G, let c > 0 be the constant of Lemma 3.5.1. Then for all 2 V and g 2 C we have Z Z k.g/kV D j.g 1 /JP ./j d D j./J.g /j d ckkV : P

P

This shows that each map .g/ W V ! V is bounded, and that the family f.g/jg 2 C g is equicontinuous. We will now show that limg!e .g/ D for each 2 V . By the above mentioned equicontinuity, it suffices to do this for a dense subspace of V . We take the dense subspace V0 WD Cc .P /. Then for each 2 V0 we have by the principle

37

3.


of uniform continuity that .g/ ! uniformly and with supports in a compact set as g ! e. This in turn implies that .g/ ! in V . It now follows by application of the principle of uniform boundedness that is a continuous representation. By what we proved above, we can also give the following direct argument. Let g0 2 G and 0 2 V . We will prove the continuity of the map W G V ! V at .g0 ; 0 /. Let > 0. Then there exists a compact neighborhood C of e in G such that k.g/0 0 k < =2 for g 2 C . Let M > 0 be such that k.g/k M for all g 2 Cg0 and let ı < =2.M C 1/. Then for all g 2 Cg0 and all 2 V with k 0 k < ı we have: k.g/

.g0 /0 k k.g/.

0 /k C k.g/0

0 k C k0

k

< M ı C =2 C ı < :

Lemma 3.5.3. Let P be a ı -stable parabolic subgroup containing A. Then with suitably normalized measures, we have Z Z Z ./JP ./ d D .kl P / dLP \H l d k P

K

LP =.LP \H /

for every 2 L1 .P ; JP /. Proof. We leave it to the reader to check that all measures that appear in this proof may be normalized such that the equalities hold. If 2 L1 .G/, then Z Z Z .g/ dg D .kp/P .p/ dp d k G ZK ZP Z D .kls/P .ls/ ds d.LP \H / l d k: K

LP =.LP \H /

.LP \H /NP

Here dp denotes left invariant measure, and P the modular function of P . If m 2 MP \ K, a 2 Aq , h 2 LP \ H and n 2 NP , then P .mahn/ D a2P . Therefore P is right .LP \ H /NP -invariant. Hence Z Z .kl P /P .l/ d.LP \H / l d k 2 L1 .P / K

38

LP =.LP \H /

3.5.


defines a G-invariant Radon measure on P . Note that G-invariant Radon measures on P are unique up to multiplication by a constant. Therefore, under the assumption that the measure are suitably normalized, Z Z Z ./JP ./ d D P .l/.kl P /JP .kl P / dLP \H l d k: K

P

LP =.LP \H /

for every 2 L1 .P ; JP /. The pull-back of JP under the map K LP =.LP \ H / ! P I

k; l .LP \ H / 7! kl P

equals K LP =.LP \ H / ! RI

k; l .LP \ H / 7!

1 : P .l/

Hence, Z Z

Z P

./JP ./ d D

K

LP =.LP \H /

.kl P / dLP \H l d k:

We define E 1 .P ; JP / to be the subspace of E .P / consisting of functions that represent a smooth vector for the representation of G on L1 .P ; JP / defined in (3.5.4). We endow this space with the Fréchet topology induced by the natural bijection E 1 .P ; JP / ! L1 .P ; JP /1 : The space E 1 .X; JG / is denoted by E 1 .X /. (Recall that JG D 1X .) Proposition 3.5.4. If 2 E 1 .P ; JP /, then vanishes at infinity. Proof. Let K be the diagonal in K K and let S D .K K/=K LP =.LP \ H / . The map ˚ W S ! P I

.k1 ; k2 / K ; l .LP \ H / 7! k1 k2 1 l P

is a surjective, smooth submersion. Since we take the measure of K to be normalized, pullback under ˚ defines an isometric embedding ˚ W L1 .P ; JP / ! L1 .S /:

39

3.


by Lemma 3.5.3. Let n 2 N and let v 2 U.g/ be of degree smaller than or equal to n. If k 2 K, then Ad.k/v can be written as a finite sum Ad.k/v D

X

cj .k/uj ;

j

where the cj are continuous functions from K to C and the uj form a basis for the subspace of U.g/ consisting of the elements of order at most n. Since K is compact, the functions cj are bounded. Therefore, if 2 E 1 .P ; JP /, u; v1 2 U.k/ and v2 2 U.lP /, then the L1 .S /-norm of .u ˝ v1 ˝ v2 /˚ can be estimated by a constant times XZ juuj ./jJP ./ d : j

P

This proves that pullback under ˚ maps E 1 .P ; JP / to the space E 1 .S / of smooth representatives for elements in L1 .S /1 . Note that S is a symmetric space of the Harish-Chandra class. According to [KS07, Theorem 3.1] every function 2 E 1 .S/ vanishes at infinity. Since pullback under ˚ maps E 1 .P ; JP / to E 1 .S / and ˚ is a continuous surjection, it follows that every function 2 E 1 .P ; JP / vanishes at infinity. Let L1 .P ; JP / be the space of equivalence classes (modulo differences on sets of measure 0) of measurable functions such that =JP is essentially bounded, i.e., 1 1 L .P ; JP / D W 2 L .P / : JP We endow this space with the norm

7! :

J 1 P L .P / Lemma 3.5.5. The pairing Z . ; / 7!

d P

induces an isometric isomorphism L1 .P ; JP / ! L1 .P ; JP /0 .

40

(3.5.5)

3.5.


Proof. Since .P ; d / is a -finite measure space, the pairing (3.5.5) induces an isometric isomorphism from L1 .P / onto L1 .P /0 . (See for example [Fri82, Theorem 4.14.6].) Now use that 7! JP defines an isometric isomorphism from L1 .P ; JP / onto L1 .P /, whereas 7! =JP induces an isometric isomorphism from L1 .P ; JP / onto L1 .P /. Let Cb .P ; JP / be the space of continuous functions such that =JP is bounded and let Eb .P ; JP / be the space of functions 2 E .P / such that u 2 Cb .XP ; JP / u 2 U.g/ : The space Eb .P ; JP / is a Fréchet space with the topology induced from the obvious set of seminorms. Note that Eb .X; JG / D Eb .X /. Although the left-regular representation of G on Cb .P ; JP / is not continuous unless P is compact (which is a rather uninteresting case), we do have the following result. Proposition 3.5.6. The left regular representation of G on the space Eb .P ; JP / is a smooth Fréchet representation. Proof. We denote the left regular representation of G on Eb .P ; JP / by . Let u 2 U.g/ be of order n. Let fuk W 1 k mg be a basis for the subspace of U.g/ consisting of elements of order atPmost n. Then there exist continuous functions ck W G ! C such that Ad.g/u D m kD1 ck .g/uk . Let C be a compact subset of G and 2 Eb .P ; JP /. By Lemma 3.5.1 there exists a constant c > 0 such that for every g 2 C m X juk .g /j ju..g//./j sup c sup J ./ JP .g / P 2P 2P kD1

This implies in particular that .g/ is continuous for every g 2 G and every seminorm of .g/ is locally uniformly bounded in g. Furthermore, if g D exp.Y / for some element Y 2 g, then again by Lemma 3.5.1 there exists a constant c such that ˇ Z 1ˇ ˇd ˇ j.g/ j 1 ˇ ˇ sup sup ˇ dt exp.t Y / ˇ dt J J P P 0 P P Z 1 j exp.t Y / .Y /j jY j sup dt c sup : JP 0 P P JP Therefore limg!e .g/ D . We conclude that the assumptions in [War72, Proposition 4.1.1.1] are satisfied and hence that is a continuous representation.

41

3.


In order to prove that the representation is smooth, it suffices to show that for every 2 Eb .P ; JP / and Y 2 g we have ˇ ˇ ˇ 1 ˇˇ exp.t Y / ˇ (3.5.6) lim sup Y ˇ D 0: ˇ ˇ t !0 P JP ˇ t Since Z

t

.t

s/

sD0

d2 exp.sY / ds D exp.t Y / 2 ds

t Y ;

it follows that ˇ ˇ ˇ exp.t Y / ˇ ˇ ˇ Y ˇ ˇ ˇ ˇ t Z t ˇ 1 t s ˇˇ d 2 sup exp.sY / ˇ ds t ds 2 P JP sD0 ˇ d2 ˇ Z t ˇ 2 exp.sY / ˇ t s ds sup ds: t P JP sD0

1 sup P JP

By Lemma 3.5.1 there exists a constant c such that the latter is smaller than or equal to Z jY 2 j t t s jY 2 j t c sup ds D c sup : JP t JP 2 sD0 P P This implies (3.5.6).

3.6

Extensions of the Radon transforms

Recall that for a parabolic subgroup P of G and an element g 2 G the conjugate parabolic subgroup g 1 P g is denoted by P g . Furthermore, recall that WMP denotes a set of representatives for the quotient of the Weyl groups WMP \K and WMP \K\H . (See Section 2.2.) Lemma 3.6.1. Let P0 2 P .aq / and let P be a parabolic subgroup containing P0 . Then with a suitable normalization of the measure d on P , Z X Z Z Z ./JP ./ d D .kan P / d n da d k P

for all 2 L1 .P ; JP /.

42

w2WMP

K Aq

NPPw 0

3.6.


Proof. From [Óla87, Theorem 1.2] applied to LP =.LP \H / and the minimal ıstable parabolic subgroup P0 \ LP D MP0 AP0 NPP0 of LP , it follows that Z .kl P / dLP \H l Z Z X Z

LP =.LP \H /

D c

MP0 \K Aq

w2WMP

NPPw

.mwan P / d n da d m;

0

for some constant c > 0. Using Lemma 3.5.3 and the invariance of the measure on K, we obtain Z ./JP ./ d P Z Z X Z Z .kmwan P / d n da d m d k D c w2WMP

D c

X w2WMP

K

MP0 \K Aq

Z Z K Aq

NPPw 0

Z NPPw

.kan P / d n da d k:

0

The measure d can be normalized such that c D 1. From now on we assume that the measure on P is normalized such that the identity in Lemma 3.6.1 holds. As before, let P and Q be ı -stable parabolic subgroups of G with A P Q. Lemma 3.6.2. If 2 L1 .Q ; JQ /, then Z P

Z Q

NP

Z j.gn Q /j d n JP .g P / d.LP \H /NP g

Q

j./jJQ ./ d :

Proof. Since Aq LP LQ , we can choose the sets of representatives WMP and WMQ such that WMP WMQ . Let P0 2 P .aq / be a minimal ı -stable

43

3.


parabolic subgroup with P0 P . By Lemma 3.6.1, Z Z j.gn Q /j d n JP .g P / d.LP \H /NP g D Q

P

NP

X

Z Z

Z

NPPw

K Aq

w2WMP

X

Z 0

Z Z

j.kannP Q /j d nP d n da d k D

Z Q

K Aq

w2WMP

Q

NP

NP w

j.kan Q /j d n da d k:

0

Here we used (3.3.2). Now we use that WMP WMQ and apply Lemma 3.6.1 once more, to obtain X Z Z Z j.kan Q /j d n da d k w2WMP

K Aq

X w2WMQ

Q

NP w 0

Z Z K Aq

Z

Z Q

NP w

j.kan Q /j d n da d k D

Q

j./jJQ ./ d :

0

Q

Recall the definition of the map RP W D.Q / ! E .P / from (3.4.1). Q

Proposition 3.6.3. The transform RP defines a continuous map from D.Q / to E 1 .P ; JP /. Q

Proof. By Lemma 3.6.2, the Radon transform RP defines a continuous map from D.Q / to L1 .P ; JP /. Due to continuity and equivariance, it is a continuous transform between the spaces of smooth vectors for the left-regular representation of G, i.e., it is a continuous map from D.Q / to L1 .P ; JP /1 . The proposition now Q follows from the fact that RP maps elements in D.X/ to smooth functions. The image of the injection D.Q / ,! L1 .Q ; JQ / is dense. Hence, by Lemma 3.6.2 and Proposition 3.6.3 there exists a unique continuous transform Q TP W L1 .Q ; JQ / ! L1 .P ; JP /

44

3.6.


such that Q

RP

D. Q /

/ E 1 . ; J / P _ P

_

/ L1 . ; J / P P

Q

TP

L1 .Q ; JQ / Q

is a commuting diagram. Note that TP is equivariant and if 2 L1 .Q ; JQ /, then Q TP .g

Z P / D

Q

NP

.gn Q / d n

Q

for almost every g P 2 P . Since TP is an equivariant and continuous transform, it maps L1 .Q ; JQ /1 continuously to L1 .P ; JP /1 . Q

Proposition 3.6.4. The map RP W D.Q / ! E .P / has a unique continuous linear extension to a map Q

RP W E 1 .Q ; JQ / ! E 1 .P ; JP / Furthermore, if 2 E 1 .Q ; JQ /, then for every g 2 G Z Q .gn Q / d n RP .g P / D Q

(3.6.1)

NP

with absolutely convergent integral. Q

Proof. There exists a unique transform RP such that the diagram Q

L1 .Q ; JQ /1

TP

/ L1 . ; J /1 P P

'

E 1 .Q ; JQ /

' Q RP

/ E 1 . ; J / P P

is a commuting diagram. This transform clearly is continuous and extends the transQ form RP W D.Q / ! E 1 .P ; JP /. It remains to be shown that the integrals in Q (3.6.1) are absolutely convergent and that RP is given by these integrals.

45

3.


Let 2 D.G/ and ' 2 E 1 .Q ; JQ /. Since TP ' 2 L1 .P ; JP /1 and since JP is smooth and non-vanishing, the integral Z .g P / WD

G

Z

Q . /TP '. 1 g

P / d D

Q

G

.g /TP '.

1

P / d

is absolutely convergent for every g 2 G. Furthermore, by Fubini, Z .g P / D

Q

NP

. '/.gn Q / d n;

where ' denotes the convolution of ' with , i.e., Z '.g Q / D

.g /'. G

1

Q / d :

By application of Lebesgue’s dominated convergence theorem one sees that is a smooth function on P . The function ' is an element of E 1 .Q ; JQ /, hence Q is a smooth representative for the smooth vector TP . '/ 2 L1 .P ; JP /1 . Q This implies that D RP . '/. We conclude that the second statement in the proposition holds for functions D ' with 2 D.G/ and ' 2 E 1 .P ; JP /. By Proposition 3.5.2 the left regular representation of G on L1 .Q ; JQ / is a Banach representation. By [DM78, Théorème 3.3] the space of smooth vectors for this representation equals the Gårding subspace. This implies that every element of E 1 .Q ; JQ / can be written as a finite sum of convolutions , with 2 D.G/ and 2 E 1 .Q ; JQ /. This proves the proposition. Lemma 3.6.5. There exists a normalization of the invariant measure on .LQ \ H /=.LP \ H / such that for every 2 L1 .Q ; JQ / and 2 Cb .P ; JP / the integral Z Q

Z .LQ \H /=.LP \H /

.gl P /.g Q / d.LP \H / l d.LQ \H /NQ g

is absolutely convergent and equal to Z P

46

Z Q

NP

.g P /.gn Q / d n d.LP \H /NP g:

3.6.


Proof. The distributions on G=.LP \ H /NQ , for 2 D.G=.LP \ H /NQ / given by Z Z gl .LP \ H /NQ d.LP \H / l d.LQ \H /NQ g Q

.LQ \H /=.LP \H /

and

Z

Z

P

Q

NP

gn .LP \ H /NQ d n d.LP \H /NP g;

respectively, both define a G-invariant Radon measure on G=.LP \ H /NQ . These measures can therefore only differ by a multiplicative positive constant (see for example [Kna02, Theorem 8.36]), and for a suitable normalization for the invariant measure on .LQ \ H /=.LP \ H / they are equal. Since 2 Cb .P ; JP / the function j JP j is bounded on P . Hence, by Lemma 3.6.2 Z Z .g P /.gn Q / d n d.LP \H /NP g D P

Q

NP

Z

Z

P

Q NP

.g P / .gn Q /JP .g P / d n d.LP \H /NP g JP .g P /

is absolutely convergent. By the first part of the proof combined with Fubini’s Theorem, Z Z .gl P /.g Q / d.LP \H / l d.LQ \H /NQ g Q

.LQ \H /=.LP \H /

is absolutely convergent as well and we see that the claimed equality holds. From now on we assume that the LQ \ H -invariant measure on the homogeneous space .LQ \ H /=.LP \ H / is normalized such that the equality in Lemma 3.6.5 holds. 2 Eb .P ; JP /, then for every g 2 G the integral Z .g Q / WD .gl P / d.LP \H / l

Proposition 3.6.6. If Q

SP

(3.6.2)

.LQ \H /=.LP \H /

Q

is absolutely convergent and the associated function SP the space Eb .Q ; JQ /. Furthermore, the transform Q

SP W Eb .P ; JP / ! Eb .Q ; JQ /

W Q ! C belongs to (3.6.3)

thus obtained, is continuous.

47

3.


