Hardy spaces for non-compactly causal symmetric spaces and the most continuous spectrum

Simon Gindikin^{∗}, Bernhard Krötz^{∗∗} and Gestur Ólafsson^{∗∗∗}

Abstract

Let G/H be a semisimple symmetric space. Then the space L^{2}(G/H) can be decomposed into a finite sum of series representations induced from parabolic subgroups of G. The most continuous part of the spectrum of L^{2}(G/H) is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions living on a tube type bounded symmetric space, containing G/H as a boundary component. A boundary value map is constructed and we show that it induces an G-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part L_{mc}^{2}(G/H) of L^{2}(G/H). We also relate our Hardy space with the classical Hardy space on the tube domain.

Introduction

When we transfer from harmonic analysis of Riemannian symmetric spaces to non Riemannian semisimple symmetric spaces the most important new phenomenon is the existence of different series of representations. This paper lies in the framework of a geometrical understanding of the different series of representations. The idea can be traced back to the paper [GeGi77], where a program was suggested to connect different series with a complex geometry of the complexification of symmetric spaces. The first step in this program was the realization of the holomorphic discrete series for groups of Hermitian type G in appropriate Hardy spaces in some tubes in G_{ℂ} which have the group G as Shilov boundary. This was accomplished by G.I. Olshanski and R. Stanton [Ol82, St86].

It turned out that there is a natural class of symmetric spaces G/H, the compactly causal symmetric spaces, on which it is possible to generalize the constructions for Hermitian groups. In [ÓØ88] the holomorphic discrete series for G/H were constructed and the appropriate Hardy space was defined and investigated in [HÓØ91]. The explicit form of the Plancherel density and

2 Simon Gindikin^{∗}, Bernhard Krötz^{∗∗} and Gestur Ólafsson^{∗∗∗}

∗ Supported in part by the NSF-grant DMS-0070816 and the MSRI ∗∗ Supported in part by the NSF-grant DMS-0097314 and the MSRI ∗∗∗ Supported in part by the NSF-grant DMS-0070607 and the MSRI the corresponding projection operators for the holomorphic discrete series were determined by the two last named authors [Kr01,KrÓl99, Ól00].

There are in principle two different classes of causal symmetric spaces, the compactly causal symmetric spaces and the non-compactly causal symmetric spaces. Let us mention that these two classes of symmetric spaces look completely different from the point of view of harmonic analysis.

For compactly causal symmetric spaces G/H one has a rich well understood complex geometry: There exist G-invariant tubes in G_{ℂ}/H_{ℂ} which have G/H as Shilov boundary; secondly, most of the compactly causal symmetric spaces can be realized as an open dense orbit in the Shilov-boundary of a bounded Hermitian symmetric space of tube type [Ber96, Bet97, ÓØ99]. For non-compactly causal symmetric spaces it was believed for a long time that a similar complex geometrical picture does not exist. In [Gi98] it was conjectured that appropriate tubes (related to complex crowns of Riemannian symmetric spaces) might also exist for non-compactly causal symmetric spaces. Also it was conjectured that some Hardy spaces on these tubes are connected with the most continuous spectrum (contrary to the compactly causal symmetric spaces where Hardy spaces correspond to the holomorphic discrete series).

In the last year substantial progress was achieved on the complex geometry for noncompactly causal symmetric spaces. This gives us now the possibility to start the corresponding analytical investigations – the subject proper of this paper.

Now a few remarks on geometric results. The complex crown of a Riemannian symmetric space G/K, first studied in [AkGi90], is a certain open connected G-invariant Stein domain in G_{ℂ}/K_{ℂ} which contains G/K as a totally real submanifold (cf. Section 1 for the definition). It is universal in many ways: It can be used to parametrize the compact cycles in complex flag manifolds (cf. [GiMa01]). Further it has the property that all eigenfunctions on G/K for the algebra of G-invariant differential operators holomorphically extend to Ξ [KrSt01a]. For a more detailed account on the complex crown Ξ we refer to the introduction of [GiKr01a].

By the recent work of the two first named authors [GiKr01b] we know that all noncompactly causal symmetric spaces G/H appear in the distinguished boundary ∂_{d}Ξ ⊆ G_{ℂ}/K_{ℂ} of Ξ, and, moreover, the non-compactly causal spaces are the only symmetric spaces which can be realized as G-orbits in ∂_{d}Ξ. Let us emphasize that one can interpret this as a connection of the elliptic analysis of the Riemannian symmetric space G/K with the hyperbolic analysis on the non-compactly symmetric spaces G/H ”through the complex crown Ξ”. Such kind of connections are well known in classical mathematical physics (transformation of solutions of Laplace equations in solutions of wave equations and back etc).

In this paper we consider those complex crowns which admit a realization as a symmetric space of Hermitian type [KrSt01b]. By this we mean that there exists a Hermitian group S ⊇ G with maximal compact subgroup U ⊇ K such that Ξ is G-biholomorphic to S/U. For these cases the distinguished boundary ∂_{d}Ξ = G(z_{1}) ≃ G/H is exactly one G-orbit and, moreover, the distinguished boundary is equal to the Shilov boundary ∂_{s}Ξ of Ξ in G_{ℂ}/K_{ℂ} [GiKr01b]. Under the additional assumption that G/H is symmetric, hence a non-compactly causal space, we arrive at the following list of our concern:

𝔤 = Lie(G)  𝔥 = Lie(H)  𝔰 = Lie(S)
𝔰𝔭(n, ℝ)  𝔤𝔩(n, ℝ)  𝔰𝔭(n, ℝ) ⊕ 𝔰𝔭(n, ℝ)
𝔰𝔲(n, n)  𝔰𝔩(n, ℂ) ⊕ ℝ  𝔰𝔲(n, n) ⊕ 𝔰𝔲(n, n)
𝔰𝔬^{∗}(4n)  𝔰𝔩(n, ℍ) ⊕ ℝ  𝔰𝔬^{∗}(4n) ⊕ 𝔰𝔬^{∗}(4n)
𝔰𝔬(2, n)  𝔰𝔬(1, n − 1) ⊕ ℝ  𝔰𝔬(2, n) ⊕ 𝔰𝔬(2, n)
𝔢_{7(−25)}  𝔢_{6(−26)} ⊕ ℝ  𝔢_{7(−25)} ⊕ 𝔢_{7(−25)}
𝔰𝔬(1, n)  𝔰𝔬(1, n − 1)  𝔰𝔬(2, n)
𝔰𝔭(n, n)  𝔰𝔭(n, ℂ)  𝔰𝔲(2n, 2n)
𝔰𝔭(n, ℂ)  𝔰𝔭(n, ℝ)  𝔰𝔭(2n, ℝ)

We call the triples (𝔰, 𝔤, 𝔥) from the above list causally symmetric triples. They can be defined axiomatically starting from a Hermitian Lie algebra 𝔰 of tube type with two involutions τ and σ such that:

(CST1) 𝔰^{τ} = 𝔤 and (𝔰, τ) is compactly causal. (CST2) 𝔤^{σ} = 𝔥 and (𝔤, σ |_{𝔤}) is non-compactly causal.

Following É. Cartan the Hermitian symmetric space S/U admits a canonical embedding in the dual compact Hermitian symmetric space and can be realized as a bounded circled domain in ℂ^{n} (the realization of disc type where the isotropy subgroup acts linearily). We will use Harish-Chandra’s construction of the realization of S/U as a bounded symmetric domain D. There is also the realization of S/U as an affine homogeneous tube domain T_{Ω} = ℝ^{n} +iΩ (the realization of halfplane type). The realization Ξ we use is different from the two mentioned above– it lies inside G_{ℂ}/K_{ℂ} and is connected with the non-transitive action of the group G.

With the different realizations of S/U one becomes different Shilov boundaries: A compact Shilov boundary ∂_{s}D for D which is a single S -orbit, and a vector space ℝ^{n} for the Shilov boundary of the tubes T_{Ω} . For the Ξ-realization we have a single G-orbit for the Shilov boundary.

Generally, Hardy spaces correspond to Hilbert spaces of holomorphic functions which have L^{2} -boundary values on the Shilov boundary. Since we have in the three cases mentioned above different Shilov boundaries, this necessarily leads to different definitions of Hardy spaces. In the first two cases the constructions of Hardy spaces H^{2}(D) and H^{2}(T_{Ω}) are well known; they are isomorphic as irreducible S -modules. We refer to them as classical Hardy spaces.

For the definition of a Hardy space on Ξ one encounters an essential geometric difficulty, namely that G does not act transitively. We use an open compression subsemigroup of D inside of S_{ℂ}, discovered by Olshanski. We denote by Γ^{−1} the intersection of this semigroup with G_{ℂ}. Since Ξ is biholomorphic to D we obtain an action of Γ on Ξ by compressions. We now can define a Hardy space H^{2}(Ξ) on Ξ by

H^{2}(Ξ): = {f ∈ O(Ξ): ∥f∥^{2}: = sup_{γ∈Γ} ∫_{G/H} |f(γ^{−1}gz_{1})|^{2} dμ_{G/H}(gH) < ∞}.

It turns out that the Hardy space H^{2}(Ξ) is a Hilbert space of holomorphic functions on Ξ with a natural unitary action L of G by left translations in the arguments. In particular, H^{2}(Ξ) admits a reproducing kernel K_{Ξ}(z, w), the so-called Cauchy-Szegö kernel, holomorphic in the first and antiholomorphic in the second variable.

One of our tools is to compare this space with the classical Hardy space H^{2}(D) which is an irreducible G-spherical unitary highest weight module for the big group S. The action of S on H^{2}(D) is given through a cocycle representation

(π_{h}(s)f)(z) = J_{h}(s^{−1}, z)^{−1}f(s^{−1}z) (f ∈ H^{2}(D), s ∈ S, z ∈ D).

Theorem 5.7 now reads:

Theorem A. There exists an explicit zero free holomorphic function ψ on Ξ ≃ D such that the mapping

Ψ: (π_{h} |_{G}, H^{2}(D)) → (L, H^{2}(Ξ)), f ↦ ψf

is a G-equivariant isomorphism of Hilbert spaces.

The fact that G/K is a totally real submanifold of S/U can now be used to construct the generalized Segal-Bargmann transform, which is a unitary G-isomorphism L^{2}(G/K) → H^{2}(Ξ), [DÓZ01,Ner99,Ól00,ÓØ96,Zh01]. This implies that the continuous branching of the Hardy space representation. Combining Theorem 5.5 and Theorem 5.8 we come to our main result:

Theorem B. The boundary value mapping

b: H^{2}(Ξ) → L^{2}(G/H), f ↦ (gH ↦ lim_{{γ→1 \atop γ∈Γ}} f(γ^{−1}gz_{1}))

is a G-equivariant isometric embedding. Furthermore the image of b is a multiplicity free full subspace of the most continuous spectrum L^{2}(G/H)_{mc} (cf. Definition 4.1).

Finally we compare the reproducing kernel K_{Ξ} of H^{2}(Ξ) with the Cauchy-Szegö kernel K_{h} from the classical Hardy space. Theorem 5.9 from below says:

Theorem C. The Cauchy-Szegö kernels K_{Ξ} and K_{h} are related through

K_{Ξ}(z, w) = ψ(z)\overline{ψ(w)}K_{h}(z, w).

To illustrate Theorem C consider the example where S = G × G. Then Ξ is Gbiholomorphic to D_{G} × D_{G}^{opp} where D_{G} is the Harish-Chandra realization of G/K and D_{G}^{opp} refers to D_{G} equipped with the opposite complex structure. Now if K_{h}^{G}(z, w) denotes the Cauchy-Szegö kernel of the classical Hardy space H^{2}(D_{G}), then Theorem C boils down to (see Example 5.10 below)

K_{Ξ}((z_{1}, w_{1}), (z_{2}, w_{2})) = \frac{K_{h}^{G}(z_{1}, z_{2})K_{h}^{G}(w_{2}, w_{1})}{K_{h}^{G}(z_{1}, w_{1})K_{h}^{G}(w_{2}, z_{2})} (∀(z_{1}, w_{1}), (z_{2}, w_{2}) ∈ Ξ ≃ D_{G} × D_{G}^{opp}) .

Note that the formula of K_{Ξ} is explicit, since the classical Hardy space kernels K_{h}^{G} are well known.

In the Plancherel decomposition of L^{2}(G/H) the most continuous spectrum L_{mc}^{2}(G/H) corresponds to one series of representations, the one which is induced off from a minimal parabolic subgroup. However, the space L_{mc}^{2}(G/H) is not multiplicity free and we encounter the next problem to separate the different multiplicities in L_{mc}^{2}(G/H) geometrically. Theorem B implies an embedding

H^{2}(Ξ) ⊕ H^{2}(Ξ^{opp}) ↪ L^{2}(G/H)_{mc}

which realizes a multiplicity two subspace of the most continuous spectrum of G/H. Presently it is unclear how to obtain the other multiplicities in L^{2}(H/H)_{mc} missed by H^{2}(Ξ) ⊕ H^{2}(Ξ^{opp}). In this context notice the striking similarity with the picture for discrete series representations. Only two out of the discrete series, the holomorphic and the antiholomorphic, can be realized as a Hardy space.

