Oct 14, 2013 - K = GÎ¸ and let Ko = K â©Go be the maximal compact subgroup corre- sponding to Î¸. Then X = Go/Ko is a Riemannian symmetric space of...

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THE RADON TRANSFORM AND ITS DUAL FOR LIMITS OF SYMMETRIC SPACES ´ JOACHIM HILGERT AND GESTUR OLAFSSON Abstract. The Radon transform and its dual are central objects in geometric analysis on Riemannian symmetric spaces of the noncompact type. In this article we study algebraic versions of those transforms on inductive limits of symmetric spaces. In particular, we show that normalized versions exists on some spaces of regular functions on the limit. We give a formula for the normalized transform using integral kernels and relate them to limits of double fibration transforms on spheres.

1. Introduction Let Go be a classical noncompact connected semisimple Lie group and G its complexification. We fix a Cartan involution θ : Go → Go on Go and denote the holomorphic extension to G by the same letter. Let K = Gθ and let Ko = K ∩ Go be the maximal compact subgroup corresponding to θ. Then X = Go /Ko is a Riemannian symmetric space of the noncompact type. The space X is contained in its complexification Z = G/K. The subscript o will be used to denote subgroups in Go . Dropping the index will then stand for the corresponding complexification in G. Let Po = Mo Ao No be a minimal parabolic subgroup of Go with Ao ⊂ {a ∈ Go | θ(a) = a−1 } and Mo = ZKo (Ao ). The space Ξo = Go /Mo No is the space of horocycles in X. We denote base point in X by xo = {Ko } and the base point {Mo No } in Ξo by ξo . The (horospherical) Radon transform is the integral transform, initially defined on compactly supported functions on X, given by Z f (gn · xo ) dn R(f )(g · ξo ) = No

for a certain normalization of the invariant measure dn on No . The dual transform R∗ maps continuous functions on Ξo to continuous functions Date: October 13, 2013. ´ The research of G. Olafsson was supported by DMS-1101337. 1

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´ JOACHIM HILGERT AND GESTUR OLAFSSON

on X and is given by ∗

R (ϕ)(g · xo ) =

Z

ϕ(gk · ξo ) dk, Ko

where dk denotes the invariant probability measure on Ko . If f and ϕ are compactly supported, then Z Z R(f )(ξ)ϕ(ξ) dξ = f (x)R∗ (ϕ)(x) dx Ξ

X

for suitable normalizations of the invariant measures on X, respectively Ξo . This explains why R∗ is called the dual Radon transform. For more detailed discussion we refer to Section 5.2. For a complex subgroup L ⊂ G we call a holomorphic function f : G/L → C regular if the orbit G · f with respect to the natural representation spans a finite dimensional subspace. We denote the Gspace of regular functions by C[G/L]. If Lo ⊂ Go is a subgroup such that Go /Lo can be viewed as a real subspace of its complexification G/L, then one calls a smooth function on Go /Lo regular, if its Go -orbit spans a finite dimensional space. Since there is a bijection between regular functions on Go /Lo and G/L we restrict our attention to C[G/L]. The dual Radon transform can be extended to the space of regular functions on Ξ but the integral defining the Radon transform is in general not defined for regular functions. In fact, a regular function on Z can be No -invariant so the integral is infinite. This problem was first discussed in [HPV02] and then further developed in [HPV03]. Let us describe the main idea from [HPV02] here. We refer to main body of the article for more details. Denote the spherical representation of Go and G with highest weight µ ∈ a∗o by (µµ , Vµ ), and its dual by (πµ∗ , Vµ∗ ). The duality is written hw, νi. Note that (πµ , Vµ ) is unitary on a compact real form U which we choose so that U ∩ Go = Ko . We fix a highest weight vector uµ ∈ Vµ of length one and a K-fixed vector e∗µ ∈ Vµ∗ such that huµ , e∗µ i = 1. Fix a highest weight vector u∗µ in Vµ∗ such that huµ , πµ∗ (so )u∗µ i = 1, where so ∈ K represents the longest Weyl group element. For w ∈ Vµ and g ∈ G let fw,µ(g · xo ) := hw, πµ∗ (g)e∗µ i

and

ψw,µ (g · ξo) := hw, πµ∗ (g)u∗µi .

Every regular function on X is a finite linear combination of functions of the form fw,µ and similarly for Ξ. The normalized Radon transform on the space C[Z] of regular functions on Z can now be defined by Γ(fw,µ ) := ψw,µ .

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

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The transform Γ−1 (ψw,µ ) := fw,µ defines a G-equivariant map C[Ξ] → C[X] which is inverse to Γ. Restricted to each G-type, the transform Γ−1 is, up to a normalization given in Lemma 5.5, the dual Radon transform R∗ . It is also shown in [HPV03], the dual Radon transform on C[Ξ] can be described as a limit of Radon transform over spheres, see Section 5.3 for details. Our aim in this article is to study the normalized transforms Γ and −1 Γ as well as their unnormalized counterparts for certain inductive limits of symmetric spaces Xj ⊂ Zj , called propagations of symmetric spaces, introduced in Section 3.1. This study is based on results ´ ´ ´ from [OW11a, OW11b] and [DOW12] on inductive limits of spherical representations, which we use to study spaces of regular functions on the limit. More precisely, in Section 4 we consider two such spaces of regular functions, the projective limit lim C[Zj ] and the inductive ←− limit Ci [Z∞ ] = lim C[Zj ]. The first main result is Theorem 4.19 which −→ describes how the graded version of Γ extends to the projective limit. We introduce the Radon transform and its dual in Section 5 and in Section 5.3 we recall the results from [HPV03] about the Radon transform as a limit of a double fibration transform associated the spheres in X. In Section 5.4 we show that the normalized Radon transform and its dual can be represented as an integral transform against kernel functions. Here the integral is taken over the compact group U. The corresponding result for the direct limit is Theorem 5.16. Many of the results mentioned so far are valid for propagations of symmetric spaces of arbitrary rank, which means that they apply also to the case of infinite rank. For some results, however, we have to require that the rank of the symmetric space lim Xj is finite. This is −→ the case in particular in Section 5.5, where we define the dual Radon transform R∗ for spaces of finite rank and connect it to the normalized dual Radon transform Γ−1 , see Theorem 5.22. Moreover, we define the Radon transform over spheres in this context, and connect it to the dual Radon transform in Theorem 5.25. acknowledgement. We would like to thank E. B. Vinberg, who suggested to us that dual horospherical Radon transforms may exist also for limits of symmetric spaces. Contents 1. Introduction 2. Finite dimensional geometry 2.1. Lie groups and symmetric spaces

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2.2. Group spheres 2.3. Spherical representations 2.4. Regular functions 3. Limits of symmetric spaces and spherical representations 3.1. Propagation of symmetric spaces 3.2. Inductive limits of spherical representations 4. Regular functions on limit spaces 4.1. Regular functions on Z∞ 4.2. Regular functions on Ξ∞ 4.3. The projective limit 5. The Radon transform and its dual 5.1. The double fibration transform 5.2. The horospherical Radon transform and its dual 5.3. The Radon transform as limit of integration over spheres 5.4. The kernels defining the normalized Radon transform and its dual 5.5. The Radon transform and its dual on the injective limits References

6 7 9 10 12 13 16 16 19 20 23 23 24 26 28 30 34

2. Finite dimensional geometry In this section we recall the necessary background from structure theory of finite dimensional symmetric spaces and related representation theory. Most of the material is in this section is standard, but we use this section also to set up the notation for later sections. 2.1. Lie groups and symmetric spaces. Lie group will always be denoted by uppercase Latin letters and their Lie algebra will be denoted by the corresponding lower case German letters. If G and H are Lie groups and θ : G → H is a homomorphism, then the derived homomorphism is denoted by θ˙ : g → h. If G = H, then Gθ = {a ∈ G | θ(a) = a} and gθ = {X ∈ g | θ(X) = X} . From now on G will stand for a connected simply connected complex semisimple Lie group with Lie algebra g. Let U be a compact real form of G with Lie algebra u and let σ˙ : g → g denote the conjugation on g with respect to u. We denote by σ : G → G the corresponding involution on G. Then, as G is simply connected, U = Gσ and U is simply connected. Let θ : U → U be a nontrivial involution and Ko := U θ . Then Ko is connected and U/Ko is a simply connected symmetric space of

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

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the compact type. Extend θ˙ : u → u to a complex linear involution, ˙ on g. Denote by θ : G → G the holomorphic also denoted by θ, ˙ Write u = ko ⊕ qo where ko := uθ˙ and involution with derivative θ. ˙ qo := {X ∈ u | θ(X) = −X}. Let so := iqo and go := ko ⊕ so . Then go is a semisimple real Lie algebra. Denote by Go the analytic subgroup of G with Lie algebra go . Then Go is θ-stable, Gθo = Ko , and Go /Ko is a symmetric space of the noncompact type. We have Go = Gθσ . Let K := Gθ . Then Ko = K ∩ U = K ∩ Go . Let Z = G/K, X = Go /Ko , and Y = U/Ko . As σ and η := σθ map K into itself it follows that both involutions define antiholomorphic involutions on Z and we have X = Zη and Y = Zσ . In particular X and Y are transversal totally real submanifolds of Z. If V is a vector space over a field K, then V ∗ denotes the algebraic dual of V . If V is a topological vector space, then the same notation will be used for the continuous linear forms. If V is finite dimensional, then each α ∈ V ∗ is continuous. Let ao be a maximal abelian subspace of so and a = aCo . For α ∈ a∗o ⊂ a∗ let goα := {X ∈ go | (∀H ∈ ao ) [H, X] = α(H)X} and gα := {X ∈ g | (∀H ∈ a) [H, X] = α(H)X} . If α 6= 0 and gα 6= {0}, then gα = goα ⊕ igoα and gα ∩ u = {0}. The linear form α ∈ a∗ \ {0} is called a (restricted) root if gα 6= {0}. Denote by Σ := Σ(g, a) ⊂ a∗ the set of roots. Let Σ0 := Σ0 (g, a) := {α ∈ Σ | 2α 6∈ Σ}, the set of nonmultipliable roots. Then Σ0 is a root system in the usual sense and the Weyl group W corresponding to Σ is the same as the Weyl group generated by the reflections sα , α ∈ Σ0 . The Riemannian symmetric spaces X and Y are irreducible if and only if the root system L Σ0 is irreducible. Let n := α∈Σ+ gα , m := zk (a) = {X ∈ k | [X, a] = {0}} and p := m ⊕ a ⊕ n. All of those algebras are defined over R and the subscript o will indicate the intersection of those algebras with go . This intersection can also be described as the η˙ fixed points in the complex Lie algebra. Define the parabolic subgroups Po := NGo (po ) ⊂ P := NG (p). We can write Po = Mo Ao No (semidirect product) where Mo := ZK (Ao ), Ao := exp ao , and No := exp no . Similarly we have P = MAN. Let F := K ∩ exp iao . Then each element of F has order two and Mo = F (Mo )◦ where ◦ denotes the connected component containing