Proof. Let 2 Eb .P ; JP / and 2 D.G/. It follows from Lemma 3.6.5 that the integral (3.6.2) is absolutely convergent for almost every g 2 G and the associated Q almost everywhere defined function g 7! SP .g Q / on G is locally integrable. Therefore, for every g 2 G, the integral Z Z .g / . 1 l P / d.LP \H / l d G

.LQ \H /=.LP \H /

is absolutely convergent. Furthermore, the integral depends smoothly on g and by Q Fubini’s Theorem it is equal to SP ./.l Q /. Here denotes the convolution product between and , i.e., is the function on P given by Z .g/.g 1 / dg . 2 P /: ./ D G

Q

This proves that for every 2 D.G/ and 2 Eb .P ; JP / the function SP . / is defined by absolutely convergent integrals (3.6.2) and is smooth. Let 2 Eb .P ; JP /. By [DM78, Théorème 3.3] the space of smooth vectors for the left regular representation of G on Cb .P ; JP / equals the space of Gårding vectors. Therefore equals a finite sum of convolutions , with 2 D.G/ Q and 2 Eb .P ; JP /. We conclude from the above argument that SP is a smooth function on Q defined by absolutely convergent integrals (3.6.2). From Lemma 3.6.5 and 3.6.2 it follows that for every 2 L1 .Q ; JQ / Z Z j j Q j./jJQ ./ d : jSP ././j d sup Q P JP Q Q

Therefore SP defines a continuous map from Eb .P ; JP / to the dual space of L1 .Q ; JQ /. By Lemma 3.5.5, this dual space equals L1 .Q ; JQ /. The space Q Cb .Q ; JQ / naturally embeds onto a closed subspace of L1 .Q ; JQ /. As SP maps Eb .P ; JP / into Cb .Q ; JQ / and is continuous as a map from that space Q into L1 .Q ; JQ /, it follows that SP is continuous as a map Eb .P ; JP / ! Q Cb .Q ; JQ /. Since SP is equivariant and the left regular representation of G on Q Eb .P ; JP / is smooth, it follows that SP is a continuous map Eb .P ; JP / ! Eb .Q ; JQ /. Note that the transform (3.6.3) is an extension of the earlier defined transform (3.4.2) for compactly supported smooth functions. Thus, the notation is unambiguous. Lemma 3.6.5 has the following corollary.

48

3.7.

Relations between Radon transforms

Corollary 3.6.7. If 2 E 1 .Q ; JQ / and 2 Eb .P ; JP /, then Z Z Q Q RP ./ ./ d D ./SP ./ d : P

Q

Qt

Q

Remark 3.6.8. Let SP be the adjoint transform of SP . By Corollary 3.6.7, the following diagram commutes. Q

E 1 .Q ; JQ /

RP

_

Qt

Eb0 .Q ; JQ /

SP

/ E 1 . ; J / P _ P / E 0 .P ; JP / b

Q

This allows to extend the definition (1.3.2) of the Radon transform RP of a comQ pactly supported distribution 2 E 0 .Q / to the Radon transform RP of a distriQt Q bution 2 Eb0 .Q ; JQ /. Accordingly, from now on we will write RP for SP . If Q

2 Eb0 .Q ; JQ /, then RP is the distribution in Eb0 .P ; JP / given by Q

Q

RP . / D SP

3.7

2 Eb .P ; JP / :

Relations between Radon transforms

Let P , Q and S be three ı -stable parabolic subgroups such that P Q S. We now consider the following diagram. G=.LP \ H /N S S

SSSS SSSS SSS )

kk kkkk k k k k u kk

G=.LQ \ H /NS

S

q qqq q q q x qq q

SSS SSS SSS SSS SS)

Q

G=.LP \ H /NQ

k kkk kkk k k kk ku kk

NNN NNN NNN N&

P

Here every map is a canonical projection. This diagram describes four double fibrations of the type considered in Section 1.3. Only three of these are relevant for our purposes:

49

3.


G=.LP \ H /NQ

G=.LQ \ H /NS

S

NNN NNN NNN NNN &

pp ppp p p pp p x pp

Q Q

pp ppp p p pp px pp

NNN NNN NNN NNN &

P

and

S

G=.LP \ HN/NS NNN p NNN ppp p p NNN p p p NN' w pp p

P

The Radon transforms for these double fibrations are related to each other in the following way. Proposition 3.7.1. Q

(i) The Radon transforms of functions RP and RSQ compose to a Radon transform of functions as follows Q

RP ı RSQ D RSP W E 1 .S ; JS / ! E 1 .P ; JP /: Q

S compose to a Radon (ii) The dual Radon transforms of functions SP and SQ transform of functions as follows Q

S SQ ı SP D SPS W Eb .P ; JP / ! Eb .S ; JS /: Q

(iii) The Radon transforms of distributions RP and RSQ compose to a Radon transform of distributions as follows Q

RP ı RSQ D RSP W Eb0 .S ; JS / ! Eb0 .P ; JP /: Proof. (i): The multiplication map Q

S NP NQ ! NPS

is a diffeomorphism with Jacobian equal to the constant function 1. Therefore the identity follows from the definitions and by application of Fubini’s Theorem.

50

3.8.

Some convex geometry

(ii): The continuous linear functional on D .LS \ H /=.LP \ H / mapping a function to Z Z lS lQ .LP \ H / d.LP \H / lQ d.LQ \H / lS .LS \H /=.LQ \H /

.LQ \H /=.LP \H /

defines a LS \ H invariant measure on .LS \ H /=.LP \ H /. If the measures are normalized such that the equality in Lemma 3.6.5 holds, as we assumed, then this measure and the invariant measure on .LS \ H /=.LP \ H / are equal. This proves the claim. (iii): This is a direct corollary of (ii).

3.8

Some convex geometry

In this section we prove some results in convex geometry that are needed in the next section. Let V be a finite dimensional vector space. If B is a subset of V , we denote the convex hull of B by ch.B/, i.e., ch.B/ is the smallest convex set containing B. We call a subset of V a cone if it is closed under the action of the multiplicative group R>0 . Let S be a subset of V . The function HS W V ! R [ f˙1gI

7! sup .x/ x2S

is called the support function of S. Note that the image of HS contains 1 if and only if S D ;. We define CS D f 2 V W HS ./ < 1g:

(3.8.1)

Note that CS is a convex cone. It is a well known result from convex geometry that if S is a subset of V , then x 2 ch.S / if and only if .x/ HS ./ for all 2 CS : Lemma 3.8.1. Let B be a non-empty subset of V and let ing 0. Then the following statements hold. HBC

be a cone in V contain-

H C HB .

(i) H

H

(ii) CBC

D C \ CB D f 2 CB W .x/ 0 for every x 2

B

g.

51

3.


(iii) The functions HBC and HB have equal restriction to CBC . In particular, ˇ H ˇC D 0: Proof. (i): Let 2 V . Then H

B ./

D infB . Moreover, for every x 2 B and y 2

.y/ C inf .x C y/ .y/ C sup B

B

because B ¤ ;. The required estimate at the point now follows by taking suprema over x 2 B and y 2 . (ii): By (i) we have HBC H C HB , hence CB \ C

CBC :

To prove the converse inclusion, let 2 CBC . From B B C we see that HB ./ HBC ./, hence 2 CB . If … C , then there exists an x 2 such that .x/ > 0, hence, because is a cone, HBC ./

sup

.b C rx/ D 1:

b2B;r2R>0

This contradicts the assumption 2 CBC , and we see that 2 C . We have now established the first equality of (ii). Let 2 V . If there exists an element x 2 such that .x/ > 0, then, because is a cone, H ./ sup .rx/ D 1: r2R>0

ˇ On the other hand, if ˇ C

0, then clearly H ./ D 0. We thus see that

D f 2 V W .x/ 0 for every x 2

g:

(iii): Let 2 CBC . Then by ˇ (ii) we have .x/ 0 for every x 2 . Since is a cone, it follows that H ˇC D 0. Using subsequently (i) and the fact that BC B B C , we find HBC ./ .HB C H /./ D HB ./ HBC ./: This establishes the equality of the restrictions. The final assertion now follows by taking B D f0g.

52

3.9.

Support of a transformed function or distribution

Lemma 3.8.2. Let C be a collection of cones in V containing 0 and let S be a closed convex subset of V . [ \ If CS C ; then S D .S C /: 2C

2C

Proof. Assume that the hypothesis is fulfilled. If 2 C then 0 2 hence S S C . It follows that S \ 2C .S C /. Conversely, suppose x 2 \ 2C .S C /. Let 2 CS . By assumption there exists a 2 C such that 2 C . According to Lemma 3.8.1 (ii), 2 CS C . Since x 2 S C , .x/ HS C ./: By Lemma 3.8.1 (iii), HS C ./ D HS ./, hence .x/ HS ./. As S is closed and convex, this implies that x 2 S.

3.9


Let P and Q be ı -stable parabolic subgroups of G with A P Q. Q The support of the transform RP of a function 2 D.Q / need not be compact in general. In Section 5.4 of Appendix B we give an example of this phenomena. The aim of the present section is to give a description of the support of a transformed function or distribution in terms of the support of that function or distribution. For a subset T of ˙.g; aq /, we define X

.T / D

R0 H˛ :

˛2T

Here H˛ is given by (2.5.1). We define the cone P

D

.˙ C .g; aq I P //:

Since G D KLP NP and LP D .LP \ K/Aq .LP \ H /, the map K Aq ! P I

.k; a/ 7! ka P

is surjective. In case P D G, this statement is equivalent to X D KAq x0 .

53

3.


If B is a subset of aq , then we define X.B/ D K exp.B/ x0

and P .B/ D K exp.B/ P :

Note that X.B/ D G .B/: We recall from Section 2.2 that P .aq / denotes the collection of minimal ı stable parabolic subgroups containing A. Lemma 3.9.1. Let C be the collection of P0 2 P .aq / with P0 P and let B be a closed and convex subset of aq . Then \ BC P D .B C P0 /: P0 2C

Proof. There is a one to one correspondence between the minimal ı -stable parabolic subgroups of G contained in P and the minimal ı -stable parabolic subgroups of LP : if P0 is a minimal ı -stable parabolic subgroup of G contained in P , then R D LP \ P0 is a minimal ı -stable parabolic subgroup of LP and, vice versa, if R is a minimal ı -stable parabolic subgroup of LP , then P0 D ZG .aP0 /NR NP is a minimal ı -stable parabolic subgroup of G contained in P . Furthermore, let C be as above, and let C 0 be the set of ı -stable parabolic subgroups R of LP containing A. Then the map P0 7! P0 \ LP is a bijection from C onto C 0 . If P0 2 C and R D LP \ P0 is the corresponding parabolic subgroup of LP , then P0 D P C P IR ; where P IR

D

.˙ C .lP ; aq I R//:

The cone C P IR equals the closure of the dual Weyl chamber corresponding to R, the opposite of R. Hence, [ C P IR D aq : R2C 0

Application of Lemma 3.8.2 with S D B C \ \ .B C P0 / D .B C P0 2C

54

R2C 0

P

now yields

P

C

P IR /

DBC

P:

3.9.


We recall that if KAN is an Iwasawa decomposition for G, then the map AKAN W G ! a is given by (3.5.3). We further recall that WMP \K\H denotes the subgroup of W consisting of all elements that can be realized in MP \ K \ H . Lemma 3.9.2. Let B be a WMP \K\H -invariant convex subset of aq and g 2 G. Then the following two assertions are equivalent. (i) g P 2 P .B C

P/

(ii) For all P0 2 P .aq / and Pm 2 P.a/ with Pm P0 P , q ı AKANPm g.LP \ H / B C P0 : Proof. Let g 2 G and write g D k exp Y hn, where k 2 K, Y 2 aq , h 2 LP \ H and n 2 NP . Let C be defined as in the previous lemma. Let P0 2 C and let Pm be a minimal parabolic subgroup of G with A Pm P0 . We now observe that by Lemma 2.4.2 q ı AKANPm g.LP \ H / D q ı A.MP \K/AN P exp Y .LP \ H / ; (3.9.1) Pm

where A.MP \K/AN P W LP ! a is defined as in (3.5.3) for the Iwasawa decompoPm sition (3.9.2) LP D .K \ MP /ANPPm : Applying the convexity theorem of Van den Ban ([Ban86, Theorem 1.1]) to LP , LP \ H and the Iwasawa decomposition (3.9.2), we obtain from (3.9.1) that q ı AKANPm g.LP \ H / D ch WMP \K\H Y C ˙ C .lP ; aq I P0 \ LP / : (3.9.3) Note that WMP \K\H P D P : Now assume that (i) holds, then Y 2 B C P . The latter set is WMP \K\H invariant, so if P0 ; Pm are as in (ii), then it follows from (3.9.3) that q ı AKANPm g.LP \ H / B C P C ˙ C .lP ; aq I P0 \ LP / B C P0 ; and (ii) follows. Conversely, assume that (ii) holds. Then it follows from (3.9.3) that Y 2 B C P0 for all P0 2 C . In view of Lemma 3.9.1, this implies that Y 2 B C P , so that (i) follows.

55

3.


We recall that the map EP from P to the power set of Q is given by Q

Q

EP .g P / D gNP Q

. 2 P /:

Proposition 3.9.3. Let B be a WMQ \K\H -invariant convex subset of aq . Then the following statements hold. (i) If g 2 G and g Q 2 Q .B C

Q /,

Q

then g P 2 P .B C

(ii) f 2 P W EP ./ \ Q .B/ ¤ ;g P .B C

P /.

P /.

Proof. (i): Let g Q 2 Q .B C Q /. Then assertion (ii) of Lemma 3.9.2 holds with Q in place of P . Since P Q, every minimal ı -stable parabolic subgroup P0 that is contained in P is also contained in Q so that assertion (ii) of Lemma 3.9.2 also holds with P . By the mentioned lemma it then follows that g P 2 P .B C P /. Q (ii): Assume EP .g P / \ Q .B/ ¤ ;. There exists an n 2 NP such that gn Q 2 Q .B/. Therefore there exist k 2 K and n 2 NP such that kgn 2 exp.B/.LQ \ H /: Let P0 2 P .aq / and Pm 2 P.a/ be such that Pm P0 P . Since LP LQ , q ı AKANPm g.LP \ H / D q ı AKANPm kgn.LP \ H / q ı AKANPm exp.B/.LQ \ H / : The last of these sets is contained in q ı A.K\L

Q Q /ANPm

exp.B/.LQ \ H / ;

which is equal to B C ˙ C .lQ ; aq I P0 \ LQ / by the convexity theorem [Ban86, Theorem 1.1] of Van den Ban. As the latter is contained in B C P0 , the statement now follows by application of Lemma 3.9.2. The first part of Proposition 3.9.3 has the following corollary. Corollary 3.9.4. Assume that B is a compact WMQ \K\H -invariant convex subset of aq and that 2 Eb0 .Q ; JQ /. Then supp./ Q .B C

Q/

Q

implies supp.RP / P .B C

P /:

In particular, if B aq is compact, convex and WK\H -invariant and 2 Eb0 .X /, then supp./ X.B/ implies supp.RP / P .B C P /:

56

3.9.


Proof. Let 2 E 1 .Q ; JQ / and assume that supp./ Q .B C Q and assume that RP is non-zero at g P . As Q RP .g

Q /.

Let g 2 G

Z P / D

Q

NP

.gn Q / d n;

Q

there exists an n 2 NP such that .gn Q / ¤ 0. By assumption gn Q 2 Q .B C Q /, hence by part (i) of Proposition 3.9.3, the element g P D gn P is contained in P .B C P /. Since B is compact, the set P .B C P / is closed. Q Therefore, the support of RP is contained in P .B C P /. We conclude that the following implication holds for every 2 E 1 .Q ; JQ /, supp./ Q .B C

Q/

H)

Q

supp.RP / P .B C

P /:

(3.9.4)

Let now 2 Eb .P ; JP / and assume that supp. / \ P .B C P / D ;. Then by (3.9.4) and Corollary 3.6.7 Z Z Q Q ./RP ./ d D 0 SP ././ d D P

Q

Q for all 2 E 1 .Q ; JQ / with supp./ Q .B C Q /. Therefore, supp SP \ Q .B C Q / D ;. We conclude that for all 2 Eb .P ; JP / the following implication is valid:

supp. / \ P .B C

P/

D;

)

Q

supp SP

\ Q .B C

Q/

D ;:

Finally, let 2 Eb0 .Q ; JQ / and assume that supp./ Q .B C 2 Eb .P ; JP / is supported in the complement of P .B C is disjoint from Q .B C Q /, hence Q

Q

RP . / D SP

P /, then supp

Q /. Q SP

If

D 0:

Therefore supp./ is contained in P .B C P /. This proves the first statement. The second statement is obtained from the first by taking Q equal to G. The following proposition gives a characterization of the support of the horospherical transform of a non-negative compactly supported smooth function on X.