We would like to mention, as we know through several talks, that J. Faraut is working on another construction of Hardy spaces for certain non-compactly causal symmetric spaces.

Our paper is organized as follows:

1. Non-compactly causal symmetric spaces and boundary components of complex domains 2. Continuous branching of spherical highest weight representations

3. Analytical and geometrical constructions on Ξ

4. The classical Hardy space inside L^{2}(G/H)_{mc}

5. The Hardy space on Ξ

Appendix A: parameter calculations

Appendix B: Structure theory for causally symmetric triples

It is our pleasure to thank the MSRI, Berkeley, for its hospitality during the Integral geometry program where this work was accomplished.

1. Non-compactly causal symmetric spaces and boundary components of complex domains

Causal symmetric Lie algebras

Let 𝔰 denote a semisimple real Lie algebra with complexification 𝔰_{ℂ}. Let S be a connected Lie group with Lie algebra 𝔰 and let S_{ℂ} denote a complex Lie group with Lie algebra 𝔰_{ℂ}. We will assume – if not otherwise stated – that S ⊆ S_{ℂ}. Denote by θ a Cartan involution on 𝔰 with Cartan decomposition 𝔰 = 𝔲 ⊕ 𝔭_{∗}. Let τ: 𝔰 → 𝔰 be an involution on 𝔰 of which we may assume to commute with θ. We will also assume that τ defines an involution on S_{ℂ}. Write 𝔰 = 𝔤 ⊕ 𝔮_{∗} for the τ -eigenspace decomposition corresponding to the τ -eigenvalues +1 and −1. Then 𝔤 is reductive and θ |_{𝔤} is a Cartan involution of 𝔤. Set 𝔨 = 𝔤 ∩ 𝔲 and 𝔭: = 𝔤 ∩ 𝔭_{∗}. Then 𝔤 = 𝔨 ⊕ 𝔭 is a Cartan decomposition of 𝔤. Let G be the connected subgroup of S with Lie algebra 𝔤. The symmetric pair (𝔰, 𝔤) is called irreducible if the only τ -invariant ideals in 𝔰 are the trivial ones, {0} and 𝔰. In this case either 𝔰 is simple or 𝔰 ≃ 𝔤 ⊕ 𝔤, with 𝔤 simple, and τ(X, Y) = (Y, X) the flip. In the second case we have S/G ∋ (a, b)G ↦ ab^{−1} ∈ G defines an isomorphism of S/G with G. We say that the symmetric space S/G is irreducible if (𝔰, 𝔤) is irreducible.

Definition 1.1. Let C be an open convex subset of 𝔮_{∗}. Then C is called hyperbolic, if for all X ∈ C the operator ad(X) is semisimple with real eigenvalues. We call C elliptic, if all operators ad(X), X ∈ C are semisimple and with imaginary spectrum.

We recall now the definition of causal symmetric spaces (c.f. [HiÓl96], Chapter 3):

Definition 1.2. (Causal symmetric spaces) The symmetric space S/G is called causal if there exists a non-empty open G-invariant convex cone C , containing no affine lines, in 𝔮_{∗}.

There are two different types of causal symmetric spaces, the non-compactly causal symmetric spaces (NCC) and the compactly causal symmetric spaces (CC). In addition, there is the intersection of those two classes, the Cayley type symmetric spaces (CT).

Definition 1.3. (NCC) Assume that S/G is an irreducible causal symmetric space. Then the following two conditions are equivalent:

(a) There exists a non-empty G-invariant open hyperbolic cone C ⊆ 𝔮_{∗} which contains no affine lines; (b) There exists an element T_{0} ∈ 𝔮_{∗} ∩ 𝔭_{∗}, T_{0} ≠ 0, which is fixed by K.

If one of those equivalent conditions are satisfied, then S/G is called non-compactly causal.

Definition 1.4. (CC) Assume that S/G is an irreducible causal symmetric space. Then the following two conditions are equivalent:

(a) There exists a non-empty G-invariant open elliptic cone C ⊆ 𝔮_{∗} which contains no affine lines;

(b) There exists an element X_{0} ∈ 𝔮_{∗} ∩ 𝔲, X_{0} ≠ 0, which is fixed by K. If one of those equivalent conditions are satisfied, then S/G is called compactly causal.

Definition 1.5. (CT) Assume that S/G is an irreducible causal symmetric space. Then S/G is called a symmetric space of Cayley type, if it is both non-compactly causal and compactly causal.

Remark 1.6. (a) The elements T_{0} and X_{0} in Definition 1.3 and Definition 1.4 are unique up to multiplication by scalar. If S/G is NCC then we normalize T_{0} such that ad(T_{0}) has spectrum {0, 1, −1}. The eigenspace corresponding to 0 is exactly 𝔰^{θτ} = 𝔨 ⊕ (𝔭_{∗} ∩ 𝔮_{∗}). If S/G is CC, then we normalize X_{0} such that the spectrum of ad(iX_{0}) is {0, 1, −1}. In this case the zero eigenspace is exactly 𝔲.

(b) If S/G is compactly causal, then 𝔲 has a non trivial center 𝔷(𝔲) and 𝔷(𝔲) ∩ 𝔮_{∗} = ℝX_{0} . If S is simple then 𝔷(𝔲) = ℝX_{0} . If 𝔰 ≃ 𝔤 ⊕ 𝔤. Then X_{0} = (X′_{0}, −X′_{0}) with X′_{0} central in 𝔨 and 𝔷(𝔲) = ℝX_{0} ⊕ ℝ(X′_{0}, X′_{0}).

Denote the complex linear extension of τ to 𝔰_{ℂ} also by τ. The c-dual (𝔰^{c}, τ^{c}) of (𝔰, τ) is defined by 𝔰^{c} = 𝔤⊕i𝔮_{∗} ⊆ 𝔰_{ℂ} with involution τ|_{𝔰^{c}}. Notice that the c-dual of (𝔰^{c}, τ^{c}) is (𝔰, τ) back again. Let S^{c} denote the analytic subgroup of S_{ℂ} with Lie algebra 𝔰^{c}. Then G = (S ∩ S^{c})_{0} , where the subscript _{0} denotes the connected component containing 1.

We recall the following fact:

Proposition 1.7. Assume that S/G is an irreducible symmetric space. Then the following holds:

(i) S/G is non-compactly causal if and only if S^{c}/G is compactly causal. (ii) S/G is of Cayley type if and only if S/G ≃ S^{c}/G. (iii) Suppose that S/G is irreducible and causal. Then S/G is of Cayley type if and only if G

is not simple. In that case 𝔷(𝔤) is one dimensional and contained in 𝔭.

Example 1.8. (The group case) Let G be a connected semisimple Lie group, S = G × G, and τ(a, b) = (b, a). Then S/G is compactly causal if and only if G/K is a Hermitian symmetric space. In this case S_{ℂ} = G_{ℂ} × G_{ℂ} and S^{c}/G ≃ G_{ℂ}/G (cf. [HiÓl96], Example 1.2.2).

The following three tables list the causal symmetric pairs (cf. [HiÓl96, Th. 3.2.8]).

Table I The group case

𝔰  𝔰^{c}  𝔤
CC  NCC
𝔰𝔲(p, q) ⊕ 𝔰𝔲(p, q)  𝔰𝔩(p + q, ℂ)  𝔰𝔲(p, q)
𝔰𝔬^{∗}(2n) ⊕ 𝔰𝔬^{∗}(2n)  𝔰𝔬(2n, ℂ)  𝔰𝔬^{∗}(2n)
𝔰𝔬(2, n) ⊕ 𝔰𝔬(2, n)  𝔰𝔬(n + 2, ℂ)  𝔰𝔬(2, n)
𝔰𝔭(n, ℝ) ⊕ 𝔰𝔭(n, ℝ)  𝔰𝔭(n, ℂ)  𝔰𝔭(n, ℝ)
𝔢_{6(−14)} ⊕ 𝔢_{6(−14)}  𝔢_{6}  𝔢_{6(−14)}
𝔢_{7(−25)} ⊕ 𝔢_{7(−25)}  𝔢_{7}  𝔢_{7(−25)}

Table II 𝔰 simple and not CT

𝔰  𝔰^{c}  𝔤
CC  NCC
𝔰𝔲(p, q)  𝔰𝔩(p + q, ℝ)  𝔰𝔬(p, q)
𝔰𝔲^{∗}(2p, 2q)  𝔰𝔲^{∗}(2(p + q)  𝔰𝔭(p, q)
𝔰𝔬^{∗}(2n)  𝔰𝔬(n, n)  𝔰𝔬(n, ℂ)
𝔰𝔬(2, n)  𝔰𝔬(1, n + 1)  𝔰𝔬(1, n)
𝔰𝔬(2, p + q)  𝔰𝔬(p + 1, q + 1)  𝔰𝔬(1, p) ⊕ 𝔰𝔬(1, q) , p, q ≥ 2
𝔰𝔭(2n, ℝ)  𝔰𝔭(n, ℝ)  𝔰𝔭(n, ℂ)
𝔢_{6(−14)}  𝔢_{6(6)}  𝔰𝔭(2, 2)
𝔢_{6(−14)}  𝔢_{6(−26)}  𝔣_{4(−20)}
𝔢_{7(−25)}  𝔢_{7(7)}  𝔰𝔲^{∗}(8)

Table III 𝔰 simple and CT

\begin{matrix}𝔰 & 𝔰^{c} ≃ 𝔰 & 𝔤 \\ 𝔰𝔭(n, ℝ) & 𝔰𝔭(n, ℝ) & 𝔤𝔩(n, ℝ) \\ 𝔰𝔲(n, n) & 𝔰𝔲(n, n) & 𝔰𝔩(n, ℂ) ⊕ ℝ \\ 𝔰𝔬^{∗}(4n) & 𝔰𝔬^{∗}(4n) & 𝔰𝔲^{∗}(2n) ⊕ ℝ \\ 𝔰𝔬(2, n) & 𝔰𝔬(2, n) & 𝔰𝔬(1, n − 1) ⊕ ℝ \\ 𝔢_{7(−25)} & 𝔢_{7(−25)} & 𝔢_{6(−26)} ⊕ ℝ \end{matrix}

The domain Ξ – the complex crown of G/K

According to [GiKr01a] all non-compactly causal symmetric spaces G/H can be realized as a boundary components of a certain complex domain wich we discuss now.

Let 𝔤 be a reductive Lie algebra with complexification 𝔤_{ℂ}. We let G_{ℂ} be a connected complex Lie group with Lie algebra 𝔤_{ℂ} and take G as the analytic subgroup of G_{ℂ} corresponding to 𝔤. Let K < G be a maximal compact subgroup and 𝔨 its Lie algebra. Let 𝔤 = 𝔨 ⊕ 𝔭 the Cartan decomposition attached to 𝔨. Let 𝔞 ⊆ 𝔭 a maximal abelian subspace, set 𝔪 = 𝔷_{𝔨}(𝔞) and for α ∈ 𝔞^{∗} define

𝔤^{α} = {X ∈ 𝔤: (∀H ∈ 𝔞)[H, X] = α(H)X} .

Let Σ = Σ(𝔤, 𝔞) be the corresponding set of (restricted) roots, i.e., Σ = {α ∈ 𝔞^{∗} \{0}: 𝔤^{α} ≠ {0}}. Then

𝔤 = 𝔞 ⊕ 𝔪 ⊕ ⨁_{α∈Σ} 𝔤^{α} .

We write A for the analytic subgroup of G corresponding to 𝔞 and 𝔫. We also need the Weyl group of Σ(𝔞, 𝔤), W: = N_{K}(𝔞)/Z_{K}(𝔞).

Let

Ω = \{X ∈ 𝔞: (∀α ∈ Σ) |α(X)| < \frac{π}{2} \}.

Then the complex crown of the Riemannian symmetric space G/K is defined by

Ξ: = G exp(iΩ)K_{ℂ}/K_{ℂ} .

Note that Ξ is an open domain in G_{ℂ} and that G acts properly on it (cf. [AkGi90]). Write ∂Ξ for the topological boundary of Ξ and notice that G exp(i∂Ω)K_{ℂ}/K_{ℂ} ⊆ ∂Ξ (cf. [AkGi90]).

The distinguished boundary of Ξ

Write \overline{Ξ} for the closure of Ξ in G_{ℂ}/K_{ℂ}, and note that \overline{Ξ} = Ξ ⨿ ∂Ξ. Write ∂_{e}Ω for the extreme points of the compact convex set \overline{Ω}. Recall from [GiKr01a] that ∂_{e}Ω is a finite union of W -orbits:

∂_{e}Ω = W(X_{1}) ⨿ … ⨿ W(X_{n}).

We define the distinguished boundary of Ξ in G_{ℂ}/K_{ℂ} by

∂_{d}Ξ: = G exp(i∂_{e}Ω)K_{ℂ}/K_{ℂ}.