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the identity element. We let Ξo := Go /Mo No ⊂ Ξ := G/MN. As θσ leaves MN invariant it follows that Ξo = Ξθσ is a totally real submanifold of Ξ. Note than K ∩MN = M, so we obtain the following double fibration, which is of crucial importance for the Radon transforms: (2.1)

G/M t p ttt t t t y t t

Z = G/K

▲▲▲ ▲▲▲q ▲▲▲ ▲%

Ξ = G/MN

2.2. Group spheres. Let X = Go /Ko be as in the previous section. Denote by s the symmetry of X with respect to xo . Then θ(g) = sgs−1 for g ∈ Go . Denote by X(A) the (additively written) group × Hom(Ao , R× + ) (where R+ stands for the multiplicative group of positive numbers). Then X(A) ≃ a∗ where the isomorphism is given by µ 7→ χµ , χµ (a) = aµ . We will simply write µ(a) for χµ (a). A group sphere in X is an orbit of a maximal compact subgroup of Go . Because of the Cartan decomposition Go = Ko Ao Ko , and the fact that all maximal compact subgroups in Go are Go -conjugate, any group sphere S is of the form gKoa−1 · xo = Kog ga−1 · xo with g ∈ Go and a ∈ Ao . The point g · xo is called the center of S and a is called the radius of S. The group sphere of radius a with center at x is denoted by Sa (x). The group Go acts transitively on the set Spha X of group spheres of radius a. The stabilizer of a group sphere S := Sa (g · xo ) is a compact subgroup of G containing the stabilizer Kog of the point g · xo and hence coinciding with it. Since g·xo is the only fixed point of Kog , it is uniquely determined by S. Moreover, Sa1 (x) = Sa2 (x) if and only if a1 and a2 are W - equivalent. We shall say that a ∈ Ao tends to infinity, written a → ∞, if α(a) → ∞ for any α ∈ Σ+ . The sphere Sa := Sa (a · xo ) = K a · xo of radius a passes through xo . It is known that it converges to the horosphere ξo = No · o as a → ∞ (see, e.g., [E73], Proposition 2.6, and [E96], p. 46). Taking the sphere Sa as the base point for the homogeneous space Spha X, we obtain the representation Spha X = Go /Koa . This gives rise to the double fibration

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

Go /(Ko ∩ Koa )

7

(2.2)

❘❘❘ ❘❘❘ qa ❘❘❘ ❘❘❘ ❘(

♥ pa ♥♥♥♥ ♥ ♥ ♥ ♥ w ♥♥ ♥

Go /Koa = Spha X

X = Go /Ko

Obviously, Ko ∩ Koa = ZKo (a). If a is regular, then ZKo (a) = ZKo (Ao ) = Mo . In this case the double fibration (2.2) reduces to Go /Mo

X

p ②②② ②② ②② ② | ②

❏❏ ❏❏ qa ❏❏ ❏❏ ❏$

Spha X

2.3. Spherical representations. In this subsection we describe the set of spherical representations and the set of fundamental weights. Each irreducible finite dimensional representation π of U or Go extends uniquely to a holomorphic irreducible representation π of G and every irreducible holomorphic representation τ of G is a holomorphic extension of an irreducible representation of U and Go . We will therefore concentrate on irreducible holomorphic respresentations of G. We will denote by π U respectively π o the restriction of a holomorphic representation π to U respectively Go . For a representation π of a topological group H or a Lie algebra h we write Vπ for the vector space on which π acts. Let VπH = {u ∈ Vπ | (∀k ∈ H) π(k)u = u} . Similarly Vπh = {u ∈ Vπ | (∀X ∈ h) Xu = 0} . If H is a connected Lie group with Lie algebra h and Vπ a smooth representation of H, then h acts on Vπ and VπH = Vπh . Back to our setup, as Ko and K are connected, it follows that if π is a irreducible finite dimensional holomorphic representation of G (and hence analytic), then VπK := VπKo = Vπk = Vπko . We say that π is spherical if VπK 6= {0} and that π is conical if VπM N 6= {0}. Note that even if Mo is not connected, then VπM N = VπMo No .

´ JOACHIM HILGERT AND GESTUR OLAFSSON

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In fact, the inclusion ⊆ is trivial and for the converse it suffices to note that VπMo No ⊆ Vπmo +no =: Vπm+n . Define a representation π ∗ on Vπ∗ = Vπ∗ by hv, π ∗(g)νi := hπ(g −1)v, νi ,

g ∈ G, v ∈ Vπ , ν ∈ Vπ∗ .

For the following theorem see [H94], Thm. 4.12 and [H84], Thm. V.1.3 and Thm. V.4.1. Theorem 2.3. Let π be an irreducible holomorphic representation of G. Then the following holds: (1) π is spherical if and only if π is conical. In that case dim VπK = dim VπM N = 1 . (2) π is spherical if and only if π ∗ is spherical. Let

(µ, α) + + Λ (G, K) := µ∈ (2.4) (α, α) ∈ Z for all α ∈ Σ + + ∗ (µ, α) ∈ Z for all α ∈ Σ0 . = µ ∈ iao (α, α) +

ia∗o

We mostly write Λ+ for Λ+ (G, K). Let W = NKo (ao )/ZKo (ao ) denote the Weyl group. The parametrization of the spherical representations is given by the following theorem.

Theorem 2.5. Let π be a irreducible holomorphic representation of G, and µ its highest weight. Let wo ∈ W be such that wo Σ+ = −Σ+ . Then the following are equivalent. (1) π is spherical. (2) µ ∈ ia∗o and µ ∈ Λ+ . Furthermore, if π is spherical with highest weight µ ∈ Λ+ , then π ∗ has highest weight µ∗ := −wo µ. Proof. See [H84, Theorem 4.1, p. 535 and Exer. V.10] for the proof. If µ ∈ Λ+ , then πµ denotes the irreducible spherical representation with highest weight µ. Denote by Ψ := {α1 , . . . , αr }, r := dimC a, the set of simple roots in + Σ0 . Define linear functionals ωj ∈ ia∗o by hωi , αj i = δi,j hαj , αj i

for

1≦j≦r .

(2.6)

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

Then ω1 , . . . , ωr ∈ Λ+ and Λ+ = Z+ ω1 + . . . + Z+ ωr =

(

r X j=1

9

) nj ωj nj ∈ Z+ .

The weights ωj are called the spherical fundamental weights for (g, k). Set Ω := {ω1 , . . . , ωr }. 2.4. Regular functions. Let L be one of the groups U, Go and G. Let M be a manifold and assume that L acts transitively on M. Then L acts on functions on M by a·f (m) = f (a−1 ·m). We say that f ∈ C(M) is an L-regular function if {a·f | a ∈ L} spans a finite dimensional space which we will denote by hL · f i. We denote by CL [M] the space of Lregular functions on M. Coming back to our usual notation we remark that the restriction map defines a Go -isomorphism CG [Z] → CGo [X] and a U-isomorphism CG [Z] → CU [Y]. Similarly, restriction defines a Go -isomorphism CG [Ξ] → CGo [Ξo ]. As soon as the acting group is clear from the context we will omit it from the notation. We will mostly consider regular functions on the two complex spaces Z and Ξ. If needed, we will use results only stated or proved for the complex case also for the real cases using the above restriction maps. For µ ∈ Λ+ we denote by C[Z]µ , respectively C[Ξ]µ , the space of regular functions on Z, respectively Ξ, of type πµ . We recall the following well known fact (cf. [HPV02]): Lemma 2.7. The action of G on C[X]µ and C[Ξ]µ is irreducible. As a G-module we have M M C[Ξ]µ . C[Z]µ and C[Ξ] = C[Z] = µ∈Λ+

µ∈Λ+

Each representation πµ occurs with multiplicity one in each of those modules. Let f ∈ C[X]µ be a highest weight vector. Recall that KAN ⊂ G is open and dense. Let kan ∈ KAN. Then (kan) · f (xo ) = f (n−1 a−1 k −1 · xo ) = aµ f (xo ) where aµ = ehµ,log ai . Hence f (xo ) 6= 0. We denote by fµ the unique highest weight vector in C[X]µ with fµ (e) = 1. Let so ∈ NKo (ao ) be a representative of the longest Weyl group element wo and recall that Nso MAN is open and dense in G. Let ψ ∈ C[Ξ]µ be a highest weight vector. Then for nso man1 ∈ Nso MAN we have −1 −1 −1 −1 (nso man1 ) · ψ(ξo ) = ψ(n−1 · ξo ) = aµ ψ(s−1 1 a m so n o · ξo ) .

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2 Hence ψ(s−1 o · ξo ) 6= 0. Note that so ∈ M. As ψ is M-invariant it −1 follows that ψ(so · ξo ) = ψ(so · ξ). Let ψµ be the unique highest weight vector in C[Ξ]µ with ψµ (so · ξo) = 1. According to Lemma 2.7 there is a unique G-intertwining operator Γ : C[Z] → C[Ξ] such that Γ(fµ ) = ψµ for all µ ∈ Λ+ . For reasons which will become clear in section 5, we call Γ the normalized Radon transform and note that its inverse Γ−1 : C[Ξ] → C[Z] is the unique G-isomorphism such that ψµ 7→ fµ for −1 all µ ∈ Λ+ . Let Γµ = Γ|C[Z]µ . Then Γ−1 µ = Γ |C[Ξ]µ . The maps Γµ and Γ−1 µ have a simple description in terms of the representation (πµ , Vµ ). Fix for all ν ∈ Λ+ a K-fixed vector eν ∈ Vν and a highest weight vector uν in Vν . Further, choose the highest weight vector u∗ν ∈ Vν∗ and the spherical vector e∗µ ∈ Vν∗ according to the normalization

huν , πν∗ (so )u∗ν i = 1 and huµ , e∗µ i = 1 . Then, for v ∈ Vµ fv,µ (aK) := hv, πµ∗ (a)e∗µ i and ψv,µ (aMN) := hv, πµ∗ (a)u∗µ i .