57

3.


Proposition 3.9.5. Let P0 2 P .aq / and let B aq . Then f 2 P0 W EP0 ./\X.B/ ¤ ;g D

[

P0 ch.WK\H b/C

˙ C .g; aq I P0 / :

b2B

If 2 D.X/ is non-negative and satisfies supp./ D X.B/, then supp.RP0 / D f 2 P0 W EP0 ./ \ X.B/ ¤ ;g:

(3.9.5)

We first prove the following lemma. Lemma 3.9.6. Let P0 2 P .aq / and let a; b 2 Aq . Then KaNP0 \ KbH ¤ ;

()

log a 2 ch.WK\H log b/ C

˙ C .g; aq I P0 / : (3.9.6)

Proof. Let Pm 2 P.a/ be a minimal parabolic subgroup contained in P0 . Then, by Lemma 2.4.2, NPm decomposes as NPm D NP0 NH , where NH D NPm \ H . It follows that the first assertion of (3.9.6)is equivalent to KaNPm \ KbH ¤ ;: As A \ H normalizes NPm , the latter assertion is equivalent to Ka.A \ H /NPm \ KbH ¤ ;:

(3.9.7)

Finally, the assertion (3.9.7) is equivalent to the assertion on the right-hand side of (3.9.6) by van den Ban’s convexity theorem [Ban86, Theorem 1.1]. Proof for Proposition 3.9.5. If Pm 2 P.a/ is a minimal parabolic subgroup contained in P0 , then LP0 D .MP0 \ K/ANPPm0 is an Iwasawa decomposition for LP0 . By Lemma 2.4.2 the subgroup NPPm0 is contained in H , hence the map .k; a/ 7! ka P0 induces a diffeomorphism K=K \ MP0 Aq ! P0 :

(3.9.8)

Let 2 P0 . Then we may write D ka P0 with k 2 K and a 2 Aq . Then EP0 ./ \ X.B/ ¤ ; is equivalent to the existence of an element Y 2 B such that

58

3.9.


KaNP0 \ K exp.Y /H ¤ ;. By Lemma 3.9.6, the latter assertion is equivalent to the existence of an element Y 2 B such that log a 2 ch.WK\H Y / C

.˙ C .g; aq I P0 //:

As (3.9.8) is a diffeomorphism, this assertion is in turn equivalent to the existence of a Y 2 B such that C ka P0 2 P0 ch.WK\H Y / C ˙ .g; aq I P0 / : This completes the proof of the first statement of Proposition 3.9.5. We now turn to the proof of the second assertion. Assume that 2 D.X / is as stated. Let g 2 G and assume that EP0 .g P0 / \ X.B/ ¤ ;: Then there is an n 2 NP0 such that gn x0 2 X.B/. As the map p W G ! X, g 7! g x0 defines a fiber bundle, supp.p / D p 1 .supp /, hence gn 2 supp.p /. Since p 0 it now follows that there exists a sequence .gj /j 2N in G such that .gj x0 / > 0 and gj ! gn if j ! 1. Note that gj P0 ! gn P0 D g P0 for j ! 1. Since is continuous, for every j 2 N there exists an open neighborhood Uj of gj in G such that .g x0 / > 0 if g 2 Uj . This implies that RP0 .gj P0 / > 0 and thus we conclude that g P0 2 supp.RP0 /. This implies that the set on the left-hand side of (3.9.5) is contained in the set on the right-hand side. We will finish the proof by establishing the converse inclusion. Let g 2 G and assume g P0 2 supp.RP0 /: Since g 7! gP0 defines a fiber bundle G ! P0 we may apply a similar argument as above to see that there exists a sequence .gj /j 2N in G, converging to g such that for every j 2 N Z 0 < RP0 .gj P0 / D .gj n P0 / d n: NP0

In particular, there exists a sequence .nj /j 2N in NP0 such that .gj nj x0 / > 0:

59

3.


Note that .gj nj x0 /j 2N is a sequence in the support of , which equals X.B/. As this support is compact, there exists a convergent subsequence. Without loss of generality we may therefore assume that gj nj x0 converges to a point g0 x0 2 X.B/ if j ! 1. Now lim nj x0 D lim gj 1 gj nj x0 D g

j !1

j !1

1

g0 x 0 :

Using a local section of the bundle p W G ! X around the point g find that there exist hj 2 H such that lim nj hj D g

j !1

1

1g

0

x0 , we

g0 :

According to [Ban86, Lemma 3.4], if a sequence .mj aj nj hj /j 2N in G, with mj 2 MP0 , aj 2 Aq , nj 2 NP0 and hj 2 H , converges to a point in the boundary of the open subset P0 H of G, then faj W j 2 Ng is not relatively compact in Aq . Furthermore, by the same lemma NP0 LP0 LP0 \H H ! P0 H I

.n; l; h/ 7! nlh

is a diffeomorphism. Applying these results to the sequence .nj hj /, we see that it converges to a point in NP0 LP0 H \ NP0 H D NP0 H . We now conclude that there exist n 2 NP0 and h 2 H such that nj ! n and hj ! h, for j ! 1. Since .gj nj x0 / is a sequence in the compact set X.B/ it follows that the limit gn x0 is contained in X.B/ as well. On the other hand, gn x0 2 gNP0 x0 D EP0 .g P0 /; and we see that the intersection of EP0 .g P0 / and X.B/ is non-empty. Remark 3.9.7. Note that if ˙ C .g; aq I P0 / D ;, the Radon transform RP0 of a compactly supported smooth function 2 D.X/ is again a compactly supported smooth function. This holds in particular if X is a Riemannian symmetric space.

60

Chapter 4

Support theorem for the horospherical transform The aim of this chapter is to prove a support theorem for the horospherical transform for functions. In Section 4.1 we derive Paley-Wiener type estimates for the Fourier transform on a Euclidean space for Schwartz functions with a certain type of support. The horospherical transform is related to the so-called unnormalized Fourier transform on X . Given the support of the horospherical transform of a function, the theory from Section 4.1 yields a Paley-Wiener estimate for one component of the unnormalized Fourier transform. This is described in Section 4.2. In Section 4.3 PaleyWiener estimates for one component of the normalized -spherical Fourier transform are deduced from the estimates for the unnormalized Fourier transform. Then in Section 4.4 we introduce some subspaces of E 1 .X / that will be used in the last two sections. For the normalized -spherical Fourier transform there exists an inversion formula due to Van den Ban and Schlichtkrull, that we describe in Section 4.5. Finally, in Section 4.6, we use the inversion formula and the Paley-Wiener estimates to obtain a support theorem for the horospherical transform. Throughout this chapter, P0 denotes a minimal ı -stable parabolic subgroup containing A.

61

4.

4.1

Support theorem for the horospherical transform

The Euclidean Fourier transform and Paley-Wiener estimates

Let V be a finite dimensional real vector space equipped with a positive definite inner product. Let u 2 E .V /. For each functional 2 V we define u WD e u. Moreover, we define the set C.u/ WD f 2 V W u 2 S .V /g: This set is a convex cone in V . (See [Hör03, Section 7.4].) The Fourier transform of u is defined to be the function Fu on C.u/ C iV given by Z Z Fu./ D e .x/ u.x/ dx D e i .x/ u .x/ dx . D Ci 2 C.u/CiV /: V

V

Let P .V / denote the ring of polynomial functions V ! C. If p 2 P .V /, we use the notation p.@/ for the linear partial differential operator with constant coefficients on V determined by p.@/e D p./e

. 2 V /:

In a similar fashion, we associate to each p 2 P .V / a differential operator p.@/ on V . Since for every homogeneous polynomial ph 2 P .V / of degree 1 ph .@/u D ph .@/u C ph ./u is a Schwartz function if u is Schwartz, we see that C.u/ C.p.@/u/ for every polynomial function p W V ! C. The function 7! Fu. C i/ is a Schwartz function on V for each 2 C.u/. Furthermore, Fu is holomorphic on the interior of C.u/ C iV (see [Hör03, Theorem 7.4.2]) and there p.@/Fu D F.x 7! p. x/u.x//; for all p 2 P .V / and q 2 P .V /.

62

and F.q.@/u/ D qF.u/;

4.1.

The Euclidean Fourier transform and Paley-Wiener estimates

Lemma 4.1.1. Let S be a closed convex subset of V and let the cone CS V be defined as in (3.8.1). Let u 2 S .V /. Then supp.u/ S H)

CS C.u/:

Proof. Let 2 CS . Then .p.@/e /1S is bounded for every polynomial p 2 P .V /. By application of the Leibniz rule we now see that p.@/u D O.1 C kxk/ N for all p 2 P .V / and N 2 N. Hence u is Schwartz. Proposition 4.1.2 (Paley-Wiener estimate). Let S be a closed, convex subset of V . If u 2 S .V / with supp.u/ S, then for every N 2 N and 2 CS C iV jFu./j 2N k.1 C /N ukL1 .1 C kk/

N HS . Re.//

e

:

Proof. If v 2 S .V / satisfies supp.v/ S and 2 CS C iV , then Z jFv./j je .x/ j jv.x/j dx kvkL1 e HS . Re.// V

Let N 2 N. Then .1 C kk/N jFu./j 2N .1 C kk2 /N jFu./j D 2N jF..1 C /N u/./j; hence, by taking v D .1 C /N u, we obtain .1 C kk/N jFu./j 2N k.1 C /N ukL1 e HS .

Lemma 4.1.3. Let B be a compact subset of V and For every 0 2 CBC and for every 2 0 CBC HB . / Proof. Let 0 2

Re.//

:

a cone in V containing 0.

HBC . 0 / HBC . C 0 / HB . / C HB .0 /: CBC and suppose 2 0 2

0 C CBC

CBC . Then CBC :

By the triangle inequality for suprema HBC . / HBC . C 0 / C HBC . 0 /;

63

4.


hence HBC . /

HBC . 0 / HBC . C 0 /:

Note that HBC . 0 / is finite because 0 2 CBC and HBC . / D HB . / by Lemma 3.8.1 (iii). By the same lemma HBC . C 0 / D HB . C 0 /. Using the triangle inequality for suprema, we find HB . C 0 / HB . / C HB .0 /: Note that HB .0 / is finite because B is compact. Lemma 4.1.4. For all ; 0 2 V we have 1 C kk .1 C k C 0 k/ .1 C kk/.1 C k0 k/: 1 C k0 k Proof. The estimate on the right is a straightforward consequence of the triangle inequality. It follows that .1 C kk/ D .1 C k C 0 C . 0 /k/ .1 C k C 0 k/.1 C k0 k/: This implies the required estimate on the left. Proposition 4.1.5 (Paley-Wiener estimate). Let B be a compact subset of V and let V be a closed cone. Let 0 2 C . Then for every N 2 N there exists a constant C0 ;N > 0 with the following property. If u is a smooth function on V such that u0 2 S .V / and supp.u/ B C , then for every 2 0 C C iV , jFu./j C0 ;N k.1 C /N u0 kL1 .1 C kk/

N HB . Re.//

e

:

Proof. Let N 2 N. Since u0 2 S .V / and supp.u0 / D supp.u/ B C follows by application of Proposition 4.1.2 that jFu./j D jFu0 . N

0 /j N

, it

(4.1.1)

2 k.1 C / u0 kL1 .1 C k

0 k/

N HBC . Re. 0 //

e

for all 2 0 CBC CiV . Since Re. 0 / 2 CBC , it follows by application of Lemma 4.1.3 that HBC . Re.

64

0 // HB . Re/ C HB .0 /:

(4.1.2)

4.2.

The unnormalized Fourier transform

Finally, by application of Lemma 4.1.4 we see that .1 C k

0 k/

N

.1 C kk/

N

.1 C k0 k/N :

(4.1.3)

Substituting the estimates (4.1.2) and (4.1.3) in (4.1.1), we obtain the required estimate with C0 ;N D 2N .1 C k0 k/N e HB .0 / . Remark 4.1.6. Proposition 4.1.5 is part of the following Paley-Wiener Theorem that we state here for the sake of completeness. Let B be a compact subset of V and let V be a closed cone. Assume that 0 2 C and that v is a function from 0 C C iV to C. Then the following two statements are equivalent. (I) v equals the restriction to 0 C C iV of the Fourier transform Fu of a function u 2 E .V / such that u0 2 S .V / and supp.u/ B C (II) The function v is continuous and its restriction to 0 C iV is Schwartz. For every 2 C and 2 0 C C iV the function z 7! v.z C / is holomorphic on fz 2 C W Re z > 0g and for every N 2 N there exists a positive constant CN such that for all 2 0 C C iV jv./j CN .1 C kk/

N HB . Re.//

e

:

For every N there exists a constant C0 ;N , depending on 0 and N only, such that if .I / holds, then .II / holds with CN smaller than or equal to C0 ;N k.1 C /N u0 kL1 . The proof for the case 0 D 0 is similar to the usual proof for the Paley-Wiener theorem for D.V /. See for example [Rud73, Theorem 7.22]. For 0 ¤ 0 the theorem then follows by application of Lemmas 4.1.3 and 4.1.4.

4.2


We start with recalling several definitions and results from [Ban88], and [BS97b]. Let .; H / be a unitary representation of MP0 in a finite dimensional Hilbert space H and 2 aqC . The space E .P0 W W / of smooth vectors for the induced

65

4.


G representation IndP . ˝ e ˝ 1/ of G (by left-induction) from the representation 0 ˝ e ˝ 1 of P0 consists of the smooth functions f W G ! H satisfying

f .mang/ D aCP0 .m/f .g/

.m 2 MP0 ; a 2 AP0 ; n 2 NP0 ; g 2 G/: (4.2.1) Here P0 is defined as in (3.3.1) with P0 in place of Q. We define V ./ to be the formal direct sum of Hilbert spaces M

V ./ D

V .; w/;

w.H \MP0 /w

V .; w/ D H

1

;

w2W w.H \MP /w

1

0 is the subspace of w.H \ MP0 /w 1 -fixed vectors in H . where H cP0 /H be the set of equivalence classes of finite dimensional unitary repLet .M resentations .; H / of MP0 such that V ./ ¤ f0g. The principal series of represenG tations for X is the series of representations IndP . ˝ e ˝ 1/ with 2 aqC and 0 cP0 /H . .; H / 2 .M

G Let IndP . ˝ e ˝ 1/ be a principal series representation. The space of gen0 eralized functions G ! H satisfying (4.2.1) is denoted by C 1 .P0 W W /. Following [Ban88, Section 5] we define the map j.P0 W W /, from V ./ to the space C 1 .P0 W W /H of H -fixed elements in C 1 .P0 W W / as follows. The sets P0 wH , for w 2 W are disjoint and open in G and their union [ ˝.P0 / D P0 wH w2W

is dense in G. For 2 aq .P0 ; 0/ P0 the function is given by 8 ˆ aCP0 .m/w for x D manwh 2 ˝.P0 / with ˆ ˆ ˆ < m 2 MP0 ; a 2 AP0 ; n 2 NP0 ; j.P0 W W /./.x/ WD ˆ w 2 W and h 2 H ˆ ˆ ˆ :0 for x … ˝.P0 /: (4.2.2) It is known that for 2 aq .P0 ; 0/ P0 the function j.P0 W W /./ thus defined is continuous, see [Ban88, Proposition 5.6]. For the remaining 2 aq;C the map is defined by meromorphic continuation. For generic 2 aqC , the map j.P0 W W / is known to be a bijection V ./ ! C 1 .P0 W W /H . See [Ban88, Theorem 5.10].

66

4.2.


Lemma 4.2.1. Let B be a compact subset of aq and let that supp. / P0 .B C P0 /: For k 2 K, let

k

be the function on aq given by k exp.Y / P0 k .Y / D

If 2 aq .P0 ; 0/ then e

k

2 E 1 .P0 ; JP0 / be such

.Y 2 aq /:

is a Schwartz function for every k 2 K. The map K ! S .aq /I

k 7! e

k

thus defined is continuous. Proof. Let be fixed as above. According to Proposition 3.5.4, the function u vanishes at infinity for every u 2 U.g/. In particular, it follows that each of the functions u is bounded and uniformly continuous on P0 . For every u 2 U.g/ there exists a finite set Fu U.g/, consisting of linearly independent elements, such that Ad.k/u 2 span.Fu /, for all k 2 K. Write Ad.k/ D P v2Fu cu;v .k/v, then the cu;v are continuous, hence bounded functions on K. It follows that there exists a constant Cu > 0 such that k exp.Y / P0 j sup sup ju k j D sup sup j Ad.k/u k2K aq

k2K Y 2aq

D

X

sup sup jcu;v .k/j jv .k exp Y /j

v2Fu k2K Y 2aq

< Cu : Since K=.MP0 \ K \ H / aq ! P0 I

.k .MP0 \ K \ H /; Y / 7! k exp.Y / P0

is a diffeomorphism and supp. / is contained in .B C P0 /, the support of k is contained in B C P0 . Let 2 aq .P0 ; 0/; then < 0 on P0 n f0g. Let p be a polynomial function on aq and let u 2 S.aq /. Then by the Leibniz rule there exist finitely many elements uj 2 S.aq / (independent of k) such that X X sup jp u.e k /j sup jp e uj k j sup jp e j Cuj < 1: aq

j

aq

This proves that the functions e

BC k

P0

j

are Schwartz functions on aq .