It was shown in [GiKr01a, Th. 2.3] that ∂_{d}Ξ is a subset of ∂Ξ which satisfies the property

(1.1)  sup_{z∈Ξ} |f(z)| = sup_{z∈∂_{d}Ξ} |f(z)|

for all bounded holomorphic functions f on Ξ which continuously extend to \overline{Ξ}. Moreover, if ∂_{d}Ξ is connected, i.e., if ∂_{e}Ω = W(X_{1}) is a single orbit, then ∂_{d}Ξ is minimal with respect to the property (1.1). In this case we call ∂_{d}Ξ the Shilov boundary of Ξ and denote it by ∂_{s}Ξ.

Write z_{j}: = exp(iY_{j})K_{ℂ} ∈ ∂_{d}Ξ and write H_{j} for the isotropy subgroup of G in z_{j}. Then we have the following (cf. [GiKr01a]):

Theorem 1.9. Assume that G_{ℂ} is simply connected. Then, with the notation introduced above, the following assertions hold:

(i) ∂_{d}Ξ ≃ G/H_{1} ⨿ … ⨿ G/H_{n}. (ii) If one of the boundary components G/H_{j} is a symmetric space, then it is a non-compactly

causal symmetric space. Moreover every non-compactly causal symmetric space G/H attached to G appears G-locally in the distinguished boundary of ∂_{d}Ξ of Ξ.

Realization of Ξ as tube domains

We consider now the case where Ξ has a realization as a tube domain. The material in this section is taken from [KrSt01b]. Throughout this section (𝔰, τ) denotes a compactly causal symmetric Lie algebra. Let 𝔰_{ℂ} denote the complexification of 𝔰 and let S_{ℂ} denote the simply connected complex Lie group with Lie algebra 𝔰_{ℂ}. Let S ⊆ S_{ℂ} be the connected subgroup corresponding to 𝔰. Let G be the analytic subgroup of S corresponding to 𝔤. Write U and K for the maximal compact subgroups in S and G. Note that the embedding

G/K ↪ S/U, gK ↦ gU

realizes G/K as a totally real submanifold of the Hermitian symmetric space S/U.

Recall that 𝔷(𝔲) ∩ 𝔮_{∗} = ℝX_{0} is one dimensional and that we can normalize X_{0} such that Spec(ad iX_{0}) = {0, 1, −1}. We let 𝔱 ⊆ 𝔲 be a compact τ -stable Cartan algebra of 𝔰 and note that 𝔷(𝔲) ⊆ 𝔱. Write Δ = Δ(𝔰_{ℂ}, 𝔱_{ℂ}) for the root system with respect to 𝔱_{ℂ}. Call a root α compact if α |_{𝔷(𝔲)} = 0 and non-compact otherwise. Thus α is non-compact if and only if 𝔤_{ℂ}^{α} ⊂ 𝔭_{ℂ}. Denote by Δ_{c} the set of compact roots and by Δ_{n} the set of non-compact roots. Let Δ_{n}^{+} := {α ∈ Δ: α(iX_{0}) = 1} and fix a positive system Δ^{+} such that Δ_{n}^{+} ⊆ Δ^{+} . Write 𝔭^{±} = ⊕_{α∈Δ_{n}^{+}}𝔤_{ℂ}^{±α} ⊆ 𝔭_{ℂ}. Then we have the triangular decomposition of 𝔰_{ℂ}:

𝔰_{ℂ} = 𝔭^{+} ⊕ 𝔲_{ℂ} ⊕ 𝔭^{−}.

Let P^{±} = exp(𝔭^{±}). Let U_{ℂ} ⊆ S_{ℂ} be the complexification of U. Then it is well known that S ⊆ P^{+}U_{ℂ}P^{−} and we have the Borel embedding

S/U ↪ S_{ℂ}/U_{ℂ}P^{−}.

The map P^{+} × U_{ℂ} × P^{−} ∋ (p^{+}, u, p^{−}) ↦ p^{+}up^{−} ∈ P^{−}U_{ℂ}P^{+} is a diffeomorphism onto the open dense subset P^{+}U_{ℂ}P^{−} ⊆ S_{ℂ}. We write s = p^{+}(s)κ(s)p^{−}(s) for the unique decomposition of an element s ∈ P^{+}U_{ℂ}P^{−}. Then we have the Harish Chandra realization of S/U as a bounded symmetric domain D ⊆ 𝔭^{+} given by

S/U ∋ sU ↦ log(p^{+}(s)) ∈ 𝔭^{+}

where log = (exp |_{𝔭^{+}})^{−1} : P^{+} → 𝔭^{+} .

Write Σ = Σ(𝔤, 𝔞) for the restricted root system with respect to 𝔞 and Σ̂ = Σ(𝔰, 𝔞) for the double restricted root system. If Σ is an abstract root system on 𝔞, then we also write Ω = Ω(Σ) to indicate the dependance of Ω on Σ. Define

Ξ_{0}: = G exp(iΩ(Σ̂))K_{ℂ}/K_{ℂ},

and notice that Ξ_{0} ⊆ Ξ since Σ ⊆ Σ̂.

Theorem 1.10. (cf. [KrSt01b]) The embedding G/K ↪ S/U extends to a G-equivariant biholomorphism

Φ: Ξ_{0} → S/U ⊆ S_{ℂ}/U_{ℂ}P^{−}.

Further the following statements are equivalent:

(1) Ξ = Ξ_{0} .

(2) Ω(Σ̂) = Ω(Σ).

(3) Σ is of type C_{n} or BC_{n} for n ≥ 2 or (𝔤, 𝔰) is one of (𝔰𝔲(1, 1), 𝔰𝔲(1, 1) ⊕ 𝔰𝔲(1, 1)) or (𝔰𝔬(1, n), 𝔰𝔬(2, n)).

The non-compactly causal symmetric spaces G/H which are singled out by the additional requirement that Ξ = Ξ_{0} ≃ S/U are the ones that are most important for the rest of this article. We will therefore discuss them in more details now. The non-compactly causal symmetric spaces which satisfy in addition Ξ = Ξ_{0} are exactly those where Σ is of type C_{r}, say

Σ = \{\frac{1}{2}(±γ_{i} ± γ_{j}): 1 ≤ i, j ≤ r\}\\{0\}.

Define Y_{j} ∈ 𝔞 by γ_{i}(Y_{j}) = 2δ_{ij} and Y_{0} = \frac{1}{2}(Y_{1} + … + Y_{r}). In this case n = 1, X_{1} = \frac{π}{2} Y_{0} and

∂_{e}Ω = W(\frac{π}{2} Y_{0}) .

Hence with z_{1}: = exp(i\frac{π}{2} Y_{0})K_{ℂ} we have ∂_{s}Ξ = G(z_{1}) is a connected G-space. Write H for the isotropy subgroup of G in z_{1} and 𝔥 for the Lie algebra of H. Sometimes we also write c for z_{1} since Ad(c) is a partial Cayley transform. It follows from [GiKr01b] that

σ: G → G, g ↦ (Ad(c^{2}) ∘ θ)(g)

is an involution on G such that H = G^{σ}. Comparing Theorem 1.10 with the list in the tables I, II and III, we arrive at the following:

Table IV Ξ ≃ S/U and ∂_{s}Ξ ≃ G/H

\begin{matrix}𝔤 & 𝔥 & 𝔰 \\ 𝔰𝔭(n, ℝ) & 𝔤𝔩(n, ℝ) & 𝔰𝔭(n, ℝ) ⊕ 𝔰𝔭(n, ℝ) \\ 𝔰𝔲(n, n) & 𝔰𝔩(n, ℂ) ⊕ ℝ & 𝔰𝔲(n, n) ⊕ 𝔰𝔲(n, n) \\ 𝔰𝔬^{∗}(4n) & 𝔰𝔩(n, ℍ) ⊕ ℝ & 𝔰𝔬^{∗}(4n) ⊕ 𝔰𝔬^{∗}(4n) \\ 𝔰𝔬(2, n) & 𝔰𝔬(1, n − 1) ⊕ ℝ & 𝔰𝔬(2, n) ⊕ 𝔰𝔬(2, n) \\ 𝔢_{7(−25)} & 𝔢_{6(−26)} ⊕ ℝ & 𝔢_{7(−25)} ⊕ 𝔢_{7(−25)} \\ 𝔰𝔬(1, n) & 𝔰𝔬(1, n − 1) & 𝔰𝔬(2, n) \\ 𝔰𝔭(n, n) & 𝔰𝔭(n, ℂ) & 𝔰𝔲(2n, 2n) \\ 𝔰𝔭(n, ℂ) & 𝔰𝔭(n, ℝ) & 𝔰𝔭(2n, ℝ) \end{matrix}

Note that the first five pairs (𝔤, 𝔥) in this list are of Cayley type. We call the triple (𝔰, 𝔤, 𝔥) from table IV a causally symmetric triple. Note that we always have

rank_{ℝ}(𝔥) = rank_{ℝ}(𝔤) = \frac{1}{2} rank_{ℝ} 𝔰.

We write 𝔤 = 𝔥⊕𝔮 for the σ -eigenspace decomposition of 𝔤. Note that σ extends to an involution on 𝔰 since Spec ad_{𝔰}(Y_{0}) = {−1, 0, 1}. Our choice of the maximal abelian subspace 𝔞 ⊆ 𝔭 is then such that 𝔞 ⊆ 𝔮 ∩ 𝔭.

Causally symmetric triples can also be defined axiomatically:

Definition 1.11. (Causally symmetric triples) A triple of Lie algebras (𝔰, 𝔤, 𝔥) with 𝔥 ⊆ 𝔤 ⊆ 𝔰 is called a causally symmetric triple if the following axioms are satisfied: (CST1) 𝔰 is Hermitian and of tube type. (CST2) There exists two involutions τ and σ on 𝔰 such that:

(1) (𝔰, τ) is compactly causal and 𝔤 = 𝔰^{τ}.

(2) (𝔤, σ |_{𝔤}) is non-compactly causal and 𝔥 = 𝔤^{σ}.

If we only require that G/K embeds into S/U and that ∂_{d}Ξ has at least one boundary G/H_{j} component (cf. Theorem 1.9) which is a non-compactly causal symmetric space, then we arrive at the following more general situation (cf. [KrSt01b] and [GiKr01b]).

Theorem 1.12. Suppose that G/K is a totally real submanifold of S/U and that there exists a symmetric subgroup H of G such that G/H is a non-compactly causal symmetric space which is a G-orbit in ∂_{d}Ξ. With ∂_{d}Ξ_{0} = G exp(i∂_{e}Ω(Σ̂))K_{ℂ}/K_{ℂ} the distinguished boundary of Ξ_{0} we arrive at the following four possibilities:

(i) (𝔰, 𝔤, 𝔥) is a causally symmetric triple. (ii) G is the structure group of an Euclidian Jordan algebra, i.e., 𝔤 is isomorphic to one of the

following

𝔤𝔩(n, ℝ)  𝔰𝔩(n, ℂ) ⊕ ℝ  𝔰𝔩(n, ℍ) ⊕ ℝ  𝔰𝔬(1, n) ⊕ ℝ  𝔣_{4(−20)} ⊕ ℝ ,

and ∂_{d}Ξ_{0} = ∂_{d}Ξ ⨿ G/K ⨿ G/K. (iii) G = SO(n, n), ∂_{d}Ξ_{0} = G/ SO(n, ℂ), and

∂_{d}Ξ = ∂_{d}Ξ_{0} ⨿ G/( SO(1, n − 1) × SO(1, n − 1)) .

(iv) G = SO(2n, ℂ), ∂_{d}Ξ_{0} = G/ SO^{∗}(2n), and ∂_{d}Ξ = ∂_{d}Ξ_{0} ⨿ G/ SO(2, 2n − 2).

The compactification of the Shilov boundary ∂_{s}Ξ

For the remainder of this section we assume that (𝔰, 𝔤, 𝔥) is a causally symmetric triple.

Proposition 1.13. Assume that (𝔰, 𝔤, 𝔥) is a causally symmetric triple. Then the mapping

ι: G_{ℂ}/K_{ℂ} → S_{ℂ}/U_{ℂ}P^{−}, gK_{ℂ} ↦ gU_{ℂ}P^{−}

is an open G_{ℂ} -equivariant embedding. Proof. Since 𝔰_{ℂ} = 𝔤_{ℂ} +𝔲_{ℂ} +𝔭^{−} it is clear that the image of ι is open. It remains to show that ι is injective, which means that G_{ℂ} ∩ U_{ℂ}P^{−} = K_{ℂ}. Also denote by τ and θ the holomorphic extension of τ and θ to an involution on S_{ℂ}. Note that S_{ℂ}^{τ} = G_{ℂ} and S_{ℂ}^{θ} = U_{ℂ} since S_{ℂ} was assumed to be simply connected.

We have τ(P^{−}) = P^{+} . Thus we obtain

G_{ℂ} ∩ U_{ℂ}P^{−} = G_{ℂ} ∩ U_{ℂ}P^{−} ∩ τ(U_{ℂ}P^{−}) = G_{ℂ} ∩ (U_{ℂ}P^{−} ∩ U_{ℂ}P^{+}).

Now the well known fact U_{ℂ}P^{−} ∩ U_{ℂ}P^{+} = U_{ℂ} implies that

G_{ℂ} ∩ U_{ℂ}P^{−} = G_{ℂ} ∩ U_{ℂ}.