(2.8)

defines regular function fv,µ on Z, respectively ψv,µ on Ξ. Furthermore, Vµ ∋ v 7→ fv,µ ∈ C[Z]µ

and

Vµ ∋ v 7→ ψv,µ ∈ C[Ξ]µ

are G-isomorphisms. Note that fµ := fuµ ,µ respectively ψµ := ψuµ ,µ are normalized highest weight vectors. 3. Limits of symmetric spaces and spherical representations In this section we introduce the notion of propagation of symmetric spaces and describe the construction of inductive limits of spherical ´ ´ representations from [OW11a, OW11b]. We then recall the main result ´ from [DOW12] about the classification of spherical representations in the case where U∞ /Ko∞ has finite rank. We start with some facts and notations for limits of topological vector spaces, which will always be assumed to be complex, locally convex and Hausdorff. Similar notations for limits will be used for Lie groups and even sets without further comments. Our standard reference is Appendix B in [HY00] and the reference therein. If W1 ⊂ W2 ⊂ · · · is an injective sequence of vector spaces, then we denote the inclusion maps Wj ֒→ Wk , k ≥ j, by ιk,j . Let W∞ := lim Wj = −→

∞ [

j=1

Wj

(3.1)

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

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and denote by ι∞,j the canonical inclusion Wj ֒→ W∞ . If each of the spaces Wj is a topological vector space and each of the maps ιk,j is continuous then a set U ⊆ W∞ is open in the inductive limit topology on W∞ if and only if U ∩Wj is open for all j. Then W∞ is a (again locally convex and Hausdorff) topological vector space. If {Wj } and {Vj } are inductive sequences of topological vector spaces and Tj : Wj → Vj is a family of continuous linear maps such that ιk,j ◦ Tj = Tk ◦ ιk,j where the first inclusion is the one related to the sequence {Vj } and the second one is the one associated to {Wj }. Then there exists a unique continuous linear map T∞ = lim Tj : W∞ → V∞ such that −→ ι∞,j ◦ T∞ = T∞ ◦ ι∞,j ◦ T∞ for all j. If W is a locally convex Hausdorff complex topological vector space, then W ∗ will denote the space of continuous linear maps W → C. We provide it with the weak ∗-topology, i.e., the weakest topology that makes all the maps W ∗ → C, f 7→ hx, f i := f (x), x ∈ W , continuous. Then W ∗ is also a locally convex Hausdorff topological vector space. If {Wj } is a inductive sequence of locally convex Hausdorff topological vector spaces then {Wj∗ }, with the projections projj,k : Wk∗ → Wj∗ , projj,k (ν) = ν|Wj , k ≥ j, is a projective sequence of locally convex Hausdorff topological vector spaces. Denote the pro∗ jective limit of those spaces by lim Wj∗ = W∞ . This notation is justified ←− by the fact that the topological dual of W∞ is lim Wj∗ . We denote by ←− ∗ → Wj the restriction map. If {Wj } and {Vj } are injective projj,∞ : W∞ sequences of topological vector spaces and Tj : Wj → Vj is as above, ∗ ∗ then there exists a unique linear map T∞ = lim Tj∗ : V∞∗ → W∞ such ← − ∗ ∗ ∗ that projj,∞ ◦ T∞ = Tj ◦ projj,∞ for all j. In fact T∞ is just the adjoint of T∞ . We finish the subsection with a simple lemma that connects the inductive limit and the projective in case we have a injective sequence of Lie groups Gj and Gj -modules Vj . This will be used several times later on. We leave the simple proof as an exercise for the reader. Lemma 3.2. Let {Gj } be an injective sequence of Lie groups and let {Vj } be a projective sequence of Gj -modules with Gj -equivariant projections projj,k : Vk → Vj . Assume that we have Gj -equivariant inclusions ιk,j : Vj → Vk making {Vj } into an injective sequence and such that projj.k ◦ ιk,j = idVj . For f ∈ lim Vj , fix j such that f ∈ Vj . Define −→ ι∞ (f ) := {ιk+1,k (f )}k≥j . Then lim Vj and lim Vj are G∞ -modules and −→ ←− ι is a well defined G∞ -equivariant embedding lim Vj ֒→ lim Vj . −→ ←−

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3.1. Propagation of symmetric spaces. Assume that G1 ⊆ G2 ⊆ . . . ⊆ Gk ⊆ Gk+1 ⊆ . . . is a sequence of connected, simply connected classical complex Lie groups as in the last section. In the following an index k (respectively j) will always indicate objects related to Gk (respectively Gj ). We assume that θk |Gj = θj and σk |Gj = σj for all j ≤ k. Then Kj = Gj ∩ Kk , Uj = Gj ∩ Uk , and Gjo = Gj ∩ Gko , for k ≥ j. This gives rise to an increasing sequence {Zj = Gj /Kj }j≧1 of simply connected complex symmetric spaces such that for k ≥ j the embedding Zj ֒→ Zk is a Gj -map. We denote this inclusion by ιk,j and note that {Zj } is an injective system. Similarly we have a sequence of transversal real forms Xj = Gjo /Kjo and Yj = Uj /Kjo . We set G∞ := lim Gj , K∞ := lim Kj and Z∞ := lim Zj = G∞ /K∞ −→ −→ −→ and similarly for other[ groups and symmetric spaces. Recall that, as a set we have G∞ = Gj but the inductive limit comes also with the inductive limit topology and a Lie group structure. The space [ Z∞ = Zj is a smooth manifold and the action of G∞ is smooth. Similar comments are valid for the other groups and the corresponding symmetric spaces. In the following we will always assume that k ≥ j and m ≥ n. As θk |Gj = θj it follows that kk ∩ gj = kj and sk ∩ gj = sj . We choose the sequence {aj } of maximal abelian subspaces of sj such that ak ∩ sj = aj . Then Σj ⊆ Σk |aj \ {0}. The ordering in ia∗o is chosen so + that Σ+ j ⊆ Σk |ajo \ {0}. In case each Xj is irreducible we say that Xk propagates Xj if (i) ak = aj , or (ii) we obtain the Dynkin diagram for Ψk is obtained from the Dynkin diagram for Ψj by only adding simple roots at the left end (so the root α1 stays the same). Note, that usually the Dynkin diagram is labeled so that the first simple root is at the left and. We have here reversed that labeling. Then, in particular, Ψk = {αk,1, . . . , αk,rk } and Ψj = {αj,1, . . . , αj,rj } are of the same type. Furthermore αk,s |aj = αj,s ´ for s = 1, . . . , rj , see [OW11a]. Furthermore, if s ≥ rj + 2, then αk,s |aj = 0. In case of reducible symmetric spaces Xt = X1t × · · · Xst t we say that Xk propagates Xn , k ≥ n, sk ≥ sn and we can arrange the irreducible components so that Xjk propagates Xjn for j = 1, . . . , sn . We say that Zk propagates Zj if Zk propagates Zj . From now on we will always assume, if nothing else is clearly stated, that the sequence {Zj } is so that Zk propagates Zj for k ≥ j.

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

13

3.2. Inductive limits of spherical representations. In this section we recall the construction of inductive limits of spherical representa´ tions from [W09] and [OW11b]. As before we assume that k ≧ j and that Zk propagates Zj . Moreover, from now on we will always assume that the groups Gj are simple. Denote by rj,k : a∗k → a∗j the projection rj,k (µ) = µ|aj . As shown in ´ ´ [OW11a, OW11b] we have Lemma 3.3. If k ≥ j, then rj,k (ωk,s) = ωj,s for s = 1, . . . , rj . + This implies that the sets of highest weights Λ+ k := Λ (Gk , Kk ) form a projective system with restrictions as projections. But those sets also form an injective system as we will now describe. This will allow us to construct an injective system of representations in an unique way, starting at a given level jo . Let µj ∈ Λ+ j and write

µj =

rj X

k s ∈ N0 .

ks ωj,s ,

s=1

Define µk ∈

Λ+ k

by

µk :=

rj X

ks ωk,s .

s=1

+ The map ιk,j : Λ+ j → Λk , µj 7→ µk is well defined and injective ιk,n ◦ ιn,j = ιk,j for j ≤ n ≤ k. We also have

rj,k ◦ ιk,j = id .