67

4.


If k; k 0 2 K, then sup jp u aq

k

k0

X j sup jp e j sup juj BC

P0

aq

j

k

uj

:

k0 j

Now sup juj aq

k

uj

k0 j

X

jcuj ;v .k/j sup j lk .v / P0

v2Fuj

lk0 .v /j:

(4.2.3)

Here lg .g 2 G/ denotes the left translation P0 ! P0 given by 7! g . The second statement of the proposition now follows since the right-hand side of (4.2.3) converges to 0 if k ! k 0 by the uniform continuity of the functions v and the boundedness of the functions cuj ;v W K ! C. Let FAq be the (Euclidean) Fourier transform on Aq , i.e., the transform mapping a function 2 L1 .Aq / to the function on iaq given by Z FAq ./ D

.a/a

. 2 iaq /:

da

Aq

Note that FAq is related to the Fourier transform Faq on aq by FAq

D Faq .

ı exp/:

Proposition 4.2.2. Let B be a compact subset of aq , let 2 V .; e/ and let 2 aq .P 0 ; 0/ C P0 . If 2 EP10 .X / satisfies supp.RP0 / P0 .B C then for every g 2 G the integral Z .x/j.P0 W W X

is absolutely convergent and equals Z FAq a 7! aP0 RP0 .g MP0 \K

68

1

P0 /;

/./.g x/ dx

(4.2.4)

ma P0 / ./.m/ d m:

(4.2.5)

4.2.


Proof. Let satisfy the above hypotheses. For as stated, the function j.P0 W W /./ is continuous. We will first prove the assertions under the assumption that g D e. By Proposition 3.6.4 and Proposition 3.5.4, the function RP0 is an element of Eb .P0 /. In view of the condition on the support of this Radon transform it now follows by application of Lemma 4.2.1 that (4.2.6) aq 3 Y 7! e .Y /CP0 .Y / RP0 k exp.Y / P0 is a continuous family (with family parameter k 2 K) of functions in the Schwartz space S .aq /. Therefore, the integral Z a CP0 RP0 .k 1 ma P0 / da Aq

is absolutely convergent and depends continuously on m 2 MP0 \ K. Since MP0 \ K is compact, the integral Z Z (4.2.7) a CP0 RP0 .k 1 ma P0 /.m/ da d m MP0 \K Aq

is absolutely convergent as well. Clearly, this integral equals (4.2.5), with g D k. We will proceed to show that the integral also equals (4.2.4). Indeed, substituting the definition of the Radon transform in (4.2.7), we see that (4.2.5) equals the absolutely convergent integral Z Z Z (4.2.8) a CP0 .k 1 man P0 /.m/ d n da d m MP0 \K Aq

Z D

NP0

Z

Z .k

MP0 \K Aq

NP0

1

man P0 /j.P0 W W

/./.man x0 / d n da d m:

For the last equality we have used that 2 aq .P0 ; 0/ P0 , so that j.P0 W W /./ is the continuous function given by (4.2.2). As 2 V .; e/, this continuous function is supported by the closure of the set P0 H . As before, we denote w 1 P0 w by P0w . We now recall [Óla87, Theorem 1.2] According to this result, Z Z Z Z .x/ dx D .man x0 / d n da d m; (4.2.9) P0 x0

for every

MP0 \K Aq

NP0

2 Cc .X /. As the multiplication map .MP0 \ K/=.MP0 \ K \ H / Aq NP0 ! P0 x0

69

4.


is a diffeomorphism onto the open subset P0 x0 of X , it follows that (4.2.9) is valid for any measurable function W X ! C, provided the integral on either one of the two sides of the equation is absolutely convergent; and in that case the integral is absolutely convergent as well. Applying this result to (4.2.8) we find that (4.2.5) equals the integral Z P0 x0

.x/j.P0 W W

/./.x/ dx:

As j.P0 W W /./.x/ is supported by P0 x0 , the latter integral equals (4.2.4). This completes the proof for g D e. Next assume that g D k 2 K. Then Lk W x 7! .k 1 x/ satisfies the same conditions as . Hence (4.2.5) is absolute convergent and equals the absolute convergent integral Z .k X

1

Z x/j.P0 W W

/./.x/ dx D

X

.x/j.P0 W W

/./.k x/ dxI

for the last equality we have used invariance of the measure dx. This establishes the result for g 2 K. We now assume that g is an arbitrary element of G. Write g D mg ag ng kg , with mg 2 MP0 , ag 2 AP0 , ng 2 NP0 and kg 2 K. Then by the transformation properties of j , the integral (4.2.4) equals ag

CP0

Z .mg / X

.x/j.P0 W W

/./.kg x/ dx:

By what we proved above, this expression equals ag

CP0

Z .mg / MP0 \K

Z D

MP0 \K

Z D

MP0 \K

FAq a 7! aP0 RP0 .kg 1 ma P0 / ./.m/ d m:

FAq a 7! aP0 RP0 .kg 1 mag 1 a P0 / ./.mg m/ d m FAq a 7! aP0 RP0 .kg 1 ng 1 ag 1 mg 1 ma P0 / ./.m/ d m:

For the last equality we have used that the measures d m and da are left-invariant, that AP0 and MP0 commute, and that NP0 is normal in P0 and stabilizes P0 . We finally observe that the last obtained integral equals (4.2.5).

70

4.2.


Following [BS97b], the unnormalized Fourier transform FPun0 . W / of a function 2 D.X/ is defined to be the element of Hom.V ./; C 1 .P0 W W // given by Z FPun0 . W / W g 7!

X

.x/j.P0 W W

/./.g x/ dx;

(4.2.10)

for 2 V ./. This Fourier transform depends meromorphically on 2 aq . For 2 aq .P 0 ; 0/ C P0 , the dependence is holomorphic, and the integral in (4.2.10) is absolutely convergent. If 2 EP10 .X /, satisfies supp.RP0 / P0 .B C

P0 /

for some compact subset B of aq , then we define (the first component of) the unnormalized Fourier transform FPun0 ;e . W /, for 2 aq .P 0 ; 0/ C P0 , to be the homomorphism V .; e/ ! E .P0 W W / given by the absolutely convergent integral (4.2.10), for 2 V .; e/. We note that for 2 D.X / the Radon transform RP0 has support in a set of the mentioned form, by Corollary 3.9.4. In that case, ˇ FPun0 ;e . W / D FPun0 . W /ˇV .;e/ ; for all 2 aq .P 0 ; 0/ C P0 . Proposition 4.2.3. Let B be a compact subset of aq and let be a cone in aq;C generated by a compact subset of aq .P 0 ; 0/. For sufficiently large R > 0 there exist for every N 2 N a constant CN > 0, a finite set FN U.g/, such that the following holds. For all 2 E 1 .X / satisfying supp.RP0 / P0 .B C

P0 /;

for all k 2 K, 2 V .; e/ and all 2 with kk > R, X kukL1 .X / .1 C kk/ kFPun0 ;e . W /.k/k CN

N HB . Re.//

e

kk:

u2FN

Let 2 V .; e/. The function K aq .P 0 ; 0/ C P0 3 .k; / 7! FPun0 ;e . W /.k/

(4.2.11)

is smooth, and in addition holomorphic in the second variable.

71

4.


Proof. Let 0 2 aq .P 0 ; 0/ C P0 . Then 0 C P0 2 aq .P0 ; 0/, so that by Lemma 4.2.1 the function aq 3 Y 7! e

0 .Y / P0 .Y /

e

RP0 .exp.Y / P0 /

belongs to S .aq / and is supported in BC P0 . We now apply Proposition 4.1.5 with P P0 in place of , so that C D aq .P 0 ; 0/, and with u D e 0 RP0 .exp./P0 /. This gives the existence of a constant C0 ;N > 0, for each N 2 N, such that for all 2 0 C aq .P 0 ; 0/ ˇ ˇˇ ˇ ˇFAq a 7! aP0 RP0 .a P0 / ./ˇ CN ./ .1 C kk/

N HB . Re.//

e

:

Here CN ./ D C0 ;N k.1 C Aq /N a 7! aP0

0

RP0 .a P0 / kL1 .Aq / :

We note that the constant C0 ;N is independent of the function . For each k 2 K, the function Lk W x 7! .k 1 x/ satisfies the same hypothesis as , so that the above estimates apply. In view of the definition of the unnormalized Fourier transform, (4.2.10), and Proposition 4.2.2 we now obtain the estimate kFPun0 ;e . W /.k/k

Z

FAq a 7! aP0 RP0 .k 1 ma P0 / ./.m/ d m D MP0 \K Z ˇ ˇ ˇ ˇ ˇFAq a 7! aP0 RP0 .k 1 ma P0 / ./ˇ k.m/k d m MP0 \K

eN;k ./.1 C kk/ C

N HB . Re.//

e

kk

for all k 2 K and 2 0 C aq .P 0 ; 0/. For the last inequality we used that is unitary and we wrote Z eN;k ./ WD C

MP0 \K

CN .Lm

1k

/ d m:

Using Leibniz’ rule, the fact that MP0 \ K centralizes Aq , and the fact that the function aP0 0 is bounded on exp.B C P0 /, we infer that there exists a finite

72

4.2.

subset FN0 S2N .aq /, such that X Z eN;k ./ C e0 ;N C 0 u2FN

e0 ;N C

MP0 \K

X Z 0 u2FN


ku a 7! RP0 Lk .ma P0 / kL1 .Aq / d m Z

Z

MP0 \K Aq

NP0

ju Lk .man x0 /j d n da d m:

We now note that u.Lk / D Lk .ŒAd.k/ 1 u/ and that Ad.k/ 1 u is expressible in terms of a basis of U2N .g/, with coefficients that are continuous, hence bounded, functions of k 2 K. Combining this observation with (4.2.9) we see that there exists a finite subset FN U.g/ such that X eN;k ./ C e0 ;N kukL1 .X / : C u2FN

In view of the previous estimates, we now conclude that for all 2 0 C aq .P 0 ; 0/ e0 ;N kFPun0 ;e . W /.k/k C

X

kukL1 .X / .1 C kk/

N HB . Re.//

e

:

u2FN

Since aq .P 0 ; 0/ is a cone in aq;C generated by a compact subset, we have, for sufficiently large R > 0, that f 2

W kk > Rg 0 C aq .P 0 ; 0/:

The first statement now follows. We address the second statement. Let U be an open subset of aq;C with compact closure in aq .P 0 ; 0/ C P0 . Then it suffices to prove the smoothness and holomorphy of (4.2.11) on K U . Fix 0 2 aq .P 0 ; 0/CP0 such that U aq .P 0 ; 0/C0 . It follows from (4.2.10) and Proposition 4.2.2 that FPun0 ;e . W /.k/ Z D FAq a 7! aP0 RP0 .k MP0 \K

(4.2.12) 1

ma 0 / ./.m/ d m;

for all 2 P0 C aq .P 0 ; 0/. Put S D B C P0 . Then aq .P 0 ; 0/ CS . We note that SS .Aq / WD f 2 S .Aq / W supp. / Sg is a closed subspace of S .Aq /. As in the proof of

73

4.


Proposition 4.2.2, it follows that (4.2.6) is a continuous family in SS .Aq /, with family parameter k 2 K. As this also applies to u, for every u 2 U.k/, it actually follows that (4.2.6) is a smooth family in SS .Aq /. We now note that the Euclidean Fourier transform defines a continuous linear map FAq W SS .Aq / ! O. CS / D O.aq .P 0 ; 0//: Here O. CS / denotes the space of holomorphic functions on CS , equipped with the usual Fréchet topology. Combining this with the above assertion about smooth families we infer that k 7! FAq a 7! aP0 0 RP0 .ka 0 / is a smooth function on K with values in the Fréchet space O.aq .P 0 ; 0//. This in turn implies that .k; / 7!FAq a 7! aP0 0 RP0 .ka P0 / ./ D FAq a 7! aP0 RP0 .ka P0 / . C 0 / is smooth on K aq .P 0 ; 0/ and in addition holomorphic in the second variable. As U 0 C aq .P 0 ; 0/, this in turn implies that (4.2.12) is smooth in .k; / 2 K U and in addition holomorphic in 2 U .

4.3

The -spherical Fourier transform FP 0 ;

b be a finite set of K-types. If .; V / is a G or K representation, then we Let # K write V# for the space of K-finite vectors with isotypes contained in #. Let V D C.K/# , where the set # of isotypes is taken with respect to the leftregular representation of K on C.K/, and let D # be the representation of K on V obtained from the right-action. We equip V with the inner product induced from L2 .K/. With respect to this inner product, is unitary. As before, let P0 2 P .aq /. In this section we will consider the -spherical Fourier transform FP 0 ; as defined in Section 6 of [BS97b]. Before we can write down the definition of this Fourier transform, we need to introduce some notation. We denote the restriction of to MP0 \ K by MP0 . Let ı C . / be the formal direct sum of Hilbert spaces M 0 ı C . / D C . /w : w2W

74

4.3.


Here 0

C . /w D C 1 .MP0 =w.MP0 \ H /w

1

W MP0 /

is the finite dimensional Hilbert space of MP0 -spherical functions on the symmetric space MP0 =w.MP0 \ H /w 1 , i.e., the Hilbert space of smooth functions f W MP0 =w.MP0 \ H /w

1

! V

satisfying f .k x/ D .k/f .x/

.k 2 MP0 \ K; x 2 MP0 =w.MP0 \ H /w

1

/:

The inner product on 0 C . /w is induced from the inner product on the space of square integrable functions. cP0 /H . The space of smooth functions f W K ! H ˝ V Assume 2 .M satisfying f .mk0 k/ D .m/ ˝ .k/ 1 f .k0 / for k; k0 2 K; m 2 MP0 \ K is denoted by E .K W W /. Note that evaluation at the identity element induces a linear isomorphism f 7! f .e/;

'

E .K W W / ! H ˝ V

MP

0

\K

:

Let V ./ be the conjugate vector space of V ./. Following [BS97b, p. 528], we define a linear map E .K W W / ˝ V ./ ! 0 C . /I

T 7!

T

by

f ˝ w .m/

D hf .e/j.m/w iH

.m 2 MP0 =w.MP0 \ H /w

1

/;

for f 2 E .K W W / and 2 V ./. Here hjiH denotes the inner product on H . This inner product is taken to be anti-linear in the second variable. Let D.X W / be the space of compactly supported smooth functions f W X ! V satisfying f .k x/ D .k/f .x/. We define & W D.X /# ! D.X W / by &./.x/.k/ D .kx/: This map is a bijection. (See [BS97b, Lemma 5].)

75

4.


Restriction to K induces a bijection from E .P0 W W /# onto E .K W W /. Using this linear isomorphism and the linear isomorphism V ./ ! V ./ (defined via the Hermitian inner product on V ./) we may view FPun0 ./. W / as an element of E .K W W /˝V ./, for all 2 aq .P 0 ; 0/CP0 . It thus makes sense to consider inner products between FPun0 . W / and elements in E .K W W / ˝ V ./. The -spherical Fourier Transform FP 0 ; is the linear transform from the space D.X W / to the space of meromorphic 0 C . /-valued functions on aq;C , defined in [BS97b, (59)]. For our purposes, it is sufficient to use the following characterization in terms of the unnormalized Fourier transform discussed in the previous section. cP0 /H such cP0 /H ./ denote the finite collection of representations 2 .M Let .M that jMP0 \K and jMP0 \K have a .MP0 \ K/-type in common. Then the spherical Fourier transform is completely determined by the requirement that hFP 0 ; &././j f ˝ i D hFPun0 . W /j A.P 0 W P0 W W / 1 f ˝ i (4.3.1) cP0 /H . /, f 2 E .K W W /, 2 V ./ and generic for 2 D.X/# , 2 .M 2 aq;C . Here A.P 0 W P0 I W / W E .P0 W W / ! E .P 0 W W / is the so-called standard intertwining operator. It is initially defined for elements of aq .P 0 ; R/ with R > 0 sufficiently large by the absolutely convergent integral Z .ng/ d n; . 2 E .P0 W W /; g 2 G/: A.P 0 W P0 I W /.g/ D NP0

For the remaining 2 aq;C it is defined by meromorphic continuation. In (4.3.1) the topological linear isomorphisms E .Q W W /# ! E .K W W /; ˇ given by f 7! &.f ˇK /, were used for Q D P0 and Q D P 0 to view A.P 0 W P0 I W / as an endomorphism of the space E .K W W /. Remark 4.3.1. To see that the definition for FP 0 ; is in fact equivalent to the defining identity [BS97b, (59)], use subsequently loc. cit. (59), (50) with P and P 0 replaced by P 0 and P0 respectively, (47) and the identity similar to the one in Proposition 3 for the unnormalized Fourier transforms. The last mentioned identity is obtained from the proof of Proposition 3 by using (30) instead of (53).