It remains to show that G_{ℂ} ∩ U_{ℂ} = K_{ℂ}. Now we are going to use the fact that the pair (𝔰, 𝔤, 𝔥) is from table IV. Quick inspection shows that G_{ℂ} is simply connected. Hence G_{ℂ}^{θ} = K_{ℂ} is connected and so G_{ℂ} ∩ U_{ℂ} = G_{ℂ} ∩ S_{ℂ}^{θ} = K_{ℂ}, concluding the proof of the proposition.

Write \overline{D} for the closure of the bounded symmetric domain in 𝔭^{+} . Recall that this is also the closure from S/U in S_{ℂ}/U_{ℂ}P^{−}. Hence Proposition 1.13 implies that the isomorphism Φ: Ξ → D from Theorem 1.10 extends to a G-equivariant embedding

\overline{Φ}: \overline{Ξ} → \overline{D}.

Thus \overline{D} is the natural G-invariant compactification of Ξ. Let z_{1}: = exp(i\frac{π}{2} Y_{0})K_{ℂ} ∈ ∂_{d}Ξ as before. It is known that the stabilizer of z_{1} in S is a maximal parabolic subgroup P of S. Note that P is given by

P = cU_{ℂ}P^{−}c^{−1} ∩ S.

Furthermore the Shilov boundary ∂_{s}D of D is the S -orbit through z_{1} and

∂_{s}D = S(x_{1}) = U(x_{1}) ≃ U/K.

From the G-equivariance of \overline{Φ} it hence follows that \overline{Φ}(∂_{s}Ξ) ⊆ ∂_{s}D.

Theorem 1.14. The image of the Shilov boundary ∂_{s}Ξ under the embedding \overline{Φ} is open and dense in ∂_{s}D. In particular, ∂_{s}D is a compactification of ∂_{s}Ξ.

Proof. First note that G/H ↪ S/P is an open G-invariant embedding. Hence \overline{Φ}(∂_{s}Ξ) is open and in ∂_{s}D.

Let us now show that \overline{Φ}(∂_{s}Ξ) is dense. For that we realize \overline{Ξ} in \overline{D} via the embedding \overline{Φ}. If Y: = \overline{Φ}(∂_{s}Ξ) is not dense, then we find a continuous function f on \overline{D}, holomorphic on D = Ξ which does not attain its maximum on \overline{Y} . But this contradicts the fact that Ξ = D and that ∂_{s}Ξ is the Shilov boundary of Ξ.

2. Continuous branching of spherical highest weight representations

Throughout this section (𝔰, τ) denotes a compactly causal symmetric Lie algebra with τ - eigenspace decomposition 𝔰 = 𝔤 ⊕ 𝔮_{∗}. We let S be a simply connected Lie group with Lie algebra 𝔰 and write G: = S^{τ} for the fixed point group. Observe that G is connected since S was supposed to be simply connected.

In this section we discuss the generalized Segal-Bargmann transform introduced in [ÓØ85], see also [DÓZ01,Ner99,Ól00,Zh01]. This allows us to give a sharp criterion for a G-spherical unitary highest weight representation (π, H) of S for continuous branching of π |_{G}.

As before we write U for the analytic subgroup of S associated to 𝔲 and U_{ℂ} for its complexification. We denote by P^{±} the analytic subgroup of S_{ℂ} with Lie algebras 𝔭^{±}. Assume for a moment that S is not simply connected and that S ⊆ S_{ℂ}. By the Harish Chandra decomposition of S we have S ⊆ P^{+}U_{ℂ}P^{−}. For s ∈ S write s = p^{+}(s)κ(s)p^{−}(s) with the obvious notation. Recall that the map S ∋ s ↦ log p^{+}(s) ∈ 𝔭^{+} gives the realization of S/U as a bounded symmetric domain in 𝔭^{+} . Further we have the middle projection κ : S → U_{ℂ}. By the usual lifting argument we obtain such a map κ also for a simply connected group S. This lifting is unique by the requirement of κ(1) = 1. We denote by X ↦ \overline{X} the conjugation in 𝔰_{ℂ} with respect to the real form 𝔰. Then \overline{𝔭^{+}} = 𝔭^{−}.

Realization of unitary highest weight representations.

Our source of reference for the facts collected below is [Ne99, Ch. XII]. Let (π_{λ}, H_{λ}) be a unitary highest weight representation of S with highest weight λ ∈ i𝔱^{∗} and with respect to the positive system Δ^{+} . Write F (λ) for the finite dimensional lowest U -type in H_{λ}, i.e., the space which is generated by applying U to a highest weight vector v_{λ} of H_{λ}. We now briefly recall the realization of H_{λ} in O(D, F (λ)). For that we write σ_{λ} for the representation of U_{ℂ} on F (λ). Define

K: D × D → U_{ℂ}, (z, w) ↦ κ(exp(−\overline{w}) exp(z))^{−1}
J: S × D → U_{ℂ}, (s, z) ↦ κ(g exp(z)).

Set

(2.1)  K_{λ} = σ_{λ} ∘ K and J_{λ} = σ_{λ} ∘ J .

Note that K_{λ} is holomorphic in the first variable, antiholomorphic in the second and satisfies the cocyle identity

(2.2)  K_{λ}(s(z), s(w)) = J_{λ}(s, z)K_{λ}(z, w)J_{λ}(s, w)^{∗}

for all s ∈ S, z, w ∈ D. Here A^{∗} denotes the adjoint of an operator A ∈ B(F (λ)) = End(F (λ)).

The relation with (π_{λ}, H_{λ}) is now as follows: π_{λ} is unitary if and only if K_{λ} is positive definite. If this is the case, then (π_{λ}, H_{λ}) can be realized in O(D, F (λ)) as the reproducing kernel Hilbert space corresponding to K_{λ}. So henceforth we will assume that H_{λ} ⊆ O(D, F (λ)). The representation π_{λ} is then given by

(2.3)  (π_{λ}(s)f)(z) = J_{λ}(s^{−1}, z)^{−1}f(s^{−1}(z))

for f ∈ H, s ∈ S and z ∈ D.

G-spherical unitary highest weight representation

A unitary highest weight representation (π_{λ}, H_{λ}) is called G-spherical if there exists a nontrivial G-fixed distribution vector ν in H_{λ}^{−∞}. Here H_{λ}^{−∞} denotes the module of distribution vectors of (π_{λ}, H_{λ}). If (π_{λ}, H_{λ}) is a unitary highest weight representation of S, then we call λ regular if H_{λ} is dense in O(D, F (λ)), or, equivalently if λ|_{𝔷(𝔲)} lies on the continuous line of the Wallach set (this is made precise in Appendix A). We say that (π_{λ}, H_{λ}) if F (λ) = ℂ. This is equivalent to λ ∈ i𝔷(𝔲)^{∗}.

We now recall some facts concernig G-spherical unitary highest weight representation. For more information and an almost complete classification see [KrNe00].

· If (π_{λ}, H_{λ}) is G-spherical, then ν_{0}: = ν |_{F (λ)} is non-zero and K -fixed. In particular, F (λ)

is a K -spherical module for U. · If λ is regular, then (π_{λ}, H_{λ}) is G-spherical if and only if F (λ) is K -spherical. · If λ ∈ i𝔷(𝔲)^{∗} is regular, then (π_{λ}, H_{λ}) is G-spherical if and only if λ ∈ i(𝔷(𝔲) ∩ 𝔮_{∗})^{∗}. In particular, if S is simple, then all regular λ ∈ i𝔷(𝔲)^{∗} correspond to G-spherical representations.

· If we are in the group case S = G × G, then the G-spherical unitary highest weight representations of S are exactly the representations π_{λ} ⊗ π_{λ}^{∗} with π_{λ} a unitary highest weight representation of G and π_{λ}^{∗} its dual representation.

The generalized Segal-Bargmann transform

In this subsection (π_{λ}, H_{λ}) denotes a G-spherical unitary highest weight representation of S. Note that this means in particular that there exists a non-trivial K -fixed vector ν_{0} ∈ F (λ)^{∗}. Write ν_{0} = ⟨·, v_{0}⟩ where 0 ≠ v_{0} ∈ F (λ) is a K -fixed vector. Define

D_{λ}: G → B(F (λ)), g ↦ J_{λ}(g, 0)^{−1} = σ_{λ}(κ(g))^{−1}

and

∥D_{λ}∥: G/K → ℂ, gK → ∥D_{λ}(g)∥

where ∥ · ∥ denotes the operator norm on B(F (λ)). Write A(G/K) for the space of analytic functions on G/K. Define a restriction map

R_{λ} : H_{λ} → A(G/K), R_{λ}(f)(gK) := ⟨D_{λ}(g)f(gK), v_{0}⟩.

Note that this map is well defined, since v_{0} is K -fixed. We write L for the left regular representation of G on functions on G/K, i.e., (L(g)f)(xK) = f(g^{−1}xK) for g, x ∈ G and f a function on G/K. We have then the following Lemma, see [Ól00], Lemma 3.4:

Lemma 2.1. Let (π_{λ}, H_{λ}) be a G-spherical unitary highest weight representation of S.

(i) The map R_{λ}: H_{λ} → A(G/K) intertwines π_{λ} |_{G} with L. Moreover, if λ ∈ i𝔷(𝔲)^{∗}, then R_{λ} is injective.

(ii) If λ is regular and ∥D_{λ}∥: G/K → ℂ is square integrable, then ImR_{λ} ⊆ L^{2}(G/K) is dense and R_{λ}: H_{λ} → L^{2}(G/K) is continuous.

Assume now that ∥D_{λ}∥ ∈ L^{2}(G/K) and that λ is regular. Then R_{λ}: H_{λ} → L^{2}(G/K) is continuous with dense image. Thus we can polarize the continuous operator R_{λ} as R_{λ} = U_{λ}P_{λ} where P_{λ} = (R_{λ}^{∗}R_{λ})^{\frac{1}{2}} and U_{λ} is a partial isometry. If λ ∈ i𝔷(𝔲)^{∗}, then U_{λ} is an isometry and the unitary G-equivariant isomorphism

U_{λ}^{∗} : L^{2}(G/K) → H_{λ}

is called the generalized Segal-Bargmann transform (cf. [Ól00]).

We are going now to specify the parameters for which ∥D_{λ}∥ is square integrable. We will distinguish the two cases of S simple and S = G × G. The group case is much simpler, since then the square integrability of D_{λ} can be easily reduced to known integrals. Although we will present a general proof for a an exact criterion on λ for the square integrability of D_{λ} we will treat the group case separately since the general proof requires some knowledge of advanced structure theory the general reader might not be so familiar with.

The square integrability of ∥D_{λ}∥: group case

In this section we consider the case where S = G × G. We let (π_{λ}, H_{λ}) be a unitary representation of G. Then the S -representation we consider is π̂_{λ} = π_{λ} ⊗ π_{λ}^{∗}. Recall that these are precisely the G-spherical unitary highest weight representations of S. As explained before we can realize H_{λ} in O(G/K, F (λ)). As before we write K_{λ} for its reproducing kernel and J_{λ} for the corresponding cocycle. We recall now some well known facts on representations of the holomorphic discrete series on G.

Write Z for the center of G. We say (π_{λ}, H_{λ}) belongs to the relative discrete series if all matrix coefficients of (π_{λ}, H_{λ}) are square integrable modulo Z.

By abuse of notation we will denote here by 𝔱 a compact Cartan algebra of 𝔤 and Δ for the corresponding root system. We write β for the highest root in Δ^{+} and write ρ: = \frac{1}{2} ∑_{α∈Δ^{+}} α.

Lemma 2.2. Let (π_{λ}, H_{λ}) be a unitary highest weight representation of the simply connected Hermitian Lie group G. Then the following assertions are equivalent:

(1) (π_{λ}, H_{λ}) belongs to the relative discrete series.

(2) The highest weight λ ∈ i𝔱^{∗} satisfies the Harish Chandra condition

⟨λ + ρ, β⟩ < 0.

(3) The integral

∫_{G/K} ∥K_{λ}(z, z)∥^{−1} dμ_{G/K}(z) < ∞

is finite. Here ∥·∥ denotes the operator norm on B(F (λ)) and μ_{G/K} the invariant measure on G/K.

(4) The integral

∫_{G/Z} ∥J_{λ}(g, 0)^{−1}(J_{λ}(g, 0)^{−1})^{∗}∥ dμ_{G/Z}(gZ) < ∞

is finite. Proof. (1) ⟺ (2) is the criterion of Harish Chandra (cf. [Ne99, Th. XII.5.12]. For the equivalence of (2) and (3) see [Ne99, Th. XII.5.6]. Finally the equivalence of (3) and (4) is immediate from (2.1).

As before we realize (π̂_{λ}, H_{λ} ˆ⊗H_{λ}^{∗}) in O(G/K × \overline{G/K}, B(F (λ))). Here we used the identification B(F (λ)) = F (λ) ⊗ F (λ)^{∗}. The U = K × K -spherical vector in B(F (λ)) is the trace in this case. The function ∥D_{λ}∥ in this case is given by

∥D_{λ}∥: G/K → ℝ^{+}, gK ↦ tr(J_{λ}(g, 0)^{−1}(J_{λ}(g, 0)^{−1})^{∗}).