(3.4)

Finally, µk is the minimal element in r−1 j,k (µj ) with respect to the partial P + ordering ν−µ = j kj ωk,j , kj ∈ Z . In particular we have the following lemma: Lemma 3.5. The sequence {Λ+ j }j with the maps ιk,j :

Λ+ j

→

Λ+ k

,

rj X s=1

ks ωsj

7→

rj X

ks ωsk

s=1

is an injective sequences of sets. Furthermore, there is a canonical inclusion Λ+ Λ+ ֒→ lim Λ+ . ∞ := lim −→ j ←− j Proof. Most of the proof has been given already. For the last statement the idea is the same as in Lemma 3.2. Given j and µj ∈ Λ+ . Then, by (3.4) the sequence (µj , µj+1, . . .) is in lim Λ+ . ←− j

14

´ JOACHIM HILGERT AND GESTUR OLAFSSON

For j ∈ N and µ = µj ∈ Λ+ j define µk = ιk,j (µ), k ≥ j, and + µ∞ = lim µj ∈ Λ∞ . Let (πµj , Vµj ) be the spherical representation of Gj −→ with highest weight µj . We can and will assume that each Vµj carries a Uj -invariant inner product such that the embeddings Vµj ֒→ Vµk are isometric. ´ Theorem 3.6 (O-W). Assume that Zk propagates Zj . Let µj ∈ Λ+ j + and define µk ∈ Λk as above. Then the following holds: (1) Let uµk ∈ Vµk be a weight vector chosen as before, let Wj := < πk (Gj )uµk > and πk,j (g) := πk (g)|Wj , g ∈ Gj . Then πk,j is equivalent to πµj and we can choose the highest weight vector uµj in Vµj so that the linear map generated by πµj (g)uµj 7→ πµk (g)uµk is a unitary Gj -isomorphism. (2) The multiplicity of πµj in πµk is one. ´ Remark 3.7. Note that in [OW11a] the statement was proved for the compact sequence {Uj }. But it holds true for the complex groups Gj by holomorphic extension. It is also true for the non-compact groups Gjo by holomorphic extension and then restriction to Gjo . The second half of the above theorem implies that, up to a scalar, the only unitary Gj -isomorphism is the one given in part (1). As a consequence we can and will always think of Vµj as a subspace of Vµk such that the highest weight vector u is independent of j, i.e., uµj = uµk for all k and j. We form the inductive limit (3.8) Vµ∞ := lim Vµj . −→ Starting at a point jo the highest weight vector uµj , j ≥ jo is constant and contained in all Vµj . I particular, µµj ∈ Vµ∞ . We also note that {πµj (g)}, g ∈ Gj , forms a injective sequence of continuous linear operators, unitary for g ∈ Uj . Hence (πµk )∞ (g) : V∞ → V∞ is a well defined continuous map. Similarly for the Lie algebra. We denote those maps by πµ∞ and dπµ∞ respectively. Hence group G∞ and acts continuously, in fact smootly, on Vµ∞ . We denote the corresponding representation of G∞ by πµ∞ . We have dπµ∞ (H)uµ∞ = µ∞ (H)uµ∞

for all

H ∈ a∞ .

The representation (µµ∞ , Vµ∞ ) is (algebraically) irreducible. We can make πµ∞ |U∞ unitary by completing Vµ∞ to an Hilbert space Vˆ∞ as is ´ usually done, see [DOW12]. The dual of Vµ∞ is given by the corresponding projective limit Vµ∗∞ = lim Vµ∗j . ←−

(3.9)

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

15

Note that in this notation Vµ∗∞ 6= Vµ∗∞ . We note that the highest weight vector, which we now denote by uµ∞ in Vµ∞ . If g ∈ Gk , k ≥ j, then πµ∗j (g) forms a projective family of operators and hence lim πµ∗k is a ←− well defined continuous representation of G∞ on Vµ∗∞ . We denote this representation by πµ∗∞ . Lemma 3.10. Let the notation be as above. Then K∞ ∗ = 1. dim lim Vµj ←−

Proof. First fix jo so that Vµj is defined for all j ≥ jo , i.e. {µj } stabilizes ∗K from jo on. As dim Vµj j = 1, j ≥ jo , there exists a unique, up to scalar, Kjo -fixed element e∗µjo . We fix e∗µj now so that projjo ,j (e∗µj ) = eµ∗jo , where projj0 ,j is the dual map of Vµjo ֒→ Vµj . Then e∗µ∞ := {e∗µj }j≥jo ∈ K∞ K∞ , then . On the other hand, if {e∗µj }j≥jo ∈ lim Vµ∗j lim Vµ∗j ←− ←− ∗ K e∗µjo ∈ Vµjo j is unique up to scalar showing that the dimension is one. From now on we fix e∗µ∞ so that huµ∞ , e∗µ∞ i = 1. Theorem 3.11. Vµ∗∞ is irreducible. Proof. Assume that W ⊂ Vµ∗∞ is a closed G∞ -invariant subspace. Then W ⊥ = {u ∈ Vµ∞ | (∀ϕ ∈ W ) hu, ϕi = 0} is closed and G∞ invariant. Hence W ⊥ = {0} or W = V∞ , and since all spaces involved are reflexive, this implies that W = V∞∗ or W = {0}.

The vector uµ∞ ∈ Vµ∞ is clearly M∞ N∞ -invariant. Therefore Vµ∞ is ´ conical (see [DO13]). But it is easy to see that with exception of some trivial cases (as Gj = Gk for all j and k, which we do not consider) the representation (πµ∞ , Vµ∞ ) is not K∞ -spherical. In fact, suppose that e ∈ Vµ∞ is a non-trivial, K∞ -invariant vector. Let j be so that e ∈ Vµj . Then e has to be fixed for all Ks , s ≥ j and hence a multiple of es . This is impossible in general as will follow from Lemma 5.5. On the other ´ hand it was shown in [DOW12] that the Hilbert space completion Vˆµ∞ is K∞ -spherical if and only if the dimension of a∞ is finite. In this case we can assume that aj = a∞ for all j. Then Σj = Σk = Σ∞ , + + + + + Σ+ j = Σk = Σ∞ and Λj = Λk = Λ∞ for k ≥ j. But we still use the notation µj etc. to indicate what group we are using. ´ Theorem 3.12 ([DOW12]). Let the notation be as above and assume K∞ ˆ that µ 6= 0. Then V∞ 6= {0} if and only if the ranks of the compact Riemannian symmetric spaces Xk are bounded. Thus, in the case where Xj is an irreducible classical symmetric space, we have V∞K∞ 6= {0}

16

´ JOACHIM HILGERT AND GESTUR OLAFSSON

only for SO(p + ∞)/SO(p) × SO(∞), SU(p + ∞)/S(U(p) × U(∞)) and Sp(p + ∞)/Sp(p) × Sp(∞) where 0 < p < ∞. Let as usually ιk,j : Vµj → Vµk be the inclusion defined in Theorem 3.6 and projj,k : Vµk → Vµj the orthogonal projection. Then, as Vµj ≃< πµk (Gj )uµk >, it follows that projj,k ◦ ιk,j = idVµj . By Lemma 3.2 there is a canonical G∞ -inclusion Vµ∞ ֒→ lim Vµj . ←− Define u∗µj ∈ Vµ∗j by hπµj (so,j )uµj , u∗µj i = 1

and

u∗µj |(πµ (Gj )uµj )⊥µ = 0 j

where so,j ∈ NKj (aj ) is so that Ad(s∗,j ) maps the set of positive roots into the set of negative roots and (πµ (Gj )uµj )⊥ µj is the orthogonal complement in Vµj . We use the inner product to fix embeddings Vµ∗j ֒→ Vµ∗k for j ≤ k. Then again, we can take u∗µj independent of j, which defines an M∞ N∞ -invariant element in lim Vµ∗j ⊂ lim Vµ∗j = Vµ∗∞ . As e∗µ∞ is −→ ←− K∞ -invariant, it follows that Vµ∗∞ is both spherical and conical. We now give another description of the representations (πµ∞ , Vµ∞ ). ´ This material is based on [DO13]. We say that a representation (π, V ) of G∞ is holomorphic if π|Gj is holomorphic for all j. ´ Theorem 3.13 ([DO13]). Assume that X∞ has finite rank. If µ∞ ∈ + + Λ∞ = Λ , then (πµ∞ , Vµ∞ ) is irreducible, conical and holomorphic. Conversely, if (π, V ) is an irreducible conical and holomorphic representation of G∞ , then there exists a unique µ∞ ∈ Λ+ ∞ such that (π, V ) ≃ (πµ∞ , Vµ∞ ). 4. Regular functions on limit spaces In this section we study the spaces lim C[Zj ] and lim C[Zj ] as well −→ ←− as their analogs for Ξ∞ . Our main discussion centers around the injective limits. We only discuss the limits of the complex cases. The corresponding results for the algebras Ci [X∞ ] = lim C[Xj ] and −→ Ci [Y∞ ] := lim C[Y∞ ] can be derived simply by restricting functions −→ from Z∞ to the real subspaces X∞ respectively Y∞ . 4.1. Regular functions on Z∞ . In this section {Zj } = {Gj /Kj } is a propagated system of symmetric spaces as before. There are two natural ways to extend the notion of a regular function on finite dimensional symmetric spaces to the inductive limit of those spaces. One is to consider the projective limit C[Z∞ ] := lim C[Zj ]. The other possi←− ble generalization would be to consider the space of functions on Z∞

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

17

which are algebraic finite sums of algebraically irreducible G∞ -modules and such that each f is locally finite in the sense that for each j the space hGj · f i is finite dimensional. But as very little is known about those spaces and not all of our previous discussion about the Radon transform and its dual generalize to those spaces we consider first the space Ci [Z∞ ] := lim C[Zj ] . −→ That this limit in fact exists will be shown in a moment. Let x∞ = {K∞ } ∈ Z∞ be the base point in Z∞ . Then all the spaces S Zj embeds into Z∞ via aKj 7→ a · x∞ and in that way Z∞ = Zj . Recall from our previous discussion and Lemma 3.5 that the sets Λ+ j of highest spherical weights form an injective system and Λ+ = lim Λ+ . ∞ −→ j Each µ∞ = lim µj determines a unique algebraically irreducible (see −→ below for proof) G∞ -module Vµ∞ = lim Vµk such that the dual space −→ Vµ∗∞ = lim Vµ∗j contains a (normalized) K∞ -fixed vector e∗µ∞ normalized ←− by the condition huµ∞ , e∗µ∞ i = 1 as after Lemma 3.10. As before, we denote the G∞ -representation on Vµ∗∞ by πµ∗∞ and consider the G∞ -map Vµ∞ ֒→ space of continuous functions on G∞ given by w 7→ fw,µ∞ ,

where fw,µ∞ (a · x∞ ) := hw, πµ∗∞ (a)e∗µ∞ i .

(4.1)

Denote the image of the map (4.1) by C[Z∞ ]µ∞ . Thus C[Z∞ ]µ∞ = {fw,µ∞ | w ∈ Vµ∞ } ≃ Vµ∞ .

(4.2)

Note that the restriction of (4.1) to Vµj and Zj is the Gj -map fw,µj (x) = hw, πµ∗j (a)e∗µj i = hw, πµ∗j (a)e∗µ∞ i ,

x = a · x∞ ,

introduced in (2.8). That this is possible follows from the proof of Lemma 3.10. Hence we have a canonical Gj -map C[Zj ]µj ֒→ C[Zk ]µk for k ≥ j such that the following diagram commutes: Vµj

C[Zj ]µj /

/

Vµk

C[Zk ]µk /

/

...