76

4.3.


If 2 E 1 .X /# satisfies supp.RP0 / P0 .B C

P0 /;

then by definition the unnormalized Fourier transform FPun0 ;e . W / is an ele ment of Hom V .; e/; E .P0 W W / for 2 aq .P 0 ; 0/ C P0 . In accordance with (4.3.1) we now define the (first component of the) spherical Fourier transform FP 0 ;;e &././ of such a function to be the meromorphic 0 C . /e -valued function on aq .P 0 ; 0/ C P0 which is given by hFP 0 ;;e &././j

f ˝ i

D hFPun0 ;e . W /j A.P 0 W P0 W W /

1

f ˝ i (4.3.2)

cP0 /H . /, f 2 E .K W W /, 2 V .; e/ and generic 2 aq .P 0 ; 0/ C for 2 .M P0 . Note that this definition is compatible with (4.3.1). If 2 D.X /# , then FP 0 ;;e &././ D pre FP 0 ; &././; for generic 2 aq .P 0 ; 0/ C P0 . Here pre denotes the projection 0 C . / ! 0 C ./e . Let A.P 0 W P0 W W / denote the standard intertwining operator, viewed as an endomorphism of E .K W W /. Then it follows from [Ban92, Lemma 16.6] that the End.E .K W W //-valued meromorphic function 7! A.P 0 W P0 W W / 1 is of ˙.g; aq /-polynomial growth on aq .P 0 ; 0/. This means that there exists a polynomial function W aq;C ! C, which is a product of functions of the form 7! h; ˛i c with ˛ 2 ˙.g; aq / and c 2 R, such that 7! ./A.P 0 W P0 W W / 1 is a holomorphic End.E .K W W // valued function on aq .P 0 ; 0/ and satisfies a polynomial estimate of the form k./A.P 0 W P0 W W / 1 k C.1 C kk/N

. 2 aq .P 0 W 0//;

with C > 0 and N 2 N. Here we note that the space End.E .K W W // is finite dimensional. The following proposition is now a direct corollary of Proposition 4.2.3 and (4.3.2).

77

4.


Proposition 4.3.2. Let B be a compact subset of aq . If 2 E 1 .X /# satisfies supp.RP0 / P0 .B C

P0 /:

(4.3.3)

Then the map aq .P 0 ; 0/ C P0 ! 0 C . /e I

7! ./FP 0 ;;e &././

is holomorphic. Furthermore, let be any cone in aq;C generated by a compact subset of aq .P 0 ; 0/. Then there exists a constant R > 0 and for every N 2 N a constant CN > 0 and a finite subset FN U.g/ such that for all 2 E 1 .X /# satisfying (4.3.3) the estimate X k./FP 0 ;;e &././k CN kukL1 .X / .1 C kk/ N e HB . Re.// u2FN

is valid for all 2

4.4

with kk > R.

Function Spaces

As before, we assume that P0 is a minimal ı -stable parabolic subgroup containing A. Let PW0K\H be the maximal WK\H -invariant subcone of P0 , i.e., WK\H P0

\

D

P0w :

w2WK\H

For a subset S of aq , we define ˚ E 1 .X I S / D 2 E 1 .X / W supp./ X.S / : We further define EP10 .X / to be the subspace of E 1 .X / given by EP10 .X / D S .X / C

[

E 1 .XI B C

WK\H /: P0

Baq Bcompact

Here S .X/ denotes the space of rapidly decreasing functions on X , which is the intersection of the Harish-Chandra Lp -Schwartz spaces C p .X / for p > 0. (See [Ban92, Chapter 17].)

78

4.4.

Function Spaces

Remark 4.4.1. If X is a Riemannian symmetric space (hence H D K) or X is a Lie group (i.e., G D G0 G0 for some reductive Lie group G0 of the HarishChandra class and H D diag.G0 /), then WK\H equals the full Weyl group W . In these cases the cone PW0K\H is the trivial cone f0g so that EP10 .X / D S .X / is independent of P0 . We say that a cone in a finite dimensional real vector space V is finitely generated if there exists a finite set f!k 2 V W 1 k ng such that D

n X

R0 !k :

(4.4.1)

kD1

A cone is said to be polyhedral if it equals the intersection of finitely many closed halfspaces. According to [Roc70, Theorem 19.1] every finitely generated cone is polyhedral and, vice versa, every polyhedral cone is finitely generated. If is a finitely generated cone (4.4.1), then C equals the polyhedral cone C

D f 2 V W .!k / 0 for all 1 k ng:

Therefore C is finitely generated as well. Note that every finitely generated cone is closed and convex. Lemma 4.4.2. Let V be a finite dimensional real vector space and let n 2 N. For k 2 N with k n, let k be a finitely generated cone in V and let Bk be a compact subset of V . Then there exists a compact subset B of V such that n \

Bk C

k

n \

BC

k:

kD1

kD1

Proof. Fix a positive definite inner product on V . This inner Tnproduct induces a dual inner product on V in the usual manner. We write 0 D kD1 k . Since the cones k are closed and convex, D fx 2 V W .x/ 0 for all 2 C 1 C C n g: P Hence, C 0 equals the closure of nkD1 C k . As the cones C k are finitely generated, so is their sum. In particular the sum is closed and we conclude that 0

C

0

D

n X

C

k

:

kD1

79

4.


We define the continuous functions sWC

1

C

n

! C 0I

.k /nkD1

7!

n X

k

kD1

and W .V /n ! RI

.k /nkD1 7!

n X

kk k:

kD1

Note that s is a surjection. Since is proper and non-negative, has a minimum on s 1 ./. We define the function H0 W C If ; 2 C 0 , then s deduce that H0 . C /

1 ./

min

s

1 ./Cs

! RI

0

Cs

1 ./

7!

1 ./

s

min : 1 .fg/

s

1 .

C /. Since is subadditive, we

min C min D H0 ./ C H0 ./: s

1 ./

s

1 ./

Hence H0 is subadditive. Furthermore, if r > 0 and 2 C 0 , then s 1 .r/ D rs 1 ./ and thus it follows that H0 is positively homogeneous of degree 1. This implies in particular that H0 is a convex function on C 0 . Since C 0 is finitely generated, the intersection of C 0 with any finitely generated cone is again finitely generated. If we fix an orthonormal basis for V , then this is in particular the case for the intersection of C 0 with any orthant (hyperoctant). Such an intersection is a proper cone in the sense that if is a non-zero element of the cone, then is not. We can thus conclude that there exists a finite collection fCj W 1 j mg of proper finitely generated cones Cj such that C

0

D

m [

Cj :

j D1 j

For 1 j m, let f!k Cj W 1 k nj g be a finite set such that Cj D

nj X kD1

80

j

R0 !k :

4.4.

Function Spaces

Since Cj is a proper closed cone, Cj n f0g is contained in an open halfspace. This implies that sj W

nj Y

j R0 !k

j nj .rk !k /kD1

! Cj I

7!

kD1

nj X

j

rk !k

kD1

is a proper map. Therefore, there exist rk > 0, for 1 k nj , such that the intersection of the unit sphere with Cj is contained in nj [ j ch Œ0; rk !k : kD1

Since H0 is convex, the supremum of H0 over the intersection of the unit sphere with Cj is by [Roc70, Theorem 32.2] smaller than or equal to the supremum of j H0 over the sets Œ0; rk !k . The latter is finite because H0 is homogeneous and j H0 .!k / is finite for every 1 k nj . Therefore, there exists an Rj > 0 such that H0 ./ Rj kk for every 2 Cj . Let R be the maximum of the Rj . Then H0 ./ Rkk

. 2 C 0 /:

T

Let now x 2 k .Bk C k /. We will use that by compactness of each set Bk , for 1 k n, we have C k D CBk C k and HBk C k D P HBk on the latter set (see Lemma 3.8.1 (iii)). Let 2 C 0 I then we may write D nkD1 k with k 2 C k , and we see that n n X X HBk .k /: k .x/ .x/ D kD1

kD1

Again, by compactness of the sets Bk , there exists an r > 0 such that HBk rk k and we finally see that n X kk k: .x/ r kD1

This inequality holds for every n-tuple .k /nkD1 2 C n s .k /kD1 is equal to . Therefore,

1

C

n

such that

.x/ rH0 .X / rRkk: Let B.0; rR/ is the closed ball centered at the origin with radius rR. From Lemma 3.8.1 it follows that CB.0;rR/C 0 D C 0 and the restriction of HB.0;rR/C 0 to C 0 equals HB.0;rR/ . The latter support function is given by HB.0;rR/ ./ D rRkk

. 2 V /:

81

4.


We can thus conclude that x 2 B.0; rR/C with B D B.0; rR/.

0.

This establishes the desired inclusion

Remark 4.4.3. The lemma does not hold true if “finitely generated” is replaced by “closed and convex”. Bart van den Dries showed us the following counterexample. Let 1

D f.x; 0; z/ 2 R3 W 0 x zg;

2

D f.x; y; z/ 2 R3 W 0 x z;

If B D f.0; y; 0/ W

x2 y xg: z

y 12 g, then for every t > 1, ( 1 2 .t; 0; t 2 / C .0; 12 ; 0/ .t; ; t / D 2 .t; 1; t 2 / C .0; 21 ; 0/ 1 2

is contained in the intersection of B C subset B 0 of R3 such that

1

and B C

2,

but there exists no compact

1 f.t; ; t 2 / W t > 1g B 0 C f.0; 0; z/ W z 0g D B 0 C . 2

1

\

2 /:

When going through the proof for Lemma 4.4.2 in this particular case, the first serious obstruction encountered is that the sum of C 1 and C 2 is not closed and therefore C 1\ 2 ¤ C 1 C C 2 : Proposition 4.4.4. EP10 .X / is a G-invariant subspace of E 1 .X/. Proof. Let B aq be a compact subset. As S .X/ and E 1 .X/ are G-invariant, it suffices to show that for every g 2 G there exists a compact subset B 00 aq such that (4.4.2) g X.B C PW0K\H / X.B 00 C PW0K\H /: As Kg X.B C PW0K\H / is a K-invariant subset of X , there exists a unique WK\H invariant compact subset C aq such that Kg X.B C

WK\H / P0

D X.C /:

We will finish the proof by showing that C is contained in a set of the form B 00 C WK\H . P0

82

4.4.

Function Spaces

Let a 2 A be such that g 2 KaK. Furthermore, let Pm 2 P.a/ be a minimal parabolic subgroup contained in P0 and let AKANPm W G ! a be defined as in (3.5.3). Then by the convexity theorem ([Ban86, Theorem 1.1]) of Van den Ban, ch.C / C

WK\H /H : P0

˙ C .g; aq I P0 / D q ı AKANPm aK exp.B C

The set on the right-hand side equals q AKANPm .aK/ C q AKANPm .exp.B C

:

WK\H /H / P0

If we apply the convexity theorem of Kostant ([Kos73, Theorem 4.1]) to the first and the convexity theorem of Van den Ban ([Ban86, Theorem 1.1]) to the second term, we obtain ˙ C .g; aq I P0 / D q W .a/ log a C ch.WK\H B/ C

ch.C / C

WK\H P0

C

(4.4.3) ˙ C .g; aq I P0 / :

Here W .a/ denotes the Weyl-group NK .a/=ZK .a/. Put B 0 D q ch W .a/ log a C ch.WK\H B/ and note that each cone on the right-hand side of (4.4.3) is contained in it follows from (4.4.3) that C B 0 C P0 :

P0 .

Hence,

By WK\H -invariance of C this implies that C

\ w2WK\H

w .B 0 C

P0 /

D

\

.B 0 C

P0w /:

w2WK\H

The cones P0w are finitely generated and B 0 is compact, so that we may complete the proof by application of Lemma 4.4.2. Remark 4.4.5. By inspection of the above proof, one readily sees that for every compact subset G0 G there exists a compact B 00 aq such that (4.4.2) is valid for all g 2 G0 .

83

4.

4.5


Inversion formula

We continue in the setting of Section 4.3. Let XC be the union of disjoint open subsets of X [ w XC D X aC q .P0 / : w2W

Let EC .P 0 W W / W XC ! Hom.0 C . /; V / be defined by EC .P 0 W W kaw x0 / W

7! .k/˚P 0 ;w . W a/

w .e/

K\MP0 \wH w 1 where ˚P 0 ;w .; / is the End V -valued function on AC q .P 0 / de fined in [BS97a, Section 10]. Let 2 aq .P 0 ; 0/, with kk D 1. By [BS99, Theorem 4.7], Z .x/ D jW j EC .P 0 W W x/FP 0 ; &././ d t Ci a q

if 2 S .X/# , x 2 XC and if t > 0 is sufficiently large. This result can be partially extended to the K-finite functions in the larger space EP10 .X /. Proposition 4.5.1 (Inversion Formula). If 2 EP10 .X /# and 2 aq .P 0 ; 0/, then for k 2 K, a 2 AC q .P 0 / and sufficiently large t > 0 Z .k/˚P 0 ;e . W a/ FP 0 ;;e &././.e/ d : .ka x0 / D jW j t Cia q

Proof. It suffices to prove the proposition for 2 E 1 .XI B C B is a compact subset of aq . As E 1 .XI B C

WK\H /# , P0

E 1 .XI B

C

WK\H /# P0

there exists a sequence .j /j 2N in

WK\H /# , P0

where

\ D.X /# is dense in 1 E .XI B C PW0K\H /# \

D.X/# converging to in E 1 .XI B C PW0K\H /# . According to [BS97a, Theorem 9.1], the function iaa 3 7! ˚P 0 ;e .t C W a/ is bounded if t > 0 is sufficiently large. The Paley-Wiener estimate in Proposition 4.3.2 therefore implies that, for t > 0 sufficiently large, Z ˇ ˇ ˇ ˇ lim ˇ .k/˚P 0 ;e . W a/ FP 0 ;;e &. j /./.e/ ˇ d D 0: j !1

84

t Cia q

4.6.

4.6

A support theorem for the horospherical transform for functions


Lemma 4.6.1. Let B be a compact subset of aq . Then the set ˚ Y 2 aq W .Y / C HB . / 0 for all 2 aq .P 0 ; 0/ \ aq D ch.B/ C Proof. The set ch.B/ C

P0

P0 :

is convex and closed. Therefore it equals

fY 2 aq W .Y / Hch.B/C

P0

./ for all 2 Cch.B/C

P0

g:

By Lemma 3.8.1 (ii) the cone Cch.B/C P0 equals C P0 , which in turn equals the closure of aq .P 0 ; 0/\aq . Furthermore, by Lemma 3.8.1 (iii) the support function Hch.B/C P0 and Hch.B/ D HB coincide on that set. We thus obtain that ch.B/ C P0 equals fY 2 aq W .Y / HB ./ for all in the closure of

aq .P 0 ; 0/ \ aq g:

Since B is compact, the function HB is continuous, hence the last equals fY 2 aq W

aq .P 0 ; 0/ \ aq g:

.Y / C HB ./ 0 for all 2

This proves the lemma. We can now prove a preliminary support theorem for the Radon transform RP0 associated with the minimal ı -stable parabolic subgroup P0 2 P .aq /. The proof is based on a Paley-Wiener type shift argument. At the end of this section we will sharpen the preliminary result by invoking the equivariance of the Radon transform. Proposition 4.6.2. Let B be a convex compact subset of aq and let 2 EP10 .X/. If supp.RP0 / P0 .B C then

supp./ \ AC q .P 0 / x0 exp.B C

P0 /;

P0 /

\ AC q .P 0 / x0 :

Proof. Because of equivariance and continuity of the Radon transform it suffices to prove the claim for K-finite functions . We will therefore assume that is K-finite b with isotypes contained in a finite subset # of K.

85

4.


Assume that a 2 AC q .P 0 /, but log a … B C a 2 aq .P 0 ; 0/ \ aq such that

P0 .

By Lemma 4.6.1 there exists

.log a/ C Hlog.B/ . / < 0: According to [BS97a, Theorem 9.1] there exists a constant c > 0 such that k˚P 0 ;e . W a/k < cat for all sufficiently large t > 0 and 2 t C iaq . The Paley-Wiener estimate for FP 0 ; (Proposition 4.3.2) and the inversion formula (Proposition 4.5.1) imply that for every integer N there exists a constant CN > 0 such that for sufficiently large t >0 Z t ..log a/CHB . // .1 C kk/ N d : j.a x0 /j CN e t Cia q

Let N dim aq C 1I then by taking the limit for t ! 1 we find .a x0 / D 0:

Lemma 4.6.3. Let S aq , let a 2 A and let g 2 KaK. Then g P0 .S / P0 q ch W .a/ log a C S : Proof. Let Pm 2 P.a/ be a minimal parabolic subgroup contained in P0 . By Kostant’s convexity theorem ([Kos73, Theorem 4.1]), gK K exp ch W .a/ log a NPm : Using that NPm D NPPm0 NP0 and that NPPm0 MP0 \ H (see Lemma 2.4.2) we now find that g P0 .S/ D gK exp.S / P0 K exp ch W .a/ log a NPm exp.S / P0 D K exp ch W .a/ log a C S P0 D P0 q ch W .a/ log a C S :

86

4.6.