If G is a locally compact group, then we write μ_{G} for a left invariant Haar measure on G.

Lemma 2.3. Let (π_{λ}, H_{λ}) be a unitary highest weight representation of G. Then D_{λ} is square integrable if and only if

⟨2λ + ρ, β⟩ < 0

for β the highest root in Δ^{+} . Proof. That D_{λ} is square integrable means that

(2.4)  ∫_{G/Z} ∥J_{λ}(g, 0)^{−1}(J_{λ}(g, 0)^{−1})^{∗})∥^{2} dμ_{G/Z}(gZ) < ∞.

Assume for a moment that λ ∈ i𝔷(𝔨)^{∗}, i.e., F (λ) ≃ ℂ. Then J_{λ}(g, 0)^{2} = J_{2λ}(g, 0) and (2.2) becomes equivalent to

(2.5)  ∫_{G/Z} ∥J_{2λ}(g^{−1}, 0)J_{2λ}(g^{−1}, 0)^{∗}∥ dμ_{G/Z}(gZ) < ∞.

From Lemma 2.1 (2) ⟺ (4) we know obtain that (2.5) is equivalent to

⟨2λ + ρ, β⟩ < 0

for β the highest root in Δ^{+} . Finally the technique to estimate the integrals in Lemma 2.2 (4) as presented in [Wa79, Lemma 3.1] shows that the assumption that λ ∈ i𝔷(𝔨)^{∗} is no restriction to the general case.

Lemma 2.4. If the highest weight λ of (π_{λ}, H_{λ}) satisfies the condition of ⟨2λ+ρ, β⟩ < 0, then λ is regular. Proof. This follows from the classification of unitary highest weight modules (cf. [EHW83]). In the scalar case this is proved in Lemma A.2 (iv) in Appendix A.

Assume now that ⟨2λ+ρ, β⟩ < 0 holds. In view of Lemma 2.3 and Lemma 2.4, this implies in particular that D_{λ} is square integrable. In view of Lemma 2.1 we have thus proved:

Theorem 2.5. Let (π_{λ}, H_{λ}) be a unitary highest weight representation with highest weight λ satisfying the condition ⟨2λ + ρ, β⟩ < 0 with β the highest root. Then we have an onto G-equivariant partial isometry

U_{λ}: H_{λ} ˆ⊗H_{λ}^{∗} → L^{2}(G/K)

which is an isomorphism when λ ∈ i𝔷(𝔨)^{∗}. In particular, for λ ∈ i𝔷(𝔨)^{∗} the branching of the irreducible unitary representation π_{λ} ⊗ π_{λ}^{∗} of G × G to the diagonal subgroup G is completely continuous and multiplicity free.

The square integrability of ∥D_{λ}∥: general case

In this subection we will prove a theorem which will give us the exact range of parameters for the square integrability of D_{λ}. For that recall some results from [ÓØ88] applied to the symmetric Lie algebra (𝔰, θτ), which is also compactly causal. The proof of the statements uses simple 𝔰𝔲(1, 1)-reduction (cf. Appendix B). Choose the Cartan subspace 𝔱 such that 𝔢: = 𝔱∩𝔮_{∗} is maximal abelian in 𝔮_{∗}. Write Δ_{𝔢} = Δ(𝔰_{ℂ}, 𝔢_{ℂ}) for the restricted root system with respect to 𝔢_{ℂ}. Note that Δ_{𝔢} is always of type C_{n} or BC_{n}. Define a positive system Δ_{𝔢}^{+} by Δ_{𝔢}^{+}: = Δ^{+} |_{𝔢}\{0}. Further set Δ_{𝔢,n}^{+}: = Δ_{n}^{+} |_{𝔢}. Note that γ |_{𝔢} ≠ 0 for all γ ∈ Δ_{n}.

Let γ_{1}, …, γ_{r} be a maximal set of long strongly orthogonal roots in Δ_{𝔢,n}^{+} . Then we can choose E_{±j} ∈ 𝔰_{ℂ}^{±γ_{j}} such that with X_{j} = −i(E_{j} − E_{−j}) the space 𝔞 = ⨁_{j=1}^{r} ℝX_{j} is maximal abelian in 𝔭. Let H_{j} ∈ i𝔢 be such that γ_{i}(H_{j}) = 2δ_{ij} then

κ(exp(∑_{j=1}^{r} t_{j}X_{j})) = exp (−\frac{1}{2} ∑_{j=1}^{r} log(cosh(2t_{j}))H_{j})) .

Hence

∥D_{λ}(exp(∑_{j=1}^{r} t_{j}X_{j}))∥ = exp (\frac{1}{2}⟨λ, ∑_{j=1}^{r} log(cosh(2t_{j}))H_{j}⟩) .

Thus with a = exp(∑_{j=1}^{r} t_{j}X_{j}) and k_{1}, k_{2} ∈ K:

(2.6)  ∥D(k_{1}ak_{2})∥^{2} = ∏_{j=1}^{r} cosh(2t_{j})^{⟨λ,H_{j}⟩} .

The root vectors E_{j} -determine a Cayley transform c such that c(∑_{j=1}^{r} ℝH_{j}) = 𝔞. Recall that Σ = Δ(𝔤, 𝔞) is the set of restricted roots of 𝔞 in 𝔤. Choose Σ^{+} such that Σ^{+} ⊆ Δ^{+} ∘ c^{−1} . Let ρ_{G} be the half sum of positive roots counted with multiplicities and let ρ_{G}^{c} = ρ_{G} ∘ c^{−1} ∈ i𝔢^{∗}.

Theorem 2.6. We have D_{λ} ∈ L^{2}(G/K) if and only if

⟨λ + ρ_{G}^{c}, β⟩ < 0

for β the highest root in Δ^{+} .

Proof. First note that ⟨λ + ρ_{G}^{c}, β⟩ < 0 is equivalent to

⟨λ + ρ_{G}^{c}, α⟩ < 0 (∀α ∈ Δ_{n}^{+}).

Further the fact that λ, ρ_{G}^{c} ∈ i𝔢^{∗} implies that this is equivalent to

⟨λ + ρ_{G}^{c}, α⟩ < 0 (∀α ∈ Δ_{𝔢,n}^{+}).

Let

Δ_{G}(k_{1}ak_{2}) := ∏_{α∈Σ^{+}} sinh(α(log a))^{m_{α}} k_{1}, k_{2} ∈ K, a ∈ A

where m_{α} = dim 𝔤^{α}. Then for f ∈ L^{1}(G):

∫_{G} f(g) dg = ∫_{K} ∫_{A^{+}} ∫_{K} f(k_{1}ak_{2}) Δ_{G}(a) dμ_{K}(k_{1}) dμ_{A}(a) dμ_{K}(k_{2})

There exists a non-zero polynomial function p(X) on 𝔞 such that for all a ∈ A^{+} and some costants c_{1}, c_{2} > 0 one has

c_{1}|p(log a)|a^{2ρ_{G}^{c}} ≤ Δ(a) ≤ c_{2}a^{2ρ_{G}^{c}} .

From this and (2.6) we get for a = exp(∑_{j=1}^{r} t_{j}X_{j}) ∈ A^{+} some constants C_{1}, C_{2} > 0:

C_{1}|p(log a)| ∏_{j=1}^{r} exp(2t_{j}⟨λ + ρ_{G}^{c}, H_{j}⟩) ≤ ∥D(k_{1}ak_{2})∥^{2}Δ(a) ≤ C_{2} ∏_{j=1}^{r} exp(2t_{j}⟨λ + ρ_{G}^{c}, H_{j}⟩) .

The claim now follows as t_{j} > 0 for a ∈ A^{+} .

From Theorem 2.6 and Lemma 2.1 we now obtain the following result:

Theorem 2.7. Let (π_{λ}, H_{λ}) be a G-spherical unitary highest weight representation of S. Suppose the highest weight λ is regular and satisfies the condition

⟨λ + ρ_{G}^{c}, β⟩ < 0

for β the highest root in Δ^{+} . Then we have an onto G-equivariant partial isometry

U_{λ}: H_{λ} → L^{2}(G/K)

which is an isomorphism when λ ∈ i(𝔷(𝔲) ∩ 𝔮_{∗})^{∗}. In particular, for λ ∈ i(𝔷(𝔲) ∩ 𝔮_{∗})^{∗} the representation π_{λ} |_{G} of G is multiplicity free with completely continuous spectrum.

Remark 2.8. (a) The condition in Theorem 2.7 is the same as the one in Lemma 2.3 in the group case. The condition on the regularity of λ in Theorem 2.7 is probably superfluous, but we did not check it.

(b) The results in [GrKo01] for 𝔰 = 𝔰𝔩(2, ℝ) ⊕ 𝔰𝔩(2, ℝ) show that the condition ⟨λ + ρ_{G}^{c}, β⟩ < 0 on λ in Theorem 2.7 is also necessary for continuous branching. It would be interesting to know if this is generally true.

Example 2.9. Let us consider S = S̃l(2, ℝ) with G = SO(1, 1). In this case ρ_{G}^{c} = 0 and this means that all unitary highest weight representations π_{λ} have continuous branching for π_{λ} |_{G}.

On the other hand consider the group case S = S̃l(2, ℝ) × S̃l(2, ℝ). Here the condition ⟨λ + ρ_{G}^{c}, β⟩ < 0 gives a real restriction. Further by the results in [GrKo01] we do not have continuous branching for π_{λ} ⊗ π_{λ}^{∗} to G on the full half line.

Applications to the classical Hardy space

Write ∂_{s}D for the Shilov boundary of D ≃ S/U and recall that ∂_{s}D is the U -orbit through a certain E ∈ ∂_{s}D. Note that D is a circular domain. In particular we have r∂_{s}D ⊆ D for all 0 ≤ r < 1.

The Hardy space parameter λ_{h} ∈ i(𝔷(𝔲) ∩ 𝔮_{∗})^{∗} is defined by

λ_{h} = −ρ_{n}: = −\frac{1}{2} ∑_{α∈Δ_{n}^{+}} α.

The corresponding Hilbert space H_{λ_{h}} is the classical Hardy space on D:

H^{2}(D): = {f ∈ O(D): sup_{0≤r<1} ∫_{U} |f(ru(E))|^{2} dμ_{U}(u) < ∞}.

Here we identified the Shilov boundary of ∂_{s}D with U(E) as explained before. We write π_{h}: = π_{λ_{h}} for the unitary irredducible representation of S on H^{2}(D).

Lemma 2.10. Assume that (𝔰, 𝔤, 𝔥) is a causally symmetric triple. Then λ_{h} is regular and satisfies the condition ⟨λ_{h} + ρ_{G}^{c}, β⟩ < 0. Proof. See Proposition A.3 in the appendix.

From this we obtain the following important corollary to Theorem 2.8:

Corollary 2.11. Let (𝔰, 𝔤, 𝔥) be a causally symmetric triple and H^{2}(D) be the classical Hardy space on D ≃ S/U. Then the inverse of the Segal-Bargmann transform

U: (π_{h} |_{G}, H^{2}(D)) → (L, L^{2}(G/K))

is a G-equivariant unitary isomorphism. In particular, the branching π_{h} | _{G} is completely continuous and multiplicity free.

3. Analytical and geometrical constructions on Ξ

From now on we will assume that (𝔰, 𝔤, 𝔥) is a causally symmetric triple. We let S_{ℂ} be a simply connected Lie group with Lie algebra 𝔰_{ℂ} and let S,U, G, K, H the analytic subgroups of S_{ℂ} with Lie algebras 𝔰, 𝔲, 𝔤, 𝔨 and 𝔥. The objective of this section is to establish certain geometrical and analytical facts which will be needed later on.

The function ψ

We will construct a function ψ which will be used for comparing the classical Hardy space with our Hardy space on Ξ in Section 6.

Denote by (π_{m}, V_{m}) the irreducible representation of U_{ℂ} with lowest weight −mρ_{n}. If such representation exists it is K_{ℂ} -spherical. Note that V_{m} ≃ ℂ is one dimenisonal since ρ_{n} ∈ i(𝔷(𝔲) ∩ 𝔮_{∗})^{∗}. Also notice that the representation det Ad |_{𝔭^{−}} of U has lowest weight −2ρ_{n}. Let

ζ = \frac{1}{2}(γ_{1} + … + γ_{r}) ∈ i(𝔷(𝔲) ∩ 𝔮_{∗})^{∗}

where γ_{1}, …, γ_{r} is a maximal set of long strongly orthogonal roots in Δ_{n}^{+} . Let

d = dim 𝔰_{ℂ}^{\frac{1}{2} (γ_{i}+γ_{j})} = dim 𝔰_{ℂ}^{\frac{1}{2} (γ_{i}−γ_{j})}

(cf. Appendix A for the table for d). Then

(3.1)  ρ_{n} = (1 + \frac{d(r − 1)}{2})ζ

(cf. Appendix A).