...

P P As C[Zj ] = ⊕ C[Zj ]µj and C[Zk ] = ⊕ C[Zk ]µk one derives that the spaces C[Zj ] form an injective system. Note that lim C[Zj ]µj and −→ Ci [Z∞ ] := lim C[Zj ] carry natural G∞ -module structures. This proves −→ part of the following theorem:

18

´ JOACHIM HILGERT AND GESTUR OLAFSSON

Theorem 4.3. The space C[Z∞ ]µ∞ is an algebraically irreducible G∞ module, and C[Z∞ ]µ∞ = lim C[Zj ]µj ≃ Vµ∞ . −→ Furthermore X⊕ C[Z∞ ]µ∞ Ci [Z∞ ] = µ∞ ∈Λ+ ∞

as a G∞ -module

Proof. (See also [KS77, Thm. 1] and [O90, §1.17].) Everything is clear except maybe the irreducibility statement. For that it is enough to show that Vµ∞ is algebraically irreducible. So let W ⊂ Vµ∞ be G∞ invariant. If W 6= {0}, then we must have W ∩ Vµj 6= {0} for some j. But then W ∩ Vµk 6= {0} for all k ≥ j and W ∩ Vµk is Gk -invariant. As Vµk is algebraically irreducible it follows that Vµk ⊂ W for all k ≥ j. This finally implies that W = Vµ∞ . Remark 4.4. In the case where the real infinite dimensional space X∞ has finite rank the space Ci [Z∞ ] has a nice representation theoretic de+ scription. In this case, as mentioned earlier, we may assume Λ+ ∞ = Λj . We have also noted that each of the spaces Vµj is a unitary representation of Uj such that the embedding Vµj ֒→ Vµk is a Gj -equivariant isometry and the highest weight vector uµj gets mapped to the highest weight vector uµk . In that way we have uµ∞ = uµj for all j. Furthermore this leads to a pre-Hilbert structure on Vµ∞ so that Vµ∞ can be completed to a unitary irreducible K∞ -spherical representation Vˆµ∞ ´ ´ of G∞ (see [DOW12, Thm. 4.5]). Furthermore, it is shown in [DO13] that each unitary K∞ -spherical representation (π, Wπ ) of G∞ such that π|Uj extends to a holomorphic representation of Gj for each j is locally finite and of the form Vˆµ∞ for some µ∞ ∈ Λ+ . Moreover, each of those representations is conical in the sense that VˆµM∞∞ N∞ 6= {0}. Finally, each irreducible unitary conical representation (π, Wπ ) of G∞ , whose restriction to U∞ whose restriction to Uj extends to a holomorphic representation of Gj is unitarily equivalent to some Vˆµ∞ . The inclusions ιk,j : Zj ֒→ Zk lead to projections on the spaces of functions given by restriction. In particular, we have the projections projj,k : C[Zk ] → C[Zj ]

(4.5)

satisfying projj,n ◦ projn,k = projj,k . Hence {C[Zj ]} is a projective sequence and lim C[Zj ] is a G∞ -module, and in fact an algebra, of ←− functions on Z∞ . In fact, let f = {fj }j≥jo ∈ lim C[Zj ] and x ∈ Z∞ . Let ←−

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

19

j be so that x ∈ Zj . Define f (x) := fj (x). If k ≥ j, then projj,k (fk ) = f |Zj . In particular fk (x) = fj (x). Hence f (x) is well defined. In general we do not have projj,k (C[Zk ]µk ) ⊂ C[Zj ]µj , but projj,k ◦ ιk,j |C[Zj ]µj = idC[Zj ]µj as this is satisfied on level of representations Vµj → Vµk → Vµj as mentioned before. Hence. by Lemma 3.2, we can view lim C[Z∞ ]µj as −→ a submodule of lim C[Zj ]µj . We record the following lemma which is ←− obvious from the above discussion: Lemma 4.6. We have a G∞ -equivariant embedding Ci [Z∞ ] ֒→ lim C[Zj ] . ←− 4.2. Regular functions on Ξ∞ . In order to construct regular functions on Ξ∞ we apply the same construction to the horospherical spaces Ξj we used for the symmetric spaces Zj . As the arguments are basically the same, we often just state the results. The following can easily be proved for at least some examples of infinite rank symmetric spaces like SL(j, C)/SO(j, C), but we only have a general prove in the obvious case of finite rank. Lemma 4.7. Assume that the rank of Zj is constant. Then for k ≥ j we have Mj = Mk ∩ Gj and Nj = Nk ∩ Gj . Definition 4.8. We say that the injective system of propagated symmetric spaces Zj is admissible if Mj = Mk ∩ Gj and Nj = Gj ∩ Nk for all k ≥ j. From now on we will always assume that the sequence {Zj } of symmetric spaces is admissible. Let ξo = eM∞ N∞ be the base point of Ξ∞ and note that we can view this as the base point in Ξj ≃ Gj · ξo ⊂ Ξ∞ . For µ∞ = lim µj ∈ Λ+ , w ∈ Vµj , ξ = a · ξo ∈ Ξj , a ∈ Gj we have −→ ψw,µj (ξ) = hw, πµ∗j (a)u∗µj i = hw, πµ∗j (a)u∗µ∞ i =: ψw,µ∞ (ξ) . This defines Gj -equivariant inclusions C[Ξj ]µj ֒→ C[Ξk ]µk ֒→ C[Ξ∞ ]µ∞ := {ψw,µ∞ | w ∈ Vµ∞ } ≃ lim C[Ξj ]µj . −→ We note that Vµ∞ → C[Ξ∞ ]µ∞ , w 7→ ψw,π∞ , is a G∞ -isomorphism. With the same argument as above this leads to an injective sequence C[Ξj ] ֒→ C[Ξk ] ֒→ lim C[Ξj ] =: Ci [Ξ∞ ] . −→

20

´ JOACHIM HILGERT AND GESTUR OLAFSSON

Theorem 4.9. Assume that the sequence {Zj } is admissible. Then X⊕ Ci [Ξ∞ ] = C[Ξ∞ ]µ∞ . µ∞ ∈Λ+ ∞

Denote by Γj the normalized Radon transform Γj : C[Xj ] = C[Zj ] → C[Ξj ] introduced in Section 2.4 and set Γ∞ (fv,µ∞ ) := ψv,µ∞ .

(4.10)

Then Γ∞ defines a G∞ -equivariant map Γ∞ : Ci [Z∞ ] → Ci [Ξ∞ ] and Γ∞ = lim Γj . −→ As each Γj is invertible it follows that Γ∞ is also invertible. In fact, the inverse is Γ−1 lim Γ−1 which maps ψv,µ∞ to fv,µ∞ . As a consequence ∞ = − → j we obtain the following theorem: Theorem 4.11. Suppose that the sequence {Zj } is admissible. Let µ∞ = lim µj ∈ Λ+ ∞ , k ≥ j, and w ∈ Vµk . Then −→ ιk,j ◦ Γj fw,µj = Γk (ιk,j ◦ fw,µj ) = ψw,µ∞ and −1 ιk,j ◦ Γ−1 j ψw,µj = Γk (ιk,j ◦ ψw,µj ) = fw,µ∞ . In particular, we have the following commutative diagram:

···

C[Zj ] /

O

/

C[Zk ] /

C[Ξj ]

/

Ci [Z∞ ] /

O

Γ−1 k

Γk ιk,j

ιk,∞

O

Γ−1 j

Γj

···

ιk,j

C[Ξk ]

Γ∞ Γ−1 ∞ ι∞,k

/

Ci [Ξ∞ ]

4.3. The projective limit. We discuss the projective limit in more detail. First we need the following notation. For µ, ν ∈ a∗o write X nα α , nα ∈ N0 . (4.12) ν ≤ µ if µ − ν = α∈Σ+

If ν ≤ µ and ν 6= µ then we also write ν < µ. The main problem in studying the projective limit is the decomposition of Vµk |Gj for k ≥ j. It is not clear if these representations decompose into representations with highest weight νj ≤ µj . In case the rank of X∞ is finite that is correct. We therefore in the reminding of this subsection make the assumption that the rank of X∞ is finite. In this case we can–and will–assume that aj = a for all j and recall from earlier discussion that + Σj = Σ is constant and so are the sets of positive roots Σ+ j = Σ and + the sets of highest weights Λ+ j = Λ .

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

Write (πµk , Vµk )|Gj ≃

r M

(πs , Ws )

21

(4.13)

s=0

with (π0 , W0 ) ≃ (πµj , Vµj ) which occurs with multiplicity one. Lemma 4.14. Assume that the rank of X∞ is finite. Let µ ∈ Λ+ . Then we have the following (1) e∗µk |Ws = 0 if Ws is not spherical. (2) Assume that Ws ≃ Vν is spherical. Then ν < µj . Proof. The first claim is obvious. Let σ be a weight of Vµk . X tα α σ = µk − α∈Ψk

with tα nonnegative integers. This in particular holds if σ is a highest weight of a spherical representation of Gj proving the claim. Lemma 4.15. Assume that the rank of X∞ is finite. Let k > j and let v ∈ Vµk . Then ψv,µ∞ |Ξj = ψv,µk |Ξj = ψprojj,k (v),µj . Proof. Let x ∈ Gj . Then πµ∗ (x)u∗µk ∈ hπµk (Gj )u∗µk i = Vµ∗j = Vµ∗j . The claim now follows as u∗µj = u∗µk . We finish this section by recalling the graded version of C[Zj ] and Γ. Recall that we are assuming that the rank of Xj is finite. Note that even if {C[Zj ]} is a projective sequence, the sequence {C[Zj ]µj } is not projective in general. Consider the ordering on a∗o as above. This defines a filtration on C[Zj ] and we denote by gr C[Zj ] the corresponding graded module. Thus M M gr C[Zj ]µj = C[Zj ]νj / C[Zj ]νj (4.16) νj <µj

νj ≤µj

and

gr C[Zj ] ≃G

M

gr C[Zj ]µj .