We can now sharpen Proposition 4.6.2 by using the equivariance of the Radon transform. Theorem 4.6.4 (Support theorem for the horospherical transform). Let B be a convex compact subset of aq and let 2 EP10 .X /. If supp.RP0 / P0 .B C then supp./ X

\

P0 /;

w .B C

/ P0 :

(4.6.1)

w2WK\H

Proof. Assume that satisfies the hypothesis. We will first show that supp./ \ Aq x0 exp.B C

P0 /

x0 :

(4.6.2)

Let Y0 2 aq be such that exp Y0 2 supp./. Then there exists a Y 2 aC q .P 0 / such C that Y0 C Y 2 aq .P 0 /. By equivariance of RP0 and by application of Lemma 4.6.3 we find that supp RP0 .lexp. Y / / D exp.Y / supp.RP0 / exp.Y / P0 .B C P0 / P0 q ch W .a/ Y C B C P0 : From Propositions 4.4.4 and 4.6.2 it now follows that C exp.Y0 C Y / x0 2 supp lexp. Y / \ Aq .P 0 / x0 exp q ch W .a/ Y C B C P0 x0 : We conclude that the set q ch W .a/ Y C B C P0 contains Y0 C Y . On the other hand, since Y 2 aC q .P 0 /, q ch W .a/ Y Y C P0 : We thus see that Y0 C Y 2 Y C B C P0 , so that Y0 2 B C P0 . We have proved (4.6.2). If k 2 K, then by equivariance of the Radon transform, the function lk satisfies the same hypotheses as , so that (4.6.2) is valid with lk in place of . This implies that \ w exp.B C P0 / x0 supp./ \ Aq x0 w2NK\H .aq /

D exp

\

w .B C

P0 /

x0 :

w2WK\H

87

4.


Invoking the K-equivariance of the Radon transform once more in a similar way, we conclude that (4.6.1) holds. Remark 4.6.5. Let B be a WK\H -invariant closed convex subset of aq . If [ C WK\H D C Pw (4.6.3) P0

w2WK\H

then \

BC

P0w

DBC

0

WK\H P0

(4.6.4)

w2WK\H

by Lemma 3.8.2. If W D WK\H , then (4.6.3) holds. (See Lemma 3.9.1.) In general there exists a compact subset B 0 of aq such that \ B C PW0K\H B C P0w B 0 C PW0K\H w2WK\H

(see Lemma 4.4.2), but if (4.6.3) does not hold, then (4.6.4) is not necessarily true. The following is a counterexample. Let G be the universal covering group of SL.3; R/. Let be a Cartan involution for G and G D KAN an Iwasawa decomposition such that G D K. The root system ˙.g; a/ is of type A2 . Let ˙ C .g; a/ be a system of positive roots and let ˛ and ˇ be the simple roots in that system. Then ˙ C .g; a/ D f˛; ˇ; ˛ C ˇg:

˛

˛Cˇ ˇ Let W ˙.g; a/ ! f˙1g be given by .˙˛/ D .˙ˇ/ D

1 and ˙ .˛ C ˇ/ D 1:

Let W g ! g be the Lie algebra involution given by ( Y .Y 2 a/ .Y / D . /.Y / Y 2 g ; 2 ˙.g; a/ :

88

4.6.


Since G is simply connected, lifts to a Lie group involution of G, which we also denote by . Let K D G and let X D G=K . We claim that (4.6.4) does not hold for every compact convex Weyl-group invariant subset B of aq in this case. The group WK\K equals the Weyl group for the root system ˙C .g ı ; a/ D f˙.˛ C ˇ/g. The reflection s in ˛ C ˇ maps ˛ to ˇ and ˇ to ˛. Let P be the minimal parabolic subgroup of G such that AP D A and ˙ C .g; aI P / D f˛; ˇ; ˛ C ˇg. Then P is ı -stable and P s D P . Therefore, WK\H P

D

P

\

P

D f0g:

Let B be the closed ball in aq with radius r, centered at the origin. The angle between the root vectors H˛ and H˛Cˇ equals the angle between Hˇ and H˛Cˇ ; both are equal to 3 . Let v be a vector perpendicular to H˛Cˇ and with length r, then a straightforward calculation shows that .B C P / \ .B C P / D ch B [ f˙2vg : In pictures:

H˛

Hˇ D

T

Hˇ

H˛

89

Chapter 5

Support theorems The support theorem (Theorem 4.6.4) for the horospherical transform for functions can be generalized to a support theorem for the Radon transform RP corresponding to a (possibly non-minimal) ı -stable parabolic subgroup P for distributions in a suitable subspace of the distribution space Eb0 .X /. In Section 5.1 we describe the spaces of distributions needed to formulate the support theorem in Section 5.2. The support theorem implies injectivity of the Radon transform on these spaces of distributions. In Section 5.3 we discuss some implications of this for generalizing the support theorem to even larger spaces of distributions. In Section 5.4 some final remarks are made. Throughout this chapter P is assumed to be a ı -stable parabolic subgroup of G containing A.

91

5.

Support theorems

5.1

Spaces of distributions

We define the convolution product of 2 D.G/ and 2 Eb .X / to be the function on X given by Z .x/ D .g/.g 1 x/ dg .x 2 X /: G

Since the left-regular representation of G on the Fréchet space Eb .X / is continuous, it follows from standard representation theory that convolution with a compactly supported smooth function on G defines a continuous operator from Eb .X / to itself. For 2 D.G/ we define L to be the compactly supported smooth function given by L .g/ D .g 1 / .g 2 G/: Since h ; i D h; L i

2 D.G/; 2 D.X /; 2 Eb .X / ;

it makes sense to define the convolution product of a function 2 D.G/ and a distribution 2 Eb0 .X / by ./ D .L / 2 Eb .X / : Convolution with a compactly supported smooth function on G defines a continuous map from the distribution space Eb0 .X / to itself. Note that the distribution , with 2 D.G/ and 2 Eb0 .X /, defines a smooth function. For 2 Eb .P ; JP / and 2 D.G/ we furthermore define the convolution product to be the function on P given by Z ./ D .g/ .g 1 / dg: G

Since the left regular representation of G on the Fréchet space Eb .P ; JP / is continuous, it follows again from standard representation theory that convolution by a compactly supported smooth function on G defines a continuous map from the space Eb .P ; JP / to itself. Lemma 5.1.1. Let 2 D.G/, 2 Eb0 .X / and

2 Eb .P ; JP /, then

RP . /. / D RP .L

92

/:

5.1.


Proof. We denote the left regular representation on Eb .X / and Eb .P ; JP / both by L. Using equivariance and continuity of SP , we obtain Z L RP . /. / D . /.SP / D .SP / D .g 1 / Lg SP dg G Z D .g 1 /RP .Lg / dg D RP .L /: G

Let PWK\H be the maximal WK\H -invariant subcone of P and let EP1 .X / be the subspace of E 1 .X / given by [ EP1 .X / D S .X / C E 1 .XI B C PWK\H /: Baq B compact

For P 2 P .aq / these definitions agree with the definitions given in the beginning of Section 4.4. Proposition 5.1.2. Let C denote the (finite) collection of P0 2 P .aq / contained in P . Then the space EP1 .X / equals the intersection \ EP1 .X / D EP10 .X /: P0 2C

In particular, EP1 .X / is invariant under the left action by G. Proof. If P0 2 C then PWK\H PW0K\H , hence EP1 .X / EP10 .X /. It follows that EP1 .X/ is contained in the given intersection. For the remaining inclusion, assume that [ 2 E 1 .XI B C PW0K\H / Baq Bcompact

for each P0 2 C . Then for every such P0 there exists a compact subset BP0 of aq such that supp./ X.BP0 C PW0K\H /: Let B be a WK\H -invariant compact subset of aq containing the (finite) union of the sets BP0 . Then \ \ \ supp./ X .B C PW0K\H / X .B C P0w / : P0 P

P0 P w2WK\H

93

5.

Support theorems

In view of Lemma 3.9.1 applied to P w and C w it follows that \ supp./ X .B C P w / : w2WK\H

According to Lemma 4.4.2 there exists a compact subset B 0 of aq such that the support of is contained in X.B 0 C PWK\H / and thus we conclude that 2 EP1 .X /. The last assertion follows from the fact that each of the spaces EP10 .X / is Ginvariant by Proposition 4.4.4. We define VP .X/ D f 2 Eb0 .X / W 2 EP1 .X/ for every 2 D.G/g:

Proposition 5.1.3. The space VP .X / is a G-invariant subspace of Eb0 .X /. Furthermore, let C be as in Proposition 5.1.2. Then \ VP .X / D VP0 .X /: P0 2C

Proof. Let 2 VP .X / and let g0 2 G. We will prove that .lg0 / 2 EP1 .X / for every 2 D.G/. To do so, let 2 D.G/. If 2 Eb .X/, then by unimodularity of G Z L lg 1 . / D .g 1 /lg 1 g 1 dg 0 0 G Z D .g0 1 g 1 g0 /lg 1 g 1 dg D L g0 .lg 1 /; G

0

0

where g0 is the compactly supported smooth function on G given by g0 .g/ D .g0 1 gg0 /. Hence for every 2 Eb .X / .lg0 / ./ D lg 1 .L / D L g0 .lg 1 / 0 0 g0 g0 D lg 1 D lg0 ./: 0

Since g0 2 D.G/ and 2 VP .X /, we have g0 2 EP1 .X/. The latter space is G-invariant by Proposition 5.1.2. This proves the first statement of the proposition. The second statement is a direct corollary of Proposition 5.1.2.

94

5.1.


We finally define V .X / D f 2 Eb0 .X / W 2 S .X / for every 2 D.G/g: Let WP be a set of representatives in K for the double cosets in the double quotient WMP \K n W=WK\H . Proposition 5.1.4. The space V .X / equals the intersection \ VP w .X /: V .X / D

(5.1.1)

w2WP

In particular, V .X / is a G-invariant subspace of Eb0 .X /. Proof. It is clear that V .X / is contained in each of the spaces VP w .X /. It remains to prove that intersection on the right-hand side of (5.1.1) is contained in the lefthand side. To do this it suffices to show that if Bw is a compact subset of aq for w 2 WP , then the intersection \ .Bw C PWwK\H / (5.1.2) w2WP

is compact. Using that MP \ K normalizes, we obtain \ \ \ \ WK\H D Pw w2WP

w 00 2W

MP \K

Pw

00 ww 0

D

w2WP w 0 2WK\H

\

Pw

D f0g:

w2W

By Lemma 4.4.2, the intersection (5.1.2) is compact. This proves the first statement The second statement is now a direct corollary of Proposition 5.1.3. Remark 5.1.5. Note that the spaces E 0 .X / and S .X / are contained in both VP .X/ and V .X/. Furthermore, the spaces VP .X / and V .X/ contain all integrable functions on X that are of rapid decay, i.e., the functions with the property that if C is a compact subset of G, then for every n 2 N Z n sup kxk jlg .x/j dg < 1: x2X

C

Here k k W X ! R denotes the function given by kka x0 k D e k log ak

.k 2 K; a 2 Aq /:

The subspace of L1 .X / consisting of the functions with support contained in X.B C WK\H / for some compact subset B of aq , is a subspace of VP .X / as well. P

95

5.

5.2

Support theorems

Support theorems

Theorem 5.2.1 (Support Theorem). Let B be a WMP \K\H -invariant convex compact subset of aq and let 2 VP .X /. If supp.RP / P .B C

P /;

then supp./ X

\

/ P :

w .B C

w2WK\H

Remark 5.2.2. Note that if P D P0 is a minimal ı -stable parabolic subgroup, then any subset B of aq is WMP0 \K\H -invariant since MP0 centralizes aq . If P D G, then RP D RG equals the identity operator VG .X / ! VG .X /. In this case the support theorem reduces to the following tautology. Let B be a WK\H -invariant convex compact subset of aq and let 2 VP .X /. Then supp./ X.B/

H)

supp./ X.B/:

Proof for Theorem 5.2.1. First assume that P D P0 is a minimal ı -stable parabolic subgroup. Let BU be a closed ball in a centered at 0 and let U be the subset K exp.BU /K of G. Note that U is symmetric in the sense that U 1 D U . Let 2 D.G/ and assume that supp. / U . If 2 D.P0 / satisfies P0 /

D ;;

does not intersect P0 .B C

P0 /

supp. / \ U P0 .B C then the support of L

RP0 . /. / D RP0 .L As this holds for all

and thus we find

/ D 0:

as above, supp RP0 .L / U P0 .B C

P0 /:

(Here we used that U is compact, so that the set on the right-hand side is closed.) Let Pm 2 P.a/ be a minimal parabolic subgroup contained in P0 and let AKANPm be the map G ! a as defined in (3.5.3). By Kostant’s convexity theorem ([Kos73, Theorem 4.1]), AKANPm .exp.BU /K/ D BU :

96

5.2.

Support theorems

Using that NPm is contained in .LP0 \ H /NP0 (see Lemma 2.4.2), we now find that U P0 .B C P0 / D K exp.BU /K exp B C P0 P0 D K exp.BU C B C P0 P0 D P0 .BU \ aq / C B C P0 : Note that .BU \ aq / C B is a compact convex subset of aq , hence Theorem 4.6.4 can be applied and thus we conclude that \ w .BU \ aq / C B C P0w : supp. / X w2WK\H

For each j 2 N, let Bj be the ball of radius 1=j and center 0 in a and let Uj D K exp.Bj /K. Let .j 2 D.G//j 2N be a sequence such that supp.j / Uj and j ! ı in E 0 .G/ (with respect to the weak topology) for j ! 1. Since convolution is sequentially continuous with respect to each variable separately, the sequence j converges to in D 0 .X / (with respect to the weak topology) for j ! 1. Therefore \ w .BUj \ aq / C B C P0 supp./ X w2WK\H

for every j 2 N, and we conclude that \ supp./ X

w .B C

/ P0 :

(5.2.1)

w2WK\H

This proves the theorem for minimal ı -stable parabolic subgroups P D P0 . We now assume P to be an arbitrary ı -stable parabolic subgroup. Let 2 VP .X/. Assume the support of RP is contained in P .B C P /. Let C be the set of minimal ı -stable parabolic subgroups P0 2 P .aq / contained in P . Let P0 2 C . Proposition 5.1.3 implies that 2 VP0 .X / and from Corollary 3.9.4 and Proposition 3.7.1 it follows that the support of RP0 is contained in P0 .B C P0 /. The previous result for minimal ı -stable parabolic subgroups implies that (5.2.1) holds. Since this is true for each P0 2 C , it follows that \ \ supp./ X w .B C P0 / P0 2C w2WK\H

The theorem now follows by application of Lemma 3.9.1.

97

5.

Support theorems

Remark 5.2.3. With essentially the same proof, it is seen that the support theorem can be generalized to distributions 2 Eb0 .X / for which there exist a sequence .j 2 D.G//j 2N such that (i) supp.j / is contained in Bj (ii) j ! ı in E 0 .G/ for j ! 1 (with respect to the weak topology). (iii) j 2 EP1 .X /. It is not clear to us whether the subset of these distributions forms a subspace of Eb0 .X/, nor are we certain that the set of these distributions is actually strictly larger than VP .X/. Corollary 5.2.4. Let 2 V .X /, let B be a W -invariant convex compact subset of aq and let g 2 G. Then the following statements are equivalent. (i) supp.RP w / g P w .B C

Pw/

for every w 2 WP .

(ii) supp./ g X.B/. Proof. (i))(ii): If supp.RP w / is contained in gP w .B C is contained in P w .B C P w /, hence \ .B C supp.lg / X

P w /, then supp

P

ww 0

/

RP w .lg /

w 0 2WK\H

by Theorem 5.2.1. Since this holds for all w 2 WP , it follows that supp.lg / is contained in \ \ X .B C P ww0 / : w2WP w 0 2WK\H

Since P is stable under WMP \K , it follows that the last equals \ X .B C P w / : w2W

According to Proposition 3.8.2 the latter set equals X.B/. We thus obtain supp./ g X.B/: (ii))(i): This is a consequence of Corollary 3.9.4.

98

5.2.