Proposition 3.1. Let the notation be as above. Let m ∈ ℕ be minimal such that (π_{m}, V_{m}) exists. Then the following holds:

(i) If (𝔤, 𝔥) is of Cayley type and 𝔤 ≠ 𝔰𝔬(2, 2k + 1), 𝔰𝔭(2n, ℝ) then m = 1. (ii) If (𝔤, 𝔥) is of Cayley type and 𝔤 = 𝔰𝔬(2, 2k + 1), 𝔰𝔭(2n, ℝ) then m = 2. (iii) If (𝔰, 𝔤) = (𝔰𝔲(2n, 2n), 𝔰𝔭(n, n)) then m = 1. (iv) If (𝔰, 𝔤) = (𝔰𝔭(2n, ℝ), 𝔰𝔭(n, ℂ)) then m = 2.

(v) If (𝔰, 𝔤) = (𝔰𝔬(2, n), 𝔰𝔬(1, n)) for n > 2 then m = 1 if n is even and m = 2 for n odd. Proof. Using the table for d in Appendix A, this follows from (3.1) by straightforward computation.

Lemma 3.2. The set is G_{ℂ}U_{ℂ}P^{−} open and dense in S_{ℂ}. Furthermore S ⊆ G_{ℂ}U_{ℂ}P^{−}. Proof. This is Theorem 2.4 in [ÓØ88].

Assume now that (π_{m}, V_{m}) exists. Then V_{m} = ℂv_{m} and v_{m} is both a lowest weight vector and a K_{ℂ} -spherical vector. Normalize v_{m} such that (v_{m}, v_{m}) = 1. Let

U := {z ∈ 𝔭^{+}: exp(z) ∈ G_{ℂ}U_{ℂ}P^{−} ∩ P^{+}U_{ℂ}P^{−}} .

Since P^{+}U_{ℂ}P^{−} is open an dense in S_{ℂ} it follows from Lemma 3.2 that U is an open dense, hence locally G_{ℂ} invariant subset of 𝔭^{+} . Moreover we have D ⊆ U.

For s ∈ U define K_{ℂ}u_{G}(s) ∈ K_{ℂ}\U_{ℂ} by

(3.2)  s ∈ G_{ℂ}u_{G}(s)P^{−} .

Define a function ψ_{m} : U → ℂ by

(3.3)  ψ_{m}(z) := (π_{m}(u_{G}(exp(z))v_{m}, v_{m}) .

Proposition 3.3. Let ψ_{m} be as above. Then the following assertions hold:

(i) Let g ∈ G_{ℂ} and z ∈ U such that gz ∈ U. Then

ψ_{m}(gz) = J_{mρ_{n}}(g, z)ψ_{m}(z) .

(ii) ψ_{m} is holomorphic and has no zeros on D. Proof. (i) We have g exp(z) = exp(gz)J(g, z)p with p ∈ P^{−}. Hence

exp(gz) = g exp(z)J(g, z)^{−1}p′

for some p′ ∈ P^{−}. It follows that

ψ_{m}(gz) = (π_{m}(exp(z))π_{m}(J(g, z)^{−1})v_{m}, v_{m})) = J_{mρ_{n}}(g, z)ψ_{m}(z)

because v_{m} is a weight vector with weight −mρ_{n}. (ii) The holomorphicity of ψ_{m} is clear by construction. To see that ψ_{m}(z) ≠ 0 for z ∈ D write z = g(0) for some g ∈ G exp(iΩ). Since D ⊆ U, we obtain from (i) that

ψ_{m}(z) = ψ_{m}(g(0)) = J_{mρ_{n}}(g, 0)ψ_{m}(0).

Now ψ_{m}(0) = (v_{m}, v_{m}) = 1 ≠ 0 and J_{mρ_{n}}(g, 0) ≠ 0 by construction.

From now on we consider ψ_{m} as a function on D only. Note that ψ_{2} always exists by Proposition 3.1. Since D is simply connected and ψ_{m} is zero-free it follows from Proposition 3.3(ii) that we can define a holomorphic square root ψ(z): = \sqrt{ψ_{2}(z)} which becomes unique under the requirement ψ(0) = 1. Note that ψ = ψ_{1} in case (π_{1}, V_{1}) exists.

Write S̃ for the universal covering of S and G_{1} for the analytic subgroup of S̃ with Lie algebra 𝔤. Note that J_{ρ_{n}}(s, z) exists for all s ∈ S̃ and z ∈ D (cf. Section 2). From Proposition 3.3(i) we thus obtain that ψ(gz) = J_{ρ_{n}}(g, z)ψ(z) for all g ∈ G_{1} and z ∈ D. In particular, J_{ρ_{n}}(g, z) = \frac{ψ(z)}{ψ(gz)} and so J_{ρ_{n}}(g, z), initially defined only on G_{1} × D, factors to a function on G × D, which we denote by J_{h}^{−1} . Summarizing our discussions we have proved:

Proposition 3.4. There exist unique holomorphic function ψ: D → ℂ^{∗} and a unique analytic function J_{h}: G × D → ℂ^{∗} with the following properties:

(i) ψ(gz) = J_{h}^{−1}(g, z)ψ(z) for all g ∈ G, z ∈ D. (ii) ψ^{2} = ψ_{2} and ψ(0) = 1. (iii) J_{h} satisfies the cocycle property J_{h}(g_{1}g_{2}, z) = J_{h}(g_{1}, g_{2}z)J_{h}(g_{2}, z) for g_{1}, g_{2} ∈ G, z ∈ D

as well as J_{h}^{−2} = J_{2ρ_{n}} and J_{h}(1, z) = 1 for all z ∈ D.

Example 3.5. Let S = G × G, and G the diagonal group. Write G ⊂ P_{G}^{+}K_{ℂ}P_{G}^{−} for the triangular decomposition for G_{ℂ} and let k_{1}(g) ∈ K_{ℂ} be the corresponding Harish-Chandra projection. If D_{G} ⊆ 𝔭_{G}^{+} is the bounded realization of G/K, then D = D_{G} × D_{G}^{opp} where D_{G}^{opp} denotes D_{G} equipped with the opposite complex structure. Note that P^{+} = P_{G}^{+} × P_{G}^{−} and P^{−} = P_{G}^{−} × P_{G}^{+} . In particular we are realizing S/U inside (G_{ℂ}/K_{ℂ}P^{−}) × (G_{ℂ}/K_{ℂ}P^{+}). Recall that the conjugation g ↦ \overline{g} with respect to G satisfies \overline{P^{+}} = P^{−}.

Let (s, t) ∈ G × G. Then

(s, t) = (t, t)(t^{−1}s, 1) = (t, t)(p_{1}^{+}k_{1}(t^{−1}s)p_{1}^{−}, 1) = (tp_{1}^{+}, tp_{1}^{+})(k_{1}(t^{−1}s)p_{1}^{−}, (p_{1}^{+})^{−1}) .

Hence

G_{ℂ}u_{G}(s, t) = G_{ℂ}(k_{1}(t^{−1}s), 1).

As U_{ℂ} = K_{ℂ} × K_{ℂ} it follows that any K_{ℂ} -spherical representation π_{U} of U_{ℂ} is of the form π_{U} = π ⊗ π^{∗} where π is an irreducible representation of K_{ℂ}. Hence

(3.4)  ψ(z, w) = π_{1}(k_{1}(exp(−w̄) exp(z))) = K_{−ρ_{n}^{G}}(z, w)^{−1}  (z, w ∈ D_{G})

where ρ_{n}^{G} is the ρ_{n} for the Hermitian group G. Notice that K_{−ρ_{n}^{G}} is nothing else but the reproducing kernel of the classical Hardy space on D_{G}.

SU(1, 1)-reduction

In this subsection we will discuss the case S = SU(1, 1) even if it is not in our list of causally symmetric triples. The reason is, that many calculations can be reduced to this situation using τ -equivariant embedding of SU(1, 1) into S. Let

S = SU(1, 1) = \{( \begin{matrix}α & β \\ β̄ & ᾱ \end{matrix} ): α, β ∈ ℂ, |α|^{2} − |β|^{2} = 1\}

Define τ : S → S by conjugation with ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). On S_{ℂ} = SL(2, ℂ) the involution τ is given by

τ ( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}d & c \\ b & a \end{matrix} )

In particular

G = \{( \begin{matrix}cosh(t) & sinh(t) \\ sinh(t) & cosh(t) \end{matrix} ): t ∈ ℝ\} .

In this situation we have

Y_{0} = \frac{1}{2} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ,

X_{0} = −i\frac{1}{2} ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ,

and

Z_{0} = \frac{1}{2} ( \begin{matrix}0 & −i \\ i & 0 \end{matrix} ) .

For the general situation of a causally symmetric triple recall the 𝔰𝔲(1, 1)-triple {X_{j}, Y_{j}, Z_{j}} from Appendix B. From the relations (B.1) and (B.2) in Appendix B it now follows:

Lemma 3.6. Let the notation be as above. Then the map X_{j} ↦ ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), Y_{j} ↦ ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ),

and Z_{j} ↦ ( \begin{matrix}0 & −i \\ i & 0 \end{matrix} ) defines a Lie algebra homomorphism into 𝔰𝔲(1, 1) intertwining the invo-

lution τ on 𝔰 and the above involution on 𝔰𝔲(1, 1). Furthermore it also intertwines the Cartan involution θ on 𝔰 and the Cartan involution X ↦ −X^{∗} on 𝔰𝔲(1, 1).

On the group level we have:

U_{ℂ} = \{( \begin{matrix}γ & 0 \\ 0 & γ^{−1} \end{matrix} ): γ ∈ ℂ^{∗}\} ,

P^{+} = \{( \begin{matrix}1 & z \\ 0 & 1 \end{matrix} ): z ∈ ℂ\} ,

and

P^{−} = \{( \begin{matrix}1 & 0 \\ w & 1 \end{matrix} ): w ∈ ℂ\} .

Further

( \begin{matrix}1 & z \\ 0 & 1 \end{matrix} ) ( \begin{matrix}γ & 0 \\ 0 & γ^{−1} \end{matrix} ) ( \begin{matrix}1 & 0 \\ w & 1 \end{matrix} ) = ( \begin{matrix}γ + zwγ^{−1} & zγ^{−1} \\ γ^{−1}w & γ^{−1} \end{matrix} ) .

Thus, if d ≠ 0 we have

( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}1 & b/d \\ 0 & 1 \end{matrix} ) ( \begin{matrix}\frac{1}{d} & 0 \\ 0 & d \end{matrix} ) ( \begin{matrix}1 & 0 \\ c/d & 1 \end{matrix} ) .

For the G_{ℂ}U_{ℂ}P^{−} -decomposition we notice that

( \begin{matrix}a & b \\ b & a \end{matrix} ) ( \begin{matrix}γ & 0 \\ 0 & γ^{−1} \end{matrix} ) ( \begin{matrix}1 & 0 \\ w & 1 \end{matrix} ) = ( \begin{matrix}aγ + bwγ^{−1} & bγ^{−1} \\ bγ + aγ^{−1}w & aγ^{−1} \end{matrix} ) .

Hence, using that a^{2} − b^{2} = 1, we obtain that

( \begin{matrix}a & b \\ c & d \end{matrix} ) = ( \begin{matrix}\frac{d}{\sqrt{d^{2}−b^{2}}} & \frac{b}{\sqrt{d^{2}−b^{2}}} \\ \frac{b}{\sqrt{d^{2}−b^{2}}} & \frac{d}{\sqrt{d^{2}−b^{2}}} \end{matrix} ) ( \begin{matrix}\frac{1}{\sqrt{d^{2}−b^{2}}} & 0 \\ 0 & \sqrt{d^{2} − b^{2}} \end{matrix} ) ( \begin{matrix}1 & 0 \\ \frac{cd−ab}{d^{2}−b^{2}} & 1 \end{matrix} ) .

Thus the open sets G_{ℂ}K_{ℂ}P^{−} and P^{+}K_{ℂ}P^{−} in S_{ℂ} can now be described as:

G_{ℂ}U_{ℂ}P^{−} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ): b^{2} − d^{2} ≠ 0\}

P^{+}U_{ℂ}P^{−} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) | d ≠ 0\}

Furthermore we notice that the P^{+}, U_{ℂ}, and P^{−} components, whenever defined, are given by

s = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ u_{G}(s) = ( \begin{matrix}\frac{1}{\sqrt{d^{2}−b^{2}}} & 0 \\ 0 & \sqrt{d^{2} − b^{2}} \end{matrix} )

and

s = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ ( \begin{matrix}1 & b/d \\ 0 & 1 \end{matrix} ) ( \begin{matrix}d^{−1} & 0 \\ 0 & d \end{matrix} ) ( \begin{matrix}1 & 0 \\ c/d & 1 \end{matrix} ) .

Notice that a double covering is needed in general for u_{G}, but as an element of the coset K_{ℂ}\U_{ℂ},

with K_{ℂ} = \{±1\} it is well defined. In the following we will identify P^{+} with ℂ by z ↦ ( \begin{matrix}1 & z \\ 0 & 1 \end{matrix} )

and ℂ^{∗} with U_{ℂ} by γ ↦ ( \begin{matrix}γ & 0 \\ 0 & γ^{−1} \end{matrix} ). In particular we get the following lemma:

Lemma 3.7. Indentify 𝔭^{+}, 𝔭^{−} with ℂ and U_{ℂ} with ℂ^{∗} in the way explained above. Let

z, w ∈ ℂ and g = ( \begin{matrix}a & b \\ c & d \end{matrix} ). Then the following assertions hold:

(i) If zw̄ ≠ 1 then K(z, w) ∈ U_{ℂ} is defined and given by

K(z, w) = \frac{1}{1 − zw̄} .