(4.17)

µ∈Λ+ j

If f ∈ C[Zj ] then [f ] denotes the class of f in gr C[Zj ]. Let κj , respectively κµj , be the G-isomorphism C[Zj ] ≃ gr C[Zj ], respectively C[Zj ]µj ≃ gr C[Zj ]µj , given by f 7→ [f ]. We let gr Γj := Γj ◦ κ−1 and j −1 (gr Γ)µj = Γj ◦ κµj . Note that this construction is also valid for j = ∞. Proposition 4.18 (Prop. 7 [HPV02]). The Gj -map gr Γj is an ring isomorphism gr C[Zj ] → C[Ξj ].

´ JOACHIM HILGERT AND GESTUR OLAFSSON

22

In the following we will not distinguish between Γj and gr Γj except where necessary. Thus we will prove statements for Γ and then use it for gr Γ without any further comments. Identifying functions on Gk /Kk with right Kk -invariant functions on Gk the following is clear projj,k (fv,µk ) = hv, πµ∗k |Gk e∗µk i . Thus, the kernel of projj,k is the Gj -module ker projj,k = {v ∈ Vk | v ⊥ πk (Gj )e∗µk } . As hvµ , e∗µk i = 6 0 it follows that ker projj,k is a sum of Gj -modules with highest weight < µj . Hence gr projj,k : gr C[Zk ] → gr C[Zj ] is well defined and gr projj,k (C[Zk ]µk ) = gr C[Zj ]µj . It follows that the sequence {gr C[Zj ]µj } is projective. We can also form the graded algebra gr Ci [Z∞ ] as in (4.16) and (4.17). Again we can view elements in gr Ci [Z∞ ] as functions on Z∞ by choosing the unique element in g ∈ [f ] ∈ gr Ci [Z∞ ]µ so that g ∈ Ci [Z∞ ]µ . The inclusion ⊕ X gr ιk,j : C[Zj ] = gr C[Zj ]µj → gr C[Zk ]µk = C[Zk ] given by

X

[fvj ,µj ] →

X

[fιk,j (vj ),µj ]

satisfies the relation gr projj,k ◦ gr ιk,j = id. The graded version gr Γ∞ is also well defined by the requirement that [fv,µ∞ ] is mapped into ψv,µ∞ and both gr Γ∞ and gr Γ−1 ∞ are Gj morphisms of rings. Theorem 4.19. Assume that the rank of X∞ is finite. Suppose that k ≥ j, µ ∈ Λ+ , v ∈ Vµk and w ∈ Vµj . Denote by projVµj the projection Vµk → Vµj . Then projj,k Γµk (fv,µk ) = Γµj (fprojVµ

j

(v),µj )

= ψprojVµ

j

(v),µj

(4.20)

and ιk,j Γµj (fw,µj ) = Γµk (fw,µk ) = ψw,µk . (4.21) In particular projj,k ◦ ιk,j = id. Similar statements hold for the inverse maps. Let gr Γ∞ = lim gr Γj and gr Γ∞,−1 = lim gr Γ−1 We therefore j ←− ←− have a commutative diagram:

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

··· o

o gr C[Zk ] proj O

j,k

gr Γk gr Γ−1 k

··· o

gr C[Zj ] o proj lim gr C[Zj ] ←− O O k,∞

gr Γk gr Γ−1 j

C[Ξk ] o

23

C[Ξj ] o

projk,j

gr Γ∞ gr Γ∞,−1

lim C[Ξj ] projk,∞ ← −

5. The Radon transform and its dual For the moment we fix the symmetric spaces X, Y, and Z and leave out the index j. The Radon transform or its dual is initially defined on the space of compactly supported function. As the dual Radon transform is an integral over the compact group Ko it is well defined on C[Ξ] and C[Ξo ], the space of regular functions on Ξ. But No is noncompact, so the Radon transform cannot be defined on C[X] as an integral over No . This problem was addressed in [HPV02, HPV03], and we recall ´ the main results here. Then, based on ideas from [G06, GKO06], we introduce two integral kernels which allow us to express both the Radon transform and the dual Radon transform as an integral against an integral kernel. We start the section by recalling the double fibration transform introduced in [H66, H70]. 5.1. The double fibration transform. Assume that G is a Lie group and H and L two closed subgroups. We assume that all of those groups as well as M = H ∩ L are unimodular. We have the double fibration G/M

G/H

✇ π ✇✇✇ ✇ ✇✇ {✇ ✇

(5.1)

❋❋ ❋❋ p ❋❋ ❋❋ #

G/L

where π and p are the natural projections. We say that x = aH and ξ = bL are incident if aH ∩ bL 6= ∅. For x ∈ G/H and ξ ∈ G/L we set xˆ := {η ∈ G/L | x and η are incident } and similarly ξ ∨ := {y ∈ G/H | ξ and y are incident } . Assume that if a ∈ L and aH ⊂ HL, then a ∈ H and similarly, if b ∈ H and bL ⊂ LH then b ∈ L. Then we can view the points in G/L as subsets of G/H, and similarly points in G/H can be viewed as

´ JOACHIM HILGERT AND GESTUR OLAFSSON

24

subsets of G/L. Then xˆ is the set of all η such that x ∈ η and ξ ∨ is the set of points y ∈ G/H such that y ∈ ξ. We also have xˆ = p(π −1 (x)) = aH·ξ0 ≃ H/L and ξ ∨ = π(p−1 (ξ)) = bL·xo ≃ K/L. Fix invariant measures on all of the above groups and the homogeneous spaces G/H, G/L, G/M, H/M and L/M such that for f ∈ Cc (G) we have Z Z f (a) da = f (am) dµG/M (aH ∩ L)dm G G/M Z f (ah) dµG/H (gH)dh = G/H Z = f (an) dµG/L (aL)dn G/L

and for f ∈ Cc (H) and ϕ ∈ Cc (L) Z Z Z f (h) dh = f (am) dmdµH/M (aM) H

and

Z

H/M

f (a) da =

L

Z

L/M

M

Z

f (am) dmdµL/M (aM) .

M

The definition of the Radon transform and the dual Radon transform is now as follows. Let xo = eH and ξo = eL. If ξ = a · ξo ∈ Ξ and x = b · xo ∈ X, then Z b f (ξ) := f (ab · xo ) dL/M (bM) f ∈ Cc (G/H) (5.2) L/M

and

∨

ϕ (x) :=

Z

ϕ(bh · ξo ) dH/M (hM) .

(5.3)

H/M

Then the following duality holds: Z Z fˆ(ξ)ϕ(ξ) dµG/L(ξ) = G/L

f (x)ϕ∨ (x) dµG/H (x) .

G/H

5.2. The horospherical Radon transform and its dual. The example studied most is the case G = Go , H = Ko and L = Mo No , where we use the notation from the earlier sections. In this case we find the the horospherical Radon transform which from now on we will simply call the Radon transform and its dual. The corresponding integral transforms are Z b Rf (a · ξo) = f (a · ξo ) = f (an · xo ) dn , f ∈ Cc (X) No

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

and ∗

∨

R ϕ(b · xo ) = ϕ (b · xo ) =

Z

ϕ(bk · zo ) dk ,

25

ϕ ∈ Cc (Ξo ) .

Ko

Here dk is the invariant probability measure on Ko . As mentioned earlier the dual Radon transform Z ∗ ψ(ak · ξo) dk R ψ(a · xo ) = Ko

is well defined on C[Ξ]. It is clearly a G-intertwining operator. Thus, there exists cµ ∈ C such that R∗µ := R∗ |C[Ξ]µ = cµ Γ−1 µ ,

µ ∈ Λ+ .

(5.4)

To describe the evaluation of cµ we recall the functions fµ and ψµ from Section 2. We find Z ∗ hu, πµ∗ (a)πµ∗ (k)u∗µ i dk R ψµ (a · xo ) = Ko Z −1 πµ∗ (k)u∗µ dki = hπµ (a) u, Ko

= cµ fµ (a · xo ) .

Thus cµ is determined by Z

Ko

πµ∗ (k)u∗µ dk = cµ e∗µ .

For g ∈ Go write g = k(g)a(g)n(g) with (k(g), a(g), n(g)) ∈ Ko × Ao × No . For λ ∈ a∗ and a = exp X ∈ Ao write aλ = eλ(X) . Let 1 X 1 X 1 ρ := mα α = mα/2 + mα α 2 2 2 + + α∈Σ

α∈Σ0

where mα := dimC gα . We normalize the Haar measure on N o = θ(No ) by Z a(¯ n)−2ρ d¯ n = 1. No

Define for λ ∈ a∗ (0) = {λ ∈ a∗ | (∀α ∈ Σ+ ) α(Re (λ), α) > 0} Z c(λ) := a(n)−λ−ρ d¯ n. No

∗

Then c is holomorphic on a (0). By the Gindikin-Karpelevich formula [GK62] which expresses c as a rational function in Gamma-functions depending on the multiplicities mα , the function c has a meromorphic extension to all of a∗ . The function c, which is called the HarishChandra c-function can be used to calculate the constant cµ from (5.4).

26

´ JOACHIM HILGERT AND GESTUR OLAFSSON

Lemma 5.5. Let µ ∈ Λ+ . Then cµ =

p p c(µ∗ + ρ) = c(µ + ρ).

´ Proof. See [DOW12, Thm. 3.4] or [HPV02, Thm 9].

p

´ We note that the statement in [DOW12] is that cµp= c(µ∗ + ρ). On the other hand the result in [HPV02] is that cµ = c(µ + ρ). But the Gindikin-Karpelevich formula implies that c(λ) = c(−wo λ) . p p As −wo ρ = ρ it follows that c(µ∗ + ρ) = c(µ + ρ).