Support theorems

If X is a Riemannian symmetric space (hence H D K) and P D P0 is a minimal parabolic subgroup, Theorem 5.2.1 reduces to the support theorem [Hel73, Lemma 8.1] of Helgason for the horospherical transform on X. (See also Theorem 1.1, Corollary 1.2 and the subsequent Remark in chapter IV of [Hel94].) The support theorem can in this case be described in a purely geometrical setting as follows. Suppose X is a Riemannian symmetric space. A horosphere in X is a closed submanifold of X by Proposition 3.2.2. Therefore the Riemannian structure on X induces a Riemannian structure and thus a measure on every horosphere. Let R be the transform mapping a function 2 S .X / to the function on the set Hor.X / of horospheres in X Z .x/ dx: R W 7! x2

In this case Hor.X / is in bijection with P0 where P0 is a minimal parabolic subgroup of the identity component G of the isometry group of X. In this way Hor.X / can be given the structure of a G-manifold. Let x 2 X. Using the Iwasawa decomposition for G, it is easily seen that the stabilizer Gx of x in G (i.e., the isotropy group of G at x) acts transitively on the set of horospheres containing x. Therefore this set carries a unique normalized Gx -invariant measure d . The dual transform of R is the transform S mapping a function 2 E .X / to Z S

W x 7!

./ d : 3x

The Radon transform R is defined on V .X / to be the transpose of S. Let d.; / be the distance-function on X. For x 2 X and R 0 we define ˇR .x/ D f 2 Hor.X/ W d.; x/ Rg and BR .x/ D fx 0 2 X W d.x 0 ; x/ Rg. Corollary 5.2.5 (Riemannian case; [Hel94, Ch. IV, Corollary 1.2]). Let 2 V .X /, x 2 X and R 0. If supp.R/ ˇR .x/; then supp./ BR .x/: Proof. Every closed ball in X D G=K is of the form gX.B/, where g is an element of G and B is a closed ball in a. The statement is therefore a direct corollary of Proposition 3.9.5 and Corollary 5.2.4.

99

5.

5.3

Support theorems

Injectivity

Theorem 5.2.1 has the following corollary. Theorem 5.3.1. The Radon transform RP W VP .X / ! Eb0 .P ; JP / is injective. Remark 5.3.2. In [Krö09, Theorem 5.5] it is claimed that the horospherical transform is injective on a certain subspace of S .X /. The proof for this theorem relies in an essential way on the assumption that Hor.X / admits the structure of an analytic manifold and the horospherical transforms RP w for w 2 W together induce a transform R on Hor.X / with the property that the transform R of a real analytic vector for the left-regular representation of G on L1 .X / is a real analytic function on Hor.X/. As stated in Remark 3.4.1 we believe that there are some problems with this kind of reasoning. A natural question is whether Theorem 5.2.1 can be generalized to a support theorem for a larger subspace of D 0 .X /. If so, the Radon transform RP would be injective on that larger subspace as well. We will now show that the support theorem does not hold in general on the Harish-Chandra Schwartz spaces C p .X / for 0 < p 1. We will use the notations introduced in Chapter 4. Let P0 2 P .aq / and let 0 < p < 1. cP0 /H and let 1 < c < 2 1 and 2 V ./. If 2 C p .X / Lemma 5.3.3. Let 2 .M p then for every g 2 G and 2 cP0 C iaq the integral Z .x/j.P0 W W /./.g x/ dx X

is absolutely convergent. If 2 V .; e/, then the integral is equal to Z FAq a 7! aP0 RP0 .g 1 ma P0 / ./.m/ d m: MP0 \K

Proof. Since C p .X / is G-invariant, it suffices to prove the claim for g D e. Let be as in the lemma. Let Pm 2 P.a/ be contained in P0 . If kah D ma0 nwh0

100

(5.3.1)

5.3.

Injectivity

where k 2 K, a; a0 2 Aq , h; h0 2 H , m 2 MP0 \ K, w 2 W and n0 2 NP0 , then w log.a0 / D q ı AKANP w .ahh0 1 /: m The last is contained in ch WK\H log.a/ C P0w by the convexity theorem ([Ban86, Theorem 1.1]) of Van den Ban. Hence log.a0 / 2 ch W log.a/ C P0 and therefore j.a0 /

CP0

j max .waw w2W

1 .1 c/P0

/

:

We define the function J W Aq ! R0 I

a 7!

Y

ja˛

a

˛ mC ˛

j

.a˛ C a

˛ m˛

/

;

˛2˙ C .g;aq IP0 /

where m˙ ˛ is the dimension of the ˙1-eigenspace for ı in g˛ . Since 2 aq .P 0 ; 0/ C P0 , it follows by [Ban88, Proposition 5.6] that j.P0 W W / is continuous. In view of [Sch84, p. 149], there exists a normalization of the measure on X such that Z k.x/j.P0 W W /./.x/k dx X Z Z D k..ka x0 //j.P0 W W /./..ka x0 //kJ.a/ da d k K Aq Z Z j.ka x0 /jJ.a/ max .waw 1 /.1 c/P0 da d k: K Aq

w2W

Following Harish-Chandra, we use the notation for the elementary spherical function on G with spectral parameter 0, and we put q W X ! R>0 I x 7! x .x/ 1 : By [Ban92, Theorem 17.1] there exists a constant C > 0 such that 2

jj C p : Furthermore, by [Ban92, Corollary 17.6] it follows that for sufficiently small > 0 there exists a constant C > 0 such that J.a/ C .ka x0 /

2

:

101

5.

Support theorems

e > 0 such that We infer that there exists a constant C Z k.x/j.P0 W W /./.x/k dx X Z Z 2 e C p 2 .ka x0 / max .waw 1 / w2WK\H K Aq Z 2 e jW j p 2 .a x0 /a .1 c/P0 da: C

.1 c/P0

da d k

AC q .P0 /

By [Ban92, Corollary 17.6], for every ı > 0 there exists a constant cı > 0 such that .a x0 / cı a.ı

1/P0

a 2 AC q .P0 / ;

hence for < p2 1 c the last integral is convergent. The claimed equality follows from equation (4.2.9). b and put D # . For x 2 X and 2 a , let Let # be a finite subset of K q;C EP0 . W W x/ denote the (unnormalized) -spherical Eisenstein integral defined in [BS97b, Section 2], i.e., the element of Hom.ı C . /; V / given by Z EP0 .

f ˝

W W x/.k/ D

K

hf .l/.k/; j.P0 W W /./.l x/i d l

cP0 /H , f 2 C.K W W /, 2 V ./ and x 2 X . for 2 .M Using the K-invariance of the measure on X , we obtain the following immediate corollary of Lemma 5.3.3. cP0 /H , let 2 V ./ and let f 2 C.K W W /. FurtherCorollary 5.3.4. Let 2 .M 2 more, let 1 < c < p 1. Then the -spherical Eisenstein integral EP0 . f ˝ W W x/ is regular on cP0 C iaq . Moreover, for every 2 C p .X/# the integral Z Z X

K

&./.x/.k/ EP0 .

f ˝

W

W x/.k/ d k dx

is absolutely convergent for every x 2 X and equals Z .x/hj.P0 W W /./.x/; f i dx: X

102

5.3.

For x 2 X and 2 aq;C , let E ı . W P0

Injectivity

W x/ be the normalized -spherical

Eisenstein integral for the minimal ı -stable parabolic subgroup P 0 defined in [BS97b, Section 5], i.e., the element of Hom.ı C . /; V / given by EPı . 0

f ˝

W W x/ D EP0 .

A.P 0 WP0 W W /

1f

˝

W W x/:

(5.3.2)

For r 2 R we define Cr .X; / to be the space of continuous functions f W X ! V satisfying the identity f .k x/ D .k/f .x/

.k 2 K/

and the estimate e

sup

rj log aj

jf .ka x0 /j < 1:

k2K;a2Aq

Let R 2 R be such that P0 2 aq .P 0 ; R/ and let ! be a connected and bounded open subset of aq .P 0 ; R/ containing both 0 and P0 . Let EPı 0 . W x/ be the dual of EP0 . W W x/. Then, according to [BS99, Lemma 12.2] there exists an r 2 R such that the ı C . /-valued integral Z E ı . W x/ f .x/ dx (5.3.3) X

is absolutely convergent for every f 2 Cr .X W / and generic 2 ! C iaq , and (5.3.3) depends meromorphically on in that region. Following [BS99, Section 12] we define the (normalized) -spherical Fourier transform FP 0 ; f ./ of a function f 2 Cr .X W / for generic 2 ! C iaq given by (5.3.3). This definition coincides with the definition for compactly supported smooth functions given in Section 4.3. If 0 < p < 1 is sufficiently small, then & maps C p .X /# into Cr .X W /. Fix such a p. Proposition 5.3.5. Let 2 C p .X /# . Then ˇ pre FP 0 ; &./ˇi a D 0 if and only if RP0 D 0: q

Proof. Fix a 1 < c
0g: There are four G-orbits in the set Hor.X / of horospheres in X: (i) HorP .X/ consisting of the G-translates of N x0 D f.1

v2 v2 ; v; / W v 2 Rg D fx 2 X W h0 ; xi D 1g: 2 2

(ii) HorQ .X/ consisting of the G-translates of the line N k 2 x0 D f.v; 1; v/ W v 2 Rg; i.e. the G-translates of the connected component of fx 2 X W h0 ; xi D 0g containing the point .0; 1; 0/. (iii) HorP .X/ consisting of the G-translates of v2 v2 ; v; / W v 2 Rg 2 2 D fx 2 X W h0 ; xi D 1g:

N k x0 D k N x0 D f. 1 C

(iv) HorQ .X/ consisting of the G-translates of the line N k 3 x0 D k N k 2 x0 D f. v; 1; v/ W v 2 Rg; 2

i.e. the G-translates of the connected component of fx 2 X W h0 ; xi D 0g containing the point .0; 1; 0/. HorP .X/ and HorP .X / are in bijection with P D G=N and P D G=N respectively via the maps EP W gN x0 7! g P

and EP W gk N x0 7! gk P :

Similarly, HorQ .X / and HorQ .X / are in bijection with G=HNQ and G=H N Q respectively, via the maps EQ W gN k 2 x0 7! gk 2 NQ H

114

and EQ W gN k 3 x0 7! gk 3 N Q H: 2

2

Hyperbolic Space

Hyperboloid of one sheet with 6 horospheres: two of HorP , two of HorP , one of HorQ and one of HorQ .

The diagonal action of R>0 on the real analytic manifold C C R is free and proper. Therefore R>0 n .C C R/ has the structure of a real analytic manifold. We denote the cosets R>0 .; p/ by Œ; p. Let P.X/ be the set of intersections of X with planes of the form fx 2 R3 W h; xi D pg where 2 C C and p 2 R. The map from R>0 n .C C R/ to P.X /, given by Œ; p D fx 2 X W h; xi D pg is a bijection. Via this bijection P.X / inherits the structure of a real analytic manifold from R>0 n.C C R/. The G-action on C C induces an action on R>0 n.C C R/ that is given by g Œ; p D Œg ; p: Note that is G-equivariant with respect to this action and the natural action of G on P.X/ and hence the last is continuous. There are three G-orbits in P.X /: (i) PC .X/ D R>0 n .C C R>0 / D HorP .X /,

115

Appendix B (ii) P0 .X/ D R>0 n .C C f0g/ , (iii) P .X/ D R>0 n .C C R 0 for every k 2 K, then RP .ka t P / > 0

.k 2 K; t > t0 /:

Proof. Let t < t0 . By Proposition B.1 the horospheres parametrized by ka t P , with k 2 K, do not intersect with X.fa t W t 2 t0 C R0 g/. Therefore these horospheres do not intersect with the support of , hence RP .ka t P / D 0 for every k 2 K. This proves the first statement.

116

Hyperbolic Space

For the second statement, assume that is non-negative and .ka t0 x0 / > 0 for every k 2 K. Let t > t0 . By Proposition B.1 there exist n 2 N such that .ka t n x0 / > 0. Therefore Z RP .ka t P / D .ka t n x0 / d n > 0: N

Note that the above corollary implies that the horospherical transform of a compactly support smooth function need not compactly supported.

117

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[Ron74] L. I. Ronkin. Introduction to the theory of entire functions of several variables. American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, Vol. 44. [Ros79] W. Rossmann. The structure of semisimple symmetric spaces. Canad. J. Math., 31(1):157–180, 1979. [Rud73] W. Rudin. Functional analysis. McGraw-Hill Book Co., New York, 1973. McGraw-Hill Series in Higher Mathematics. [Sch84] H. Schlichtkrull. Hyperfunctions and harmonic analysis on symmetric spaces, volume 49 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 1984. [Var77]

V. S. Varadarajan. Harmonic analysis on real reductive groups. Lecture Notes in Mathematics, Vol. 576. Springer-Verlag, Berlin, 1977.

[War72] G. Warner. Harmonic analysis on semi-simple Lie groups. I. SpringerVerlag, New York, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 188. [Wie85] J. J. O. O. Wiegerinck. A support theorem for Radon transforms on Rn . Nederl. Akad. Wetensch. Indag. Math., 47(1):87–93, 1985. [WK85] J. Wiegerinck and J. Korevaar. A lemma on mixed derivatives and results on holomorphic extension. Nederl. Akad. Wetensch. Indag. Math., 47(3):351–362, 1985.

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Index of notation . . . . . . . . . . . . . . . . 92 A . . . . . . . . . . . . . . . . 17 a . . . . . . . . . . . . . . . . 17 A.P 0 W P0 I W / . 76 AKAK . . . . . . . . . . . 37 AKAN . . . . . . . . . . . 36 AP . . . . . . . . . . . . . . 17 AP . . . . . . . . . . . . . . 36 aP . . . . . . . . . . . . . . . 17 Aq . . . . . . . . . . . . . . . 22 aq . . . . . . . . . . . . . . . 22 aq .P; R/ . . . . . . . . . 25 aC q .P / . . . . . . . . . . . 25 B . . . . . . . . . . . . . . . . 25 C.u/ . . . . . . . . . . . . . 62 C 1 .P0 W W / . . 66 ı C ./ . . . . . . . . . . . . 74 ı C ./w . . . . . . . . . . 75 CS . . . . . . . . . . . . . . . 51 Cb .P ; JP / . . . . . . 41 ch . . . . . . . . . . . . . . . 51 D, D 0 . . . . . . . . . . . . . 4 E , E 0 . . . . . . . . . . . . . .4 E .K W W / . . . . . . 75 E .P0 W W / . . . . . 65 E 1 .X/ . . . . . . . . . . . 39 E 1 .XI S/ . . . . . . . . . 78 E 1 .P ; JP / . . . . . . 39 EP1 .X/ . . . . . . . . . . . 93 EP10 .X/ . . . . . . . . . . 78 Eb .X/ . . . . . . . . . . . . 41 Eb .P ; JP / . . . . . . 41

EP . . . . . . . . . . . . . . 30 Q EP . . . . . . . . . . . . . . 30 F . . . . . . . . . . . . . . . . 62 FAq . . . . . . . . . . . . . . 68 FP 0 ; . . . . . . . . . . . . 76 FP 0 ;;e . . . . . . . . . . . 77 FPun0 . . . . . . . . . . . . . . 71 FPun0 ;e . . . . . . . . . . . . 71 .T / . . . . . . . . . . . . 53 P . . . . . . . . . . . . . . . 53 WK\H . . . . . . . . . . 78 P0

nP . . . . . . . . . . . . . . . 18 Q NP . . . . . . . . . . . . . . 20 Q nP . . . . . . . . . . . . . . . 20 ˝.P0 / . . . . . . . . . . . 66 p . . . . . . . . . . . . . . . . 16 P.a/ . . . . . . . . . . . . . 23 P .aq / . . . . . . . . . . . 23 P g . . . . . . . . . . . . . . 25 ˚P 0 ;w .; / . . . . . . . 84 q . . . . . . . . . . . . . . . . 20

H . . . . . . . . . . . . . . . 20 h . . . . . . . . . . . . . . . . 20 H˛ . . . . . . . . . . . . . . 53 HS . . . . . . . . . . . . . . 51

R.................5 P . . . . . . . . . . . . . . . 33 RP . . . . . . . . . . . . . . 35 Q RP . . . . . . . . . . . 35, 45

j.P0 W W / . . . . . . 66 JP . . . . . . . . . . . . . . . 36

S .................5 S .X / . . . . . . . . . . . . 78 . . . . . . . . . . . . . . . . 20 ˙.g; a/ . . . . . . . . . . . 17 ˙.g; aq /. . . . . . . . . .22 ˙ C .g; a/ . . . . . . . . . 17 ˙ C .g; aq / . . . . . . . . 22 C ˙˙ .g; aq I P / . . . . . 24 SP . . . . . . . . . . . . . . . 35 Q SP . . . . . . . . . . . 35, 47

K . . . . . . . . . . . . . . . . 16 k . . . . . . . . . . . . . . . . . 16 L1 .P ; JP / . . . . . . 36 L1 .P ; JP / . . . . . 40 LP . . . . . . . . . . . . . . 17 lP . . . . . . . . . . . . . . . 17 M . . . . . . . . . . . . . . . 17 m . . . . . . . . . . . . . . . . 17 MP . . . . . . . . . . . . . . 17 cP0 . . . . . . . . . . . . . 66 M mP . . . . . . . . . . . . . . 17 cP0 /H . . . . . . . . . . 66 .M NP . . . . . . . . . . . . . . 18