(ii) If az + d ≠ 0 then J(g, z) is defined and

J(g, z) = cz + d.

(iii) Assume that |z| < 1. Then u_{G}(z) is defined and

u_{G}(z) = \frac{1}{\sqrt{1 − z^{2}}} .

Description of ∂_{s}Ξ in ∂_{s}D

In this subsection we will use the 𝔰𝔲(1, 1)-reduction to identify ∂_{s}Ξ in ∂_{s}D as the nonvanishing locus of the function ψ.

Theorem 3.8. We have:

∂_{s}Ξ = {z ∈ ∂_{s}D: ψ(z) ≠ 0} .

Proof. “⊆”: Recall the element z_{1} ∈ ∂_{s}Ξ with isotropy subgroup H, i.e., ∂_{s}Ξ = G(z_{1}) ≃ G/H. We first show that ψ(z_{1}) ≠ 0 by using SU(1, 1)-reduction. For S = SU(1, 1) we have D = {z ∈ ℂ: |z| < 1} and z_{1} = i. In particular ψ(z_{1}) ≠ 0 by Lemma 3.7(iii). Now ψ(z_{1}) ≠ 0 in the general case follows from simple SU(1, 1)-reduction (cf. Lemma 3.6 and our structural results in Appendix B). From the covariance property of ψ (cf. Proposition 3.4) we hence get that ψ(gz_{1}) ≠ 0 for all g ∈ G. In particular ψ has no zeros on G(z_{1}) = ∂_{s}Ξ. “⊇”: By Theorem 1.14 we know that ∂_{s}Ξ is open and dense in ∂_{s}D. Let Y_{j} and 𝔞 = ∑_{j=1}^{r} ℝY_{j} be as in Appendix B. Let A = exp(𝔞). Then G = KAH. Let g_{n} be a sequence in G such that g_{n}(z_{1}) → ∂_{s}D\∂_{s}Ξ. Write g_{n} = k_{n}a_{n}h_{n} with k_{n} ∈ K, h_{n} ∈ H and a_{n} = exp(∑_{j=1}^{r} t_{r,n}Y_{j}) ∈ A. Note that g_{n}(z_{1}) → ∂_{s}D\∂_{s}Ξ means precisely that a_{n} → ∞.

We have g_{n}(z_{1}) = k_{n}a_{n}(z_{1}). We also have that |J_{h}(k, z)| = 1 for all k ∈ K. Hence it follows from Proposition 3.4 and c: = |ψ(z_{1})| > 0 that

|ψ(k_{n}a_{n}(z_{1}))| = |J_{h}^{−1}(k_{n}, a_{n}(z_{1}))| · |ψ(a_{n}(z_{1}))| = |ψ(a_{n}, z_{1})| = c|J_{h}^{−1}(a_{n}, z_{1})| .

By SU(1, 1)-reduction (cf. Lemma 3.6 and Lemma 3.7(ii)) we get

J_{h}^{−1}(a_{n}, z_{1}) = ∏_{j=1}^{r}(i sinh(t_{n,j}) + cosh(t_{n,j}))^{−ρ_{n}(H_{j})} .

As a_{n} → ∞ there exists a j and subsequence s_{n} = t_{n,j} such that |s_{n}| → ∞ and therefore

∏_{j=1}^{r} (i sinh(t_{n,j}) + cosh(t_{n,j}))^{−ρ_{n}(H_{j})} → 0

because ρ_{n}(H_{j}) > 0.

L^{2} -isomorphism on the Shilov boundaries

Let μ_{∂_{s}D} be the unique (up to constant) U -invariant measure on ∂_{s}D. If λ_{h} = −ρ_{n} is analytically integral, then S acts unitarily on L^{2}(∂_{s}D, μ_{∂_{s}D}) by

(3.5)  (π_{h}(s)f)(z) = J_{h}(s^{−1}, z)^{−1}f(s^{−1}z)  (s ∈ S, z ∈ D, f ∈ H^{2}(D))

with J_{h} = J_{−ρ_{n}}. In the case where m = 2 (see Proposition 3.1) we need to go to a double covering of S. But notice, that π_{h}(s) is always defined for s ∈ G according to Proposition 3.4. One of the consequences of this observation is the following identification of L^{2} -spaces:

Lemma 3.9. The following assertions hold:

(i) The G-invariant measure on G/H as a subset of ∂_{s}D is given by

f ↦ ∫_{∂_{s}D} f(z)|ψ(z)|^{−2} dμ_{∂_{s}D}(z)

where dμ_{∂_{s}D} is the unique (up to constant) U -invariant measure on ∂_{s}D. (ii) The mapping f ↦ \frac{1}{ψ} f is a G-equivariant isomorphism of L^{2}(G/H) onto L^{2}(∂_{s}D, μ_{∂_{s}D}).

Proof. This follows from the covariance of ψ in Proposition 3.4 (cf. [BÓ01], Theorem 5.1).

4. The classical Hardy space inside L^{2}(G/H)_{mc}

In this section we will show how to realize the classical Hardy space H^{2}(D) in the most continuous spectrum of L^{2}(G/H).

The most continuous spectrum L^{2}(G/H)_{mc}

Recall the involution σ on 𝔤 which is associated to 𝔥. Let 𝔤 = 𝔥 ⊕ 𝔮 be the σ -eigenspace decomposition. Note that we can choose 𝔞 ⊆ 𝔮 ∩ 𝔭 since (𝔤, σ) is non-compactly causal (cf. [HiÓl96]). In particular, the minimal parabolic subgroup P_{min} = MAN is also a minimal θσ - stable parabolic subgroup of G. Here, as usual, M = Z_{K}(𝔞). If 𝔪 is the Lie algebra of M, then note that 𝔪 ⊆ 𝔥 since 𝔞 ⊆ 𝔮 ∩ 𝔭. Moreover we have

M = Z_{H}(𝔞) ⊆ H

as G_{ℂ} is simply connected (cf. [HiÓl96, Lemma 3.1.22]).

Write π_{δ} = Ind_{MAN}^{G}(δ) for the K -spherical unitary principal series of parameter δ ∈ i𝔞^{∗}. Notice that π_{δ} ≃ π_{wδ} for w ∈ W. The most continuous part L^{2}(G/H)_{mc} in L^{2}(G/H) is by definition the G-invariant subspace in L^{2}(G/H) which corresponds in the Plancherel formula to to all principal series induced off from a minimal θσ -stable parabolic subgroup. Denote by W_{0} = N_{K∩H}(𝔞)/Z_{K∩H}(𝔞) the very little Weyl group. Since M = M ∩ H we now obtain from [vdBS97] or [De98] that

(L, L^{2}(G/H)_{mc}) ≃ ( ∫_{i𝔞^{∗}/W}^{⊕} π_{δ} ⊗ id dμ(δ), ∫_{i𝔞^{∗}/W}^{⊕} H_{δ} ⊗ ℂ^{|W/W_{0}|} dμ(δ))

where μ is a Borel measure which is completely continuous with respect to the Lebesgue measure on i𝔞^{∗}. In particular for a measure dense subset the multiplicity of π_{δ} in L^{2}(G/H)_{mc} is |W/W_{0}|. Let ω ⊆ i𝔞^{∗}/W be an open subset. Then we write

P_{ω}: L^{2}(G/H)_{mc} → ∫_{ω} H_{δ} ⊗ ℂ^{|W/W_{0}|} dμ(δ)

for the orthogonal G-invariant projection associated to ω.

Definition 4.1. We call a closed G-invariant subspace of W ⊆ L^{2}(G/H)_{mc} a subspace with full spectrum if P_{ω}(W) ≠ {0} for all non-empty open subets ω ⊆ i𝔞^{∗}/W.

It is interesting to compare the most-continuous spectrum L^{2}(G/H)_{mc} with the spectrum of L^{2}(G/K). The Plancherel Theorem for L^{2}(G/K) has the form

(L, L^{2}(G/K)) ≃ ∫_{i𝔞^{∗}/W} π_{δ} dμ′(δ)

with μ′ a Borel measure completely continuous with respect to dδ. Hence we see that the spectrum of L^{2}(G/K) is the same as the most continuous part of the spectrum of L^{2}(G/H), but the difference is that the spectrum of L^{2}(G/K) is multiplicity free.

Embedding of H^{2}(D) in the most continuous spectrum

Recall the U -invariant measure μ_{∂_{s}D} on the Shilov boundary of D and the boundary value map of the classical Hardy space:

b: (π_{h}, H^{2}(D)) → (π_{h}, L^{2}(∂_{s}D, μ_{∂_{s}D})), f ↦ (z ↦ lim_{{r→1 \atop r<1}} f(rz))

which is an S -equivariant isometric embedding. On the other hand we have the G-equivariant isomorphism from Lemma 3.9:

Ψ: (π_{h}, L^{2}(∂_{s}D)) → (L, L^{2}(G/H)), f ↦ ψf.

Theorem 4.2. The mapping

Ψ ∘ b: (π_{h} |_{G}, H^{2}(D)) → (L, L_{mc}^{2}(G/H))

is a G-equivariant isometric embedding. Moreover the image im Ψ ∘ b is a mulitiplicity free subspace of full spectrum in L_{mc}^{2}(G/H). Proof. From Lemma 3.9 it is clear that Ψ ∘ b is a G-equivariant map from H^{2}(D) into L^{2}(G/H). It remains to specify the image. From Corollary 2.11 we obtain that π_{h} |_{G} ≃ L^{2}(G/K). Thus the remaining assertions of the theorem follow from our discussion of the most continuous spectrum in the previous subsection.

5. The Hardy space on Ξ

In order to define the Hardy space on Ξ we first have to recall some facts on semigroups compressing D, resp. Ξ.

Recall the space 𝔢 = 𝔱 ∩ 𝔮_{∗} and associated restricted root system Δ_{𝔢} from Section 2. For α ∈ Δ_{𝔢} we denote by α̌ ∈ i𝔢 the coroot of α. Define

C_{min}: = cone({α̌: α ∈ Δ_{𝔢,n}^{+}}),

where cone(·) refers to the convex cone generated by (·). Then

W_{min}: = Ad(G)C_{min}

is a minimal open convex cone in i𝔮_{∗} and

Γ: = G exp(W_{min})

is an open G-biinvariant subsemigroup of S_{ℂ} (cf. [HiÓl96, Ch. 4]). The closure of Γ in S_{ℂ} is given by \overline{Γ} = G exp(\overline{W_{min}}). The polar mapping

G × \overline{W_{min}} → \overline{Γ}, (g, X) ↦ g exp(X)

is a homeomorphism (cf. [Ne99, Th. XI.1.7]). Moreover, \overline{Γ} is an involutive semigroup with involution

\overline{Γ} → \overline{Γ}, γ = g exp(X) ↦ γ^{∗} = exp(X)g^{−1}.

Note that Γ^{−1} compresses D:

(∀γ ∈ Γ^{−1}) γ(\overline{D}) ⊆ D

(cf. [Ne99, Th. XII.3.3]). From the realization of \overline{Ξ} in \overline{D} we therefore obtain an action of Γ^{−1} on \overline{Ξ} with the property

(5.1)  (∀γ ∈ Γ^{−1})  γ(\overline{Ξ}) ⊆ Ξ.

Definition 5.1. (Hardy space on Ξ) The Hardy space on Ξ is defined by

H^{2}(Ξ) = {f ∈ O(Ξ): ∥f∥^{2}: = sup_{γ∈Γ} ∫_{G/H} |f(γ^{−1}gz_{1})|^{2} dμ_{G/H}(gH) < ∞}.

From the definition of the Hardy space it is not clear yet that H^{2}(Ξ) is a Hilbert space. This will follow from the following geometric fact and its proof:

Lemma 5.2. The semigroup orbit Γ^{−1}(z_{1}) ⊆ Ξ is open and measure dense.

Proof. Recall the elements H_{j} ∈ i𝔢 and note that ∑_{j=1}^{n} t_{j}H_{j} ∈ C_{min} whenever t_{j} ≥ 0 and not all t_{j} = 0. By simple SU(1, 1)-reduction we obtain for all j that

(5.2)  exp(−t_{j}H_{j})(z_{1}) = exp(i arctan(e^{−t_{j}})Y_{j})(0).

Write D: = {z ∈ ℂ: |z| < 1} for the unit disc and identify A exp(iΩ)(0) with D^{n} in the obvious way through SU(1, 1)-reduction. With D^{×} = D\{0}. Equation (5.2) then gives

(5.3)  AW exp(−C_{min})(z_{1}) = (D^{×})^{n}.

In particular AW exp(−C_{min})(z_{1}) is open and Zariski dense in A exp(iΩ)(0) = D^{n}. Sweeping out Ξ by G-orbits through exp(iΩ)(0) proves the lemma.

If M is a complex manifold, then we write O(M) for the Fréchet space of holomorphic functions on M.