5.3. The Radon transform as limit of integration over spheres. We have seen that the dual Radon transform and the normalized transform Γ−1 µ are the same up to a normalizing factor that depends on the K-representation µ. No such relation exists for Γµ and R|C[X]µ because the definition of the Radon transform R does not make sense for regular functions. However, in [HPV03] a solution was proposed by considering the Radon transform as a limit of Radon transform of a double fibrations transforms with both stabilizers being compact. Hence the corresponding integral transforms are well defined for regular functions. We recall the setup from [HPV03]. Since both Ko and Koa are compact, both the Radon transform associated to this double fibration and its dual transform are well-defined on regular functions and give G-equivariant linear maps Ra : C[X] −→ C[Spha X] and R∗a : C[Spha X] −→ C[X]. One can identify Spha X with X associating to each sphere its center. Then Ra and R∗a become linear endomorphisms of C[X]. By definition, Ra is then obtained by integrating over spheres of radius a, while R∗a is obtained by integrating over spheres of radius a−1 . The spaces X, Ξo , and Spha X can all be embedded into the algebraic dual space Y C[X]∗ = C[X]∗µ , µ∈Λ

which we equip with the product topology. Let vµ± ∈ C[X]∗µ be the highest (resp. lowest) weight vector uniquely determined by the normalizing condition hfµ± , vµ∓ i = 1, where fµ+ := fµ , and fµ− := sxo · fµ . Then we have vµ− = sxo · vµ+∗ where sxo is the symmetry around the base point xo . Similarly, let

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

vµ0

C[X]∗µ

27

Ko

be the Ko -invariant vector uniquely determined by ∈ the normalizing condition hfµ0 , vµ0 i = 1,

where fµ0 is the unique K-invariant function in C[X]µ such that fµ0 (xo ) = 1. Note that fµ0 is a zonal spherical function. The Go -equivariant map ιe : X → C[X]∗ defined by hf, ιe (x)i = f (x) for f ∈ C[X] is injective and satisfies ιe (o) = (vµ0 )µ∈Λ+ (see [HPV03, Section 5]). For any a ∈ Ao obtains an injective Go -equivariant map ιa : Spha X → C[X]∗ by x µ ιa (Sa (x)) = µ∗ , if ιe (x) = (xµ )µ∈Λ+ . a µ∈Λ+ In particular, a · v0 µ . ιa (Sa ) = ∗ µ a µ∈Λ+ The induced map ι∗a : C[X] → C[Spha X] is a Go -module isomorphism. Finally, ι(ξo ) := (vµ+ )µ∈Λ+ defines a G-equivariant map ι : Ξo → C[X]∗ . Since the stabilizer of ξ0 , as well the stabilizer of (vµ+ )µ∈Λ+ , is Mo No , the map ι is well-defined and injective. The induced map ι∗ : C[X] → C[Ξo ] coincides with the Go -module isomorphism Γ, where we note that C[X] = C[Z] and C[Ξo ] = C[Ξ]. We obtain the diagram Go /Mo t qa ttt t tt t zt

❊❊ ❊❊ q ❊❊ ❊❊ ❊"

❏❏ ❏❏ ιa ❏❏ ❏❏ ❏$

②② ι ②②② ②② |② ②

Spha X

Ξo

R[X]∗ which turns out to be commutative. This is part of the following proposition which is proven in [HPV03, Section 6]: Proposition 5.6. (i) lima→∞ ιa ◦ qa = ι ◦ q. (ii) lima→∞ qa∗ ◦ ι∗a = q ∗ ◦ ι∗ . (iii) lima→∞ R∗a ◦ ι∗a = R∗ ◦ ι∗ . (iv) R∗ (ι∗ (f )) = c(µ + ρ)f for all f ∈ C[X]µ . We note that those results can be applied to C[Z] and C[Ξ] by restriction and holomorphic extension. Suppose now as before that we have a propagated sequence of symmetric spaces Zj → Zk , k ≥ j. We also assume that the rank is finite.

´ JOACHIM HILGERT AND GESTUR OLAFSSON

28

Then, on each level, we have ι∗j = Γj . Therefore, we can define for f ∈ C[Z∞ ] ι∗∞ (f ) = ι∗j (f ) if f ∈ C[Zj ] . Then Proposition 5.7. If rank X∞ is finite and f ∈ C[Z∞ ]. Then Γ∞ f = ι∗∞ (f ) . 5.4. The kernels defining the normalized Radon transform and its dual. We start by stating the following version of the orthogonality relations which are usually formulated in terms of invariant inner products. The proof for this version as is the same as the usual one. The invariant measure on U is always the normalized Haar measure. The following is the usual orthogonality relation stated in form of duality. Lemma 5.8. For µ, ν ∈ Λ+ let d(µ) = dim Vµ = dim Vµ∗ . Then Z hu, πµ∗ (b)u∗ ihπν (a)v, v ∗ i db = δµ,ν d(µ)−1 hv, u∗ihu, v ∗i U

for all u, v ∈ Vµ and for all u∗ , v ∗ ∈ Vµ∗ .

´ It follows by [DOW12, Thm. 3.4] that heµ , e∗µ i = c(µ + ρ). Define X d(µ)c(µ + ρ) huµ , πµ∗ (a)e∗µ i (5.9) kZ (a) := µ∈Λ+

and kΞ (a) :=

X

d(µ) heµ, πµ∗ (a)u∗µ i .

(5.10)

µ∈Λ+

Lemma 5.11. Let O = {nak ∈ NAK | (∀j) |a−ωj | < 1}. Then O is open in G. The set O is right K-invariant and left MN-invariant. Furthermore, the sums defining kZ and kΞ converge uniformly on compact subsets of O and define holomorphic functions on O. Proof. Write µ = k1 ω1 + . . . + kr ωr . Let x ∈ O and write bj = a−ωj . Then |bj | < 1 for j = 1, . . . , r and d(µ)huµ, πµ∗ (x)u∗µ i = d(µ)hπµ (k −1 a−1 n−1 )uµ , u∗µ i = d(µ)a−µ r Y k = d(µ) bj j . j=1

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

29

Similarly, we have d(µ)c(µ +

ρ)huµ , πµ∗ (x)e∗µ i

= d(µ)c(µ + ρ)

r Y

k

bj j .

j=1

The claim follows now because d(µ) is polynomial in k1 , . . . , kr and c(µ + ρ) < 1. The function kZ is left MN-invariant and right K-invariant. Hence kZ can also be viewed as a function on Ξ × X given by kZ (a · ξo , b · xo ) := kZ (a−1 b) . The function kΞ is left K-invariant and right MN-invariant and can be viewed as a function kΞ on X × Ξ defined by kΞ (a · xo , b · ξo ) := kΞ (a−1 b) . Even if the sums (5.9) and (5.10) do not in general converge for all a ∈ G, they are well defined as linear G-maps C[X] → C[Ξ], respectively C[Ξ] → C[X], given by Z KZ (f )(ξ) := f (u · xo )kZ (ξ, u · xo ) du, U

respectively

KΞ (f )(x) :=

Z

f (u · ξo )kΞ(x, u · ξo ) du,

U

where du is the normalized Haar measure on U. On the right hand side only finitely many terms are nonzero so the sums converge. Note that the first integral can be written as an integral over the compact symmetric space U/Ko and the second integral is an integral over U/Mo . Theorem 5.12. We have KZ = Γ and KΞ = Γ−1 . Proof. It is enough to show that KZ |C[X]ν = Γν and KΞ |C[Ξ]ν = Γ−1 ν for all ν ∈ Λ+ . Thus we have to show that KZ (fν ) = ψν and KΞ (ψν ) = fν . We have X Z d(µ) KZ (fν )(a · ξo ) = hπµ (a)uµ , πµ∗ (b)e∗µ ihuν , πν∗ (b)e∗ν i du ∗i he , e µ µ U + ν∈Λ hπν∗ (a)u∗ν , uν i

= = ψµ (a · ξo) .

The statement for kΞ is proved in the same way.

´ JOACHIM HILGERT AND GESTUR OLAFSSON

30

Remark 5.13. The above we realized Γ and Γ−1 as integral operators. Similar to [G06] one could also consider the integral operator given by e · ξo , b · xo ) = e the kernel K(a k(a−1 b) where X X e huµ , πµ∗ (a)e∗µ i . (5.14) fµ (a) = k(a) = µ∈Λ∗

µ∈Λ+

Then we have the following theorem, see also [G06]:

Theorem 5.15. Let O be as in Lemma 5.11 and let x = kan ∈ O. be so that |aωj | < 1 for j = 1, . . . , r = rank X. Then r Y 1 e k(b) = 1 − aωj j=1 and e k is holomorphic on O.

Proof. Let µ = k1 ω1 + . . . kr ωr ∈ Λ+ . For x = nak ∈ O write as before bj = a−ωj . Then r Y k −µ ∗ ∗ bj j . huµ , πµ (x)eµ i = a = j=1

It follows that

e k(x) =

∞ X

bk11

k1 =0

which finish the proof.

...

∞ X

kr =0

bkr r

r Y (1 − bj )−1 = j=1

5.5. The Radon transform and its dual on the injective limits. In this section we discuss the extension of the Radon transform and its dual on the infinite dimensional spaces. The kernels defined in (5.9) and (5.10) do not define functions on Z∞ , respectively Ξ∞ , because the changes in the dimensions as we move from one space to another. But we still have the following: Theorem 5.16. Assume that the sequence {Zj } is admissible. For f ∈ Ci [Z∞ ] and ψ ∈ Ci [Ξ∞ ] the pointwise limits KZ∞ f (ξ) = lim KZj f (ξ) and KΞ∞ ψ(x) = lim KZj ψ(x) are well defined and Γ ∞ f = KZ ∞ f

and

Γ−1 ∞ ψ = KΞ∞ ψ .

Proof. If v ∈ Vµj , then v ∈ Vµk for all k ≥ j. Hence Theorem 5.12 implies that if s > k > j are so that ξ ∈ Ξk we have KZs f (ξ) = KZk f (ξ) .