. . . . . . . . . . . . . . . . 74 V .X / . . . . . . . . . . . . 95 V ./ . . . . . . . . . . . . . 66 V .; w/ . . . . . . . . . . 66 V 1 . . . . . . . . . . . . . . 25 VP .X / . . . . . . . . . . . 94 V . . . . . . . . . . . . . . . 74

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V# . . . . . . . . . . . . . . . 74 W . . . . . . . . . . . . . . . 24 WP . . . . . . . . . . . . . . 95 WS . . . . . . . . . . . . . . 24

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W . . . . . . . . . . . . . . . 24 WMP . . . . . . . . . . . . 25 X . . . . . . . . . . . . 21, 28 X.B/ . . . . . . . . . . . . 54

XC . . . . . . . . . . . . . . 84 x0 . . . . . . . . . . . . . . . 28 P . . . . . . . . . . . . . . 28 P . . . . . . . . . . . . . . . 28 P .B/ . . . . . . . . . . . 54

General index C Cartan decomposition . . . . . . . . . . . . . 16 Cartan involution . . . . . . . . . . . . . . . . . . 16 convolution product . . . . . . . . . . . . . . . 92 D decomposition Cartan . . . . . . . . . . . . . . . . . . . . . . . 16 generalized Iwasawa . . . . . . . . . . 24 Iwasawa . . . . . . . . . . . . . . . . . . . . . 18 Langlands . . . . . . . . . . . . . . . . . . . . 18 polar . . . . . . . . . . . . . . . . . . . . . . . . . 24 double fibration . . . . . . . . . . . . . . . . . . . . 3 for Q and P . . . . . . . . . . . . . . . 29 for horospheres . . . . . . . . . . . . . . . 28 E essentially connected . . . . . . . . . . . . . . 21 F Fourier transform Euclidean . . . . . . . . . . . . . . . . . . . . 62 normalized -spherical on X . . . 76 on Aq . . . . . . . . . . . . . . . . . . . . . . . . 68 unnormalized on X . . . . . . . . . . . 71 H horosphere . . . . . . . . . . . . . . . . . . . . . . . 28 horospherical transform . . . . . . . . . . . . 35

Lie group reductive of the H-C class . . . . . . 16 P Paley-Wiener estimate . . . . . . . . . 63, 64 for FPun0 ;e . . . . . . . . . . . . . . . . . . . . . 71 for FP 0 ;;e . . . . . . . . . . . . . . . . . . . 77 parabolic subgroup . . . . . . . . . . . . . . . . 17 ı -stable . . . . . . . . . . . . . . . . . . 21 polar decomposition . . . . . . . . . . . . . . . 24 R Radon transform . . . . . . . . . 5, 35, 45, 47 k-plane transform . . . . . . . . . . . . 6, 7 S standard intertwining operator . . . . . . 76 support function . . . . . . . . . . . . . . . . . . 51 support theorem for RP . . . . . . . . . . . . . . . . . . . . . . 96 for the k-plane transform . . . . . . 12 for the horospherical tr. . . . . . . . . 87 symmetric space . . . . . . . . . . . . . . . . . . 20 reductive . . . . . . . . . . . . . . . . . . . . . 21 Riemannian . . . . . . . . . . . . . . . . . . 21 T

I incidence . . . . . . . . . . . . . . . . . . . . . . . . . . 3 inversion formula for FP 0 ; . . . . . . . . . . . . . . . . . . . . 84 for the k-plane transform . . . . . . . 7 Iwasawa decomposition . . . . . . . . . . . . 18

transform Euclidean Fourier . . . . . . . . . 62, 68 horospherical . . . . . . . . . . . . . . . . . 35 normalized -sph. F. tr. on X . . 76 Radon . . . . . . . . . . . . . . 5, 35, 45, 47 unnormalized Fourier tr. on X . . 71 transversality . . . . . . . . . . . . . . . . . . . . . . 3

L Langlands decomposition . . . . . . . . . . 18

W Weyl group . . . . . . . . . . . . . . . . . . . . . . . 24

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Samenvatting voor niet-wiskundigen In dit proefschrift wordt een zogenaamde dragerstelling voor een bepaalde klasse van Radontransformaties geformuleerd en voor deze stelling wordt een wiskundig bewijs gegeven. Het voert wat te ver om in deze samenvatting het bewijs te beschrijven, maar ik zal een poging doen om duidelijk te maken wat de stelling zegt.

Figuur 1: CT-scan van een lichaam

Allereerst zal ik uitleggen wat een Radontransformatie is. De lezer heeft ongetwijfeld wel eens gehoord van een computertomogram of CT-scan. Om een dergelijke scan te maken, worden een Röntgenbron en een Röntgendetector aan weerszijden van een object geplaatst. Vanaf de bron wordt een smalle bundel Röntgenstraling uitgezonden die door het object heengaat en daar gedeeltelijk geabsorbeerd wordt. De detector meet vervolgens de intensiteit van het restant van de straling.

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Hiermee wordt bepaald hoeveel straling er geabsorbeerd is door het materiaal in het object op de lijn van de bron naar de detector. Door de bron en de detector te verplaatsen kan dit gedaan worden voor iedere lijn door het object. Op deze manier wordt aan iedere lijn een meetwaarde toegekend. De vraag is nu hoe uit deze data een weergave als in Figuur 1 kan worden verkregen. Ieder soort materie heeft een eigen Röntgendichtheid. Naarmate de Röntgendichtheid van materie groter is, absorbeert deze meer Röntgenstraling. In een voorstelling als Figuur 1 is de Röntgendichtheid weergegeven: een lage dichtheid correspondeert met zwart of donker grijs, een hoge dichtheid met licht grijs of wit. Wiskundig gezien wordt het getal, dat aan een lijn wordt toegekend door de absorptie te meten, verkregen door de Röntgendichtheid te integreren over die lijn; het is het totaal van de Röntgendichtheid over de lijn. De Röntgendichtheidsfunctie is de functie die aan ieder punt in de ruimte de bijbehorende dichtheid toekent. Deze functie wordt omgevormd tot een functie op de verzameling van lijnen door aan iedere lijn de integraal van de Röntgendichtheidsfunctie over die lijn toe Figuur 2: Johann Radon te kennen. Deze omvorming is het schoolvoorbeeld van een Radontransformatie. De data, die een CT-scanner levert, is in feite de Radontransformatie van de dichtheidsfunctie. Om een weergave als in Figuur 1 uit de meetdata te verkrijgen, moet de vraag worden beantwoord hoe een functie uit zijn Radongetransformeerde kan worden verkregen. Deze vraag werd al beantwoord ver voordat de eerste CT-scanner gemaakt werd, namelijk in 1917 door de Oostenrijkse wiskundige Johann Radon. Een probleem dat minder relevant is voor de toepassing in de theorie van computertomografie, maar op zichzelf interessant is en toepassingen heeft in de wiskunde, is het volgende. Stel dat B een bol in de ruimte is en stel dat de Radongetransformeerde van een functie gelijk is aan 0 op alle lijnen die niet door de bol B gaan. De vraag is nu of uit deze informatie conclusies getrokken kunnen worden over het gebied waar de oorspronkelijke functie gelijk aan 0 is. Specifieker: kan geconcludeerd worden dat de oorspronkelijke functie buiten de bol gelijk is aan 0?

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In de situatie die optreedt bij computertomografie, zoals hierboven beschreven, is het antwoord op deze vraag triviaal: als er straling uitgezonden wordt over een lijn die door materie gaat met een Röntgendichtheid die niet gelijk is aan 0, dan zal er straling geabsorbeerd worden. Als op een lijn geen absorptie wordt gemeten, dan kan derhalve op die lijn geen materie aanwezig zijn met een Röntgendichtheid ongelijk aan 0 en is de dichtheidsfunctie daar dan dus gelijk aan 0. De vraag is in deze situatie zo eenvoudig te beantwoorden omdat de functies onder beschouwing dichtheidsfuncties zijn en daarom nergens negatieve waarden aannemen. Als een functie ook negatieve waarden kan aannemen, dan wordt het een stuk moeilijker om de vraag te beantwoorden. De integraal van een dergelijke functie is de som van de bijdragen van de positieve waarden en de (negatieve) bijdragen van de negatieve waarden. Het totaal kan gelijk zijn aan 0 zonder dat de afzonderlijke delen gelijk zijn aan 0. Toch is het ook in dit geval zo dat de functie gelijk is aan 0 buiten de bol als zijn Radongetransformeerde gelijk is aan 0 op alle lijnen die de bol niet snijden; deze stelling werd bewezen door Sigurdur Helgason in 1965. Het gebied waar een functie ongelijk is aan 0 wordt de drager van die functie genoemd. De stelling doet een uitspraak over de drager van de functie in termen van de drager van de Radongetransformeerde functie en wordt daarom een dragerstelling genoemd.

Figuur 3: Links een bol, rechts een hyperboloïde

In dit proefschrift wordt een dragerstelling bewezen voor een andere Radontransformatie dan degene die hierboven beschreven is. In de eerste plaats moet de “gewone” driedimensionale ruimte die wij om ons heen zien vervangen worden door een zogenaamde symmetrische ruimte. Zoals de naam al doet vermoeden is een symmetrische ruimte een object met veel symmetrieën. Voorbeelden van sym-

129

Samenvatting

metrische ruimten zijn boloppervlakken en hyperboloïden. Op een symmetrische ruimte kan meetkunde worden bedreven. In iedere symmetrische ruimte bestaat een bijzondere collectie van meetkundige objecten genaamd horosferen. In veel gevallen zijn deze horosferen interessante objecten. In plaats van lijnen in de “normale” 3-dimensionale ruimte, worden in dit proefschrift horosferen in een symmetrische ruimte beschouwd. De Radontransformatie neemt een functie, die aan ieder punt in de symmetrische ruimte een getal toekent, en vormt die om in de functie op de verzameling van horosferen, die aan iedere horosfeer de waarde van de integraal van de functie over die horosfeer toekent. Deze Radontransformatie wordt ook wel de horosferische transformatie genoemd. Er is een belangrijke collectie van symmetrische ruimten waarop op een natuurlijke manier over afstanden gesproken kan worden. Dit zijn de zogenaamde Riemannse symmetrische ruimten. Niet alle symmetrische ruimten hebben echter een dergelijk natuurlijk afstandsbegrip. In 1973 bewees Sigurdur Helgason een dragerstelling voor de horosferische transformatie op een Riemannse symmetrische ruimte. De stelling luidt als volgt. Stel B is een bol in een Riemannse symmetrische ruimte. Als de horosferische transformatie van een functie gelijk is aan 0 op iedere horosfeer die B niet raakt, dan is de oorspronkelijke functie gelijk aan 0 buiten de bol B. De dragerstelling, die in dit proefschrift wordt bewezen, is een generalisatie van deze stelling voor horosferische transformaties op symmetrische ruimten die niet noodzakelijk Riemanns zijn.

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Dankwoord De totstandkoming van dit proefschrift is niet alleen het werk van de auteur; integendeel. Velen hebben bewust of onbewust een bijdrage geleverd. Een aantal van hen wil ik op deze plaats in het bijzonder bedanken. Allereerst en voornamelijk gaat mijn dank uit naar mijn promotor Erik van den Ban. De vele discussies met jou, Erik, je scherpe blik en het vele commentaar dat je geleverd hebt, hebben mij zeer geholpen. Het was voor mij een eer jou als leermeester te mogen hebben. Ook mijn vroegere leermeester Joop Kolk ben ik dank verschuldigd. Dankzij de hulp van Joop en Erik heb ik deze promotieplaats kunnen krijgen. Joop, de vele gesprekken die wij gehad hebben kon ik altijd zeer waarderen en ik denk er met genoegen aan terug. Toen Joop en Erik een subsidie van NWO toegewezen kregen voor het project dat geresulteerd heeft in dit proefschrift, bleek dat dit via NWO zou worden bekostigd door het Van Beuningen-Peterich Fonds, waarvoor ik bij deze mijn dank uitspreek. De afname van investeringen van de overheid in de wetenschap gedurende de afgelopen jaren, maakt dat promotieplaatsen die gefinancierd worden door particuliere instellingen zeer waardevol zijn. Ik hoop dat dit initiatief door velen gevolgd zal worden. Het werken aan het Mathematisch Instituut in Utrecht is altijd plezierig geweest, mede dankzij mijn collega-AiO’s. Hiervoor wil ik hen bedanken. Een aantal AiO’s wil ik bij name noemen. In mijn ochtendritueel kwam gewoonlijk, na het halen van koffie, een bezoek aan de kamer van Bart voor. Onder het genot van een koekje uit de trommel hebben wij het wel en wee van het AiO-schap en vele, vele andere zaken besproken. Ik heb nooit het gevoel gehad een nuttige bijdrage te kunnen leveren aan de wiskundige problemen waar Bart mee worstelt, maar hij heeft mij wel vaak kunnen helpen. De lezer kan een voorbeeld hiervan vinden in Remark 4.4.3. Bart, ik ben blij dat je hebt toegestemd om op te treden als mijn paranimf.

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Het is niet altijd makkelijk om met anderen over je onderzoek te spreken, zelfs niet met wiskundigen. Nadat Vincent promoveerde in december 2009 en elders ging werken, merkte ik pas hoe fijn het is vragen te kunnen stellen aan iemand die dezelfde taal spreekt. When I started as a PhD-student, I shared a room on the sixth floor of the Wiskunde Gebouw with Steven, Liesbeth, Bas and Taoufik. After some time we had to leave this office and I was moved to a different one at the fifth floor. This room I shared with Albert-Jan and Alexander. After Alexander had his defense and started working in Scotland, it was decided that we had to leave and Albert-Jan and I were moved to a room on the fourth floor, next to the one of Vincent and Charlene. A few months later the fourth floor had to be abandoned by the Mathematical institute. Hence it was decided that we had to leave our office and Albert-Jan and I were moved to the seventh floor. I would like to thank all my previous roommates for the great times I had with them in between moving. Albert-Jan is met bijna 4 jaar het langst mijn kamergenoot geweest. Hij heeft mij in die tijd kennis laten maken met zijn favoriete Chinese restaurant en van hem heb ik geleerd dat de Bruhatdecompositie van GL.n; R/ in sommige kringen de LU -decompositie wordt genoemd (hoewel strikt gezien de laatste natuurlijk alleen gedefinieerd is op de open Bruhatcel en dus zeker niet bestaat voor alle inverteerbare matrices). AJ, onze werktijden hadden een niet al te grote overlap, maar het was altijd gezellig als je er was. Hiervoor mijn dank. Ten slotte wil ik mijn vrienden en mijn familie bedanken voor hun steun de afgelopen jaren. Met name wil ik Nico noemen. Hoewel ik mij moeilijk kan voorstellen dat de werken van Aldegonde, Van den Vondel en Hooft mij ooit zullen gaan vervelen, was het mij toch altijd weer een groot genoegen om mijn boeken op hun plaats in mijn boekenkasten te laten staan en een avond bij Nico en Annet door te brengen. Het verheugt mij dan ook zeer dat Nico heeft toegezegd mij als paranimf bij te staan. Nico, mijn hartelijke dank daarvoor. Niet alleen de afgelopen jaren, maar zo lang ik mij kan herinneren hebben mijn ouders mij gesteund en gemotiveerd. De wiskunde in het voorgaande gedeelte mag jullie niet veel zeggen, maar aan jullie heb ik het in hoge mate te danken dat ik dit proefschrift heb kunnen schrijven.

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Curriculum Vitae Job Kuit werd op 13 januari 1983 geboren te Ede. Zijn Atheneumdiploma behaalde hij aan het Ichthus college in Veenendaal in 2001. Daarna studeerde hij zowel wiskunde als theoretische natuurkunde aan de universiteit van Utrecht. In 2007 studeerde hij in beide disciplines cum laude af. In zijn afstudeerwerk, dat geschreven werd onder het toeziend oog van Joop Kolk, werd de basis gelegd voor zijn latere promotieonderzoek. Omdat het Job in de laatste jaren van zijn studie zeer duidelijk was geworden dat zijn interesse niet lag in de fysica, maar des te meer in de wiskunde, begon hij in augustus 2001 zijn promotieonderzoek aan het Mathematisch Instituut van Universiteit Utrecht in de theorie van Radontransformatie op symmetrische ruimten. Dit onderzoek werd uitgevoerd onder begeleiding van Erik van den Ban en heeft geresulteerd in dit proefschrift. Gedurende de maanden augustus en september van 2011 zal Job deelnemen aan de workshop Analysis on Lie groups in het Max Planck Institute in Bonn, waarna hij in oktober 2011 zal gaan werken als postdoc aan de universiteit van Kopenhagen.

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