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

31

Hence the sequence becomes constant and the claim follows. The ar gument for KΞ∞ ψ is the same. Note that the equations (4.20) and (4.21) together with Theorem 5.12 imply that the maps gr KZ∞ = lim gr KZj and gr KΞ∞ = lim KΞ∞ ←− ←− are well defined. Here gr stands for gr KZj ([f ]) = [KZj f ] respectively gr KΞj (f ) = [KΞj f ]. Theorem 5.17. Assume that the sequence Zj is admissible. Then gr Γ∞ f = gr KZ∞ f

gr Γ∞,−1f = gr KΞ∞ f .

and

In order to make the results of Section 5.3 useful for limits of symmetric spaces, we first have to extend the notion of a sphere of radius a. So let a = lim aj ∈ A∞ := lim Aj . We call A∞ regular, if ZKj (aj ) = Mj −→ −→ for all j. This is a useful notion only in the finite rank case, so we will assume for the remainder of this section that we are in the situation of Remark 4.4. A simple calculation shows that the diagram Xj /

Xk

ιk,e

ιj,e

C[Xj ]

∗

/

C[Xk ]∗

of Gj -module morphisms is commutative. The normalizations from Section 5.3 are compatible and yield the following commutative diagram Sphaj (Xj ) ιj,aj

/

Sphak (Xk ) ιk,ak

∗

C[Xj ] O

/

C[Xk ]∗ O

ιj

Ξj,o /

ιk

Ξk,o

of Gj -module morphisms. In fact, for the commutativity of the upper µ∗ ∗ µ∗ square one uses the equality aj j = ak k = aµ , whereas the commutativity of the lower square is a consequence of Theorem 4.11. Thus the Sphaj (Xj ) and the C[Xj ]∗ form inductive systems. For the corresponding inductive limits Spha (X∞ ) and C[X∞ ]∗ we obtain the commutative

´ JOACHIM HILGERT AND GESTUR OLAFSSON

32

diagram G∞,o /M∞,o ❏❏❏ ❏❏q❏∞ ❏❏ ❏❏ %

q∞,a ♣♣♣♣ ♣ ♣♣♣ x♣♣♣

Ξ∞,o

Spha (X∞ ) ◆◆◆ ◆◆◆ι∞,a ◆◆◆ ◆◆&

t ι∞ ttt tt t y tt t

C[X∞ ]∗ Using this notation Proposition 5.6(i) remains true: lim ι∞,a ◦ q∞,a = ι∞ ◦ q∞ .

a→∞

(5.18)

Due to G-equivariance, it suffices to prove that lim ι∞,a (qa (eM)) = ι∞ (q∞ (eM)).

a→∞

This means that for fixed λ ∈ Λ+ and j ≥ jo we have to verify 0 a · vj,λ + = vj,λ . ∗ a→∞ aλ 0 Writing vj,λ ∈ C[Xj ]∗λ as a sum of weight vectors (the highest weight being λ∗ ), we see that

lim

0 a · vj,λ + = kj,λvj,λ ∗ a→∞ aλ for some constant kj,λ. The calculation

lim

∗

− − − 0 (a−1 fj,λ )(xo ) aλ fj,λ (xo ) hfj,λ , a · vj,λ i = = = 1. (5.19) ∗ ∗ ∗ λ λ λ a a a shows that kj,λ = 1. This implies (5.18). Equation (5.18) yields immediately the convergence of the induced maps of function spaces. ∗ ∗ lim q∞,a ◦ ι∗∞,a = q∞ ◦ ι∗∞ .

a→∞

´ It was shown in [DOW12, Thm. 4.7] that the limit c∞ (µ∞ ) := lim c(µj + ρj ) j→∞

exists and is strictly positive if the rank of X∞ is finite. Define R∗ : Ci [Ξ∞ ] → Ci [Z∞ ]

(5.20)

RADON TRANSFORM FOR LIMITS OF SYMMETRIC SPACES

33

by R∗ (f )(x) = lim R∗j f (x) . j→∞

(5.21)

Theorem 5.22. Assume that the rank of X∞ is finite. Let f ∈ Ci [Ξ∞ ]. Then the pointwise limit (5.21) exists and for f ∈ C[Ξ∞ ]µ∞ we have R∗ f = c∞ (µ∞ )1/2 Γ−1 f

and

R∗∞ (ι∗ (f )) = c∞ (µ∞ )f .

Proof. As every function in Ci [Ξ∞ ] is a finite sum of elements in C[X∞ ]ν we only have to show this for fixed µ∞ ∈ Λ+ ∞ . But then the claim follows from (5.4), Theorem 5.17, Proposition 5.6, part (iv), and Proposition 5.7. As we are assuming that the rank of X∞ is finite, it follows from K∞ ∗ ´ ˆ 6= {0}. Denote by proj∞ the orthogonal [DOW12] that Vµ∞ projection K∞ . proj∞ : Vˆµ∗∞ → Vˆµ∗∞

´ It follows also from the calculations in [DOW12] that the sequence ∗ ∗ ∗ ˆ ˆ {eµj } converges to eµ∞ in the Hilbert space Vµ∞ = Vµ∗∞ . Hence K∞ ⊂ lim Vµ∗j . proj∞ Vˆµ∗∞ = Vˆµ∗∞ ←−

Finally, a simple calculation shows that Z proj∞ (w) = lim πµ∞ (k)w dk . j→∞

(5.23)

Kj

If f = fw,µ∞ ∈ C[Z∞ ]µ∞ , then there exists jo such that f |Zj = fw,µj ∈ C[Zj ]µj for all j ≥ jo . We have Z Ra (f |Zj )(g · Sa ) = hw, πµ∗j (g)πµ∗j (k)πµ∗j (a)e∗µj i dk . (5.24) Kj

Theorem 5.25. Let f ∈ Ci [Z∞ ]. Then the pointwise limit Ra,∞ f (g · Sa ) := lim Ra,j f (g · Sa ) j→∞

exists and the following holds: (1) Ra,∞ f (g ·Sa ) = hw, πµ∗∞ (g)proj∞ (πµ∗∞ (a)e∗µ∞ )i if f ∈ Ci [Z∞ ]µ∞ . (2) lim R∗a,∞ ◦ ι∗a = R∗∞ ◦ ι∗ . a→∞

Proof. This follows from (5.23), (5.24), and Proposition 5.6.

´ JOACHIM HILGERT AND GESTUR OLAFSSON

34

Remark 5.26. We note that the following diagrams do not commute C[Ξj ]

ιk,j

/

C[Ξk ]

R∗j

R∗k

C[Xj ]

ιk,j

/

C[Xk ]

C[Sphaj (Xj )] R∗j,a

ιk,j

/

C[Sphak (Xk )] R∗k,a

j

C[Xj ]

ιk,j

/

k

C[Xk ]

This follows from the corresponding commutative diagrams for the normalized dual Radon transforms γj−1 and the normalizing factor that relates those two transforms. This makes the corresponding theory for infinite rank spaces problematic as in that case limj→∞ c(µj + ρj ) = 0. References ´ ´ [DO13] M. Dawson and G. Olafsson, Conical representations for direct limits of symmetric spaces. In preparation. ´ ´ [DOW12] M. Dawson, G. Olafsson and J. Wolf, Direct Systems of Spherical Functions and Representations, Journal of Lie Theory 23 (2013), 711–729. [E73] P.E. Eberlein, Geodesic flows on negatively curved manifolds. II, Trans. Amer. Math. Soc. 178 (1973), 57–82. [E96] , Geometry of nonpositively curved manifolds. Chicago Lectures in Mathematics. University of Chicago Press, 1996. [G06] S. Gindikin, The horospherical Cauchy-Radon transform on compact symmetric spaces, Mosc. Math. J. 6 (2006), 299–305. [GK62] S. Gindikin and F. I. Karpelevich, Plancherel measure for symmetric Riemannian spaces of non-positive curvaure, Dokl. Akad. Nauk. SSSR 145 (1962), 252–255, English translation, Soviet Math. Dokl. 3 (1962), 1962–1965. ´ ´ [GKO06] S. Gindikin, B. Kr¨otz, and G. Olafsson, Holomorophic horospherical transform on non-compactly causal spaces, IMRN 2006 (2006), 1–47. [H66] S. Helgason, A duality in integral geometry on symmetric spaces, Proc. U.S.Japan Seminar in Differential Geometry, Kyoto, Japan, l965, pp. 37-56. Nippon Hyronsha, Tokyo l966. : A duality for symmetric spaces with applications to group represen[H70] tations, Advan. Math. 5 (l970), 1–154. [H78] , Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, 1978. , Groups and Geometric Analysis, Academic Press, 1984. [H84] [H94] , Geometric Analysis on Symmetric Spaces, Math. Surveys Monogr. 39, Amer. Math. Soc. Providence, RI 1994. [HPV02] J. Hilgert, A. Pasquale and E. B. Vinberg, The dual horospherical Radon transform for polynomials, Mosc. Math. J. 2 (2002), 113–126, 199. [HPV03] , The dual horospherical Radon transform as a limit of spherical Radon transforms, Lie groups and symmetric spaces, 135–143, Amer. Math. Soc. Transl. Ser. 2, 210, Amer. Math. Soc., Providence, RI, 2003. [HY00] Z. Huang and J. Yan, Introduction to Infinite Dimensional Stochastic Analysis. Mathematics and its Applications. Kluwer Academic Press, 2000.

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35

[KS77] V. I. Kolomycev and J. S. Samoilenko, On irreducible representation of inductive limits of groups, Ukrainian Math. J. 29 (1977), 402–405. ´ ´ [OW11a] G. Olafsson and J. Wolf, Extension of Symmetric Spaces and Restriction of Weyl Groups and Invariant Polynomials, Contemporary Math. 544 (2011), 85–100. ´ , The Paley-Wiener Theorem and Limits of Symmetric Spaces, to [OW11b] appear in JGA. {arXiv:1101.4419}. [O90] G. Olshanskii, Unitary representations of infinite dimensional pairs (G, K) and the formalism of R. Howe. In: Eds. A. M. Vershik and D. P. Zhelobenko: Representation of Lie Groups and Related Topics. Adv. Stud. Contemp. Math. 7, Gordon and Breach, 1990. [W08] J. A. Wolf, Infinite dimensional multiplicity free spaces I: Limits of compact commutative spaces. In “Developments and Trends in Infinite Dimensional Lie Theory, ed. K.-H. Neeb & A. Pianzola, Birkh¨auser, 2013. [W09] , Infinite dimensional multiplicity free spaces II: Limits of commutative nilmanifolds, Contemporary Mathematics, 491 (2009), 179–208. ¨r Mathematik, Universita ¨t Paderborn, 33098 Paderborn, Institut fu Germany E-mail address: [email protected] Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A. E-mail address: [email protected]