Corollary 5.3. The Hardy space H^{2}(Ξ) is a Hilbert space of holomorphic functions. Moreover, the inclusion mapping H^{2}(Ξ) ↪ O(Ξ) is continuous.

Proof. If we know that H^{2}(Ξ) is complete, then the standard semigroup techniques imply that H^{2}(Ξ) is a Hilbert space (cf. [HÓØ91] or [Ne99, Ch. XIV]). The completeness and the continuity of the embedding H^{2}(Ξ) ↪ O(Ξ) is easily obtained from (5.3) and Cauchy’s Theorem (use Γ = GΓG and choose local coordinates – recall the definition of the Hardy space).

Knowing that H^{2}(Ξ) is a Hilbert space, it is straightforward from the definition of H^{2}(Ξ) that the left regular representation L of G on H^{2}(Ξ) is unitary. In the terminology of [FaTh99] or [Kr99] this means that H^{2}(Ξ) is a G-invariant Hilbert space of holomorphic functions on Ξ. Those Hilbert spaces feature the following properties:

Remark 5.4. (a) The inclusion mapping H^{2}(Ξ) ↪ O(Ξ) is continuous and so all point evaluations

ev_{z}: H^{2}(Ξ) → ℂ, f ↦ f(z) (z ∈ Ξ)

are continuous. In particular for every z ∈ Ξ there exists an element K_{z} ∈ H^{2}(Ξ) such that f(z) = ⟨f, K_{z}⟩.

(b) The Hardy space admits a reproducing kernel

K_{Ξ}: Ξ × Ξ → ℂ, (z, w) ↦ ⟨K_{w}, K_{z}⟩.

Note that K_{Ξ} is holomorphic in the first and antiholomorphic in the second variable. We call K_{Ξ} the Cauchy-Szegö kernel of Ξ.

(c) By construction the representation (L, H^{2}(Ξ)) of G is unitary. This can also be phrased as that K_{Ξ} is G-invariant:

K_{Ξ}(gz, gw) = K_{Ξ}(z, w)

for all g ∈ G, z, w ∈ Ξ.

The boundary value map

Using the standard semigroup techniques for Hardy spaces (cf. [HÓØ91] or [Ne99, Ch. XIV]) one easily shows that the prescription

L: \overline{Γ} → B(H^{2}(Ξ)), (L(γ)f)(z): = f(γ^{−1}z)

defines a strongly continuous involutive contractive representation of \overline{Γ} whose restriction to Γ is a holomorphic mapping. Here strongly continuous means that the mapping \overline{Γ} → H^{2}(Ξ), γ ↦ L(γ)f is continuous for all f ∈ H^{2}(Ξ); involutive means L(γ^{∗}) = L(γ)^{∗} for all γ ∈ \overline{Γ}, and contractive means that ∥L(γ)∥ ≤ 1 for all γ ∈ \overline{Γ}. Note that L is involutive implies in particular that L|_{G} is unitary.

In order to define the boundary value map we first have to exhibit a good G-invariant subspace of H^{2}(Ξ) whose elements extend continuously up to the boundary. A good choice herefore is the space of analytic vectors H^{2}(Ξ)^{ω} of the representation (L, H^{2}(Ξ)) of G. According to [KNÓ97, App.] we have

H^{2}(Ξ)^{ω} = ⋃_{γ∈Γ} L(γ)H^{2}(Ξ).

In particular it follows from the compression property (5.1) that all functions in H^{2}(Ξ)^{ω} holomorphically extend over the closure \overline{Ξ} of Ξ. As a consequence we obtain a well defined boundary value mapping

b^{ω}: H^{2}(Ξ)^{ω} → L^{2}(G/H), f ↦ (gH ↦ lim_{{γ→1 \atop γ∈Γ}} f(γ^{−1}gz_{1}))

which is G-equivariant and continuous.

Theorem 5.5. The boundary value mapping

b^{ω}: H^{2}(Ξ)^{ω} → L^{2}(G/H), f ↦ (gH ↦ lim_{{γ→1 \atop γ∈Γ}} f(γ^{−1}gz_{1}))

extends to a G-equivariant isometric embedding b: H^{2}(Ξ) → L^{2}(G/H).

Proof. This follows from our discussion above and the standard semigroup techniques (cf. [HÓO91] or [Ne99, Ch. XIV]).

The isomorphism between H^{2}(Ξ) and H^{2}(D)

In the sequel we will identify Ξ with D under the biholomorphism Φ: Ξ → D from Section 2. We let G act on O(Ξ) via the left regular representation L and we let S (or its double covering if necessary) act on O(D) via the cocycle action π_{h}

(π_{h}(s)f)(z) = J_{h}(s^{−1}, z)^{−1}f(s^{−1}z)

for all s ∈ S, z ∈ D and f ∈ O(D). Then we have an isomorphism of topological vector spaces

Ψ: O(D) → O(Ξ), F ↦ ψF

with inverse mapping

Ψ^{−1}: O(Ξ) → O(D), f ↦ \frac{1}{ψ} f

Note that Ψ and Ψ^{−1} are defined since ψ is a zero free holomorphic function according to Proposition 3.4.

Lemma 5.6. The maps Ψ and Ψ^{−1} are G-equivariant. Proof. Since J_{h} = J_{−ρ_{n}} this follows from Proposition 3.4.

From Lemma 5.6 we obtain immediately that the mapping

Ψ: (π_{h} |_{G}, H^{2}(D)) → (L, O(Ξ)), f ↦ ψf

is a continuous G-equivariant embedding. More precisely we have:

Theorem 5.7. The mapping

Ψ: (π_{h} |_{G}, H^{2}(D)) → (L, H^{2}(Ξ)), f ↦ ψf

is a G-equivariant isomorphism of Hilbert spaces. Proof. Recall the element X_{0} ∈ 𝔷(𝔲) ∩ 𝔮_{∗}. For s ∈] − ∞, 0[ define

γ_{s}: = exp(isX_{0})

and note that γ_{s} ∈ Z(U_{ℂ}) ∩ Γ. Furthermore we have γ_{s}^{−1}(z_{1}) = z_{e^{s}} which follows by simple 𝔰𝔲(1, 1)-reduction using the results in Appendix B. We first show that \frac{1}{ψ} f ∈ H^{2}(D) for all f ∈ H^{2}(Ξ) and that ∥f∥ = ∥\frac{1}{ψ} f∥. For every s < 0 set f_{s}(gH): = f(γ_{s}^{−1}gz_{1}) for g ∈ G. Now it follows from the definition of H^{2}(Ξ) and Lemma 3.9 that

(5.4)

∥f∥^{2} = lim_{{γ→1 \atop γ∈Γ}} ∫_{G/H} |f(γ^{−1}gz_{1})|^{2} dμ_{G/H}(gH)

= sup_{s<0} ∫_{G/H} |f(γ_{s}^{−1}gz_{1})|^{2} dμ_{G/H}(gH)

= sup_{s<0} ∫_{G/H} |f_{s}(gz_{1})|^{2} dμ_{G/H}(gH)

= sup_{s<0} ∫_{∂_{s}D} |f_{s}(z)|^{2}|ψ(z)|^{−2} dμ_{∂_{s}D}(z)

= sup_{0≤r<1} ∫_{∂_{s}D} |\frac{f(rz)}{ψ(z)}|^{2} dμ_{∂_{s}D}(z).

Now consider the function

F: [0, 1] × ∂_{s}D, (r, z) ↦ |\frac{ψ(z)}{ψ(rz)}|.

Note that F is continuous, hence bounded since defined on a compact set. In particular we find a constant C > 0 such that \frac{1}{|ψ(z)|} ≥ C \frac{1}{|ψ(rz)|} for all r, z. With this information we obtain from (5.4) that

(5.5)  ∥f∥^{2} ≥ C sup_{0≤r<1} ∫_{∂_{s}D} |\frac{f(rz)}{ψ(rz)}|^{2} dμ_{∂_{s}D}(z),

i.e., \frac{1}{ψ} f ∈ H^{2}(D). Moreover from F (1, z) = 1 for all z ∈ ∂_{s}D we now obtain from (5.4) and (5.5) that

\begin{aligned}∥f∥^{2} & = lim_{r→1} ∫_{∂_{s}D} |\frac{f(rz)}{ψ(z)}|^{2} dμ_{∂_{s}D}(z) \\ & = lim_{r→1} ∫_{∂_{s}D} |F (r, z)|^{2} |\frac{f(rz)}{ψ(rz)}|^{2} dμ_{∂_{s}D}(z). \end{aligned}

Thus we have shown that Ψ^{−1} maps H^{2}(Ξ) isometrically into H^{2}(D). Finally, reversing the arguments from above, also yields that Ψ maps H^{2}(D) isometrically into H^{2}(Ξ), concluding the proof of the theorem.

An immediate conclusion from Theorem 5.7 in conjunction with Theorem 5.5 and Theorem 4.2 now is:

Theorem 5.8. The image of the boundary value map b : H^{2}(Ξ) → L^{2}(G/H) is a mulitiplicity free subspace of L^{2}(G/H)_{mc} of full spectrum.

The Cauchy-Szegö kernel

We conclude this section with the derivation of the Cauchy-Szegö kernel for H^{2}(Ξ).

Theorem 5.9. If K_{h} is the reproducing kernel of H^{2}(D) then the Cauchy-Szegö kernel K_{Ξ} is given by

K_{Ξ}(z, w) = ψ(z)\overline{ψ(w)}K_{h}(z, w).

Proof. In this proof denote by (·|·) the scalar product on H^{2}(Ξ) and by ⟨·, ·⟩ the scalar product on H^{2}(D). Fix z ∈ Ξ and let f ∈ H^{2}(Ξ). Consider the function w ↦ \overline{ψ(z)}ψ(w)K_{h}(w, z) and note that this function lies in H^{2}(Ξ). We now have

(f|\overline{ψ(z)}ψ(·)K_{h}(·, z)) = ψ(z)(f|ψ(·)K_{h}(·, z))

= ψ(z)⟨\frac{1}{ψ(·)}f, K_{h}(·, z)⟩

= ψ(z)\frac{1}{ψ(z)}f(z) = f(z).

This concludes the proof of the theorem.

Example 5.10. Let us consider the case of S = G × G. We keep the notation of Example 3.5 and identify Ξ = D with D_{G} × D_{G}^{opp} . Write K_{h}^{G}(z, w) for the kernel of the classical Hardy space on D_{G}. Then the kernel K_{h} of the classical Hardy space on D is given by

K_{h}((z_{1}, w_{1}), (z_{2}, w_{2})) = K_{h}^{G}(z_{1}, z_{2})\overline{K_{h}^{G}(w_{1}, w_{2})}

for all (z_{1}, w_{1}), (z_{2}, w_{2}) in D. From Theorem 5.8 and 3.4 we thus obtain that

(5.6)  K_{Ξ}((z_{1}, w_{1}), (z_{2}, w_{2})) = \frac{K_{h}^{G}(z_{1}, z_{2})\overline{K_{h}^{G}(w_{1}, w_{2})}}{K_{h}^{G}(z_{1}, w_{1})\overline{K_{h}^{G}(z_{2}, w_{2})}} .

In particular for G = SU(1, 1) where D_{G} = D is the unit disc we have K_{h}(z, w) = \frac{1}{1−z\overline{w}} and so

K_{Ξ}((z_{1}, w_{1}), (z_{2}, w_{2})) = \frac{(1 − z_{1}\overline{w_{1}})(1 − \overline{z_{2}}w_{2})}{(1 − z_{1}\overline{z_{2}})(1 − \overline{w_{1}}w_{2})}.

Final remarks

It is possible to extend the results in this paper to the cases in Theorem 1.12 (ii), (iii) and (iv). Let us explain this for groups G which are structure groups of Euclidean Jordan algebras (the case in Theorem 1.12(ii)). Here one has

∂_{d}Ξ_{0} = \underbrace{∐_{j=1}^{n} G/H_{j}}_{∂_{s}Ξ} ⨿G/K ⨿ G/K

with each G/H_{j} non-compactly causal (for more precise information see [GiKr01b, Sect. 4]). For every boundary component G/H_{j} of ∂_{d}Ξ_{0} one can define a Hardy space

H_{j}^{2}(Ξ_{0}) = {f ∈ O(Ξ_{0}): sup_{γ∈Γ} ∫_{G/H_{j}} |f(γ^{−1}gz_{j})|^{2} dμ_{G/H_{j}}(gH_{j}) < ∞}.

We expect a boundary value map

b_{j}: H_{j}^{2}(Ξ_{0}) ↪ L^{2}(G/H_{j})

with image a full subspace in L^{2}(G/H_{j})_{mc} . Also we expect for each boundary component G/H_{j} a G-invariant isometric isomorphism

H_{j}^{2}(Ξ_{0}) ↪ H^{2}(D)

with the classical Hardy space.

Appendix A: parameter calculations

Unless not otherwise specified 𝔰 denots a simple Hermitian Lie algebra with compact Cartan subalgebra 𝔱 ⊆ 𝔲. Write γ_{1}, …, γ_{n} for a maximal system of strongly orthogonal roots. Define

ζ: = \frac{1}{2}(γ_{1} + … + γ_{r})

and note that ζ ∈ i𝔷(𝔲)^{∗}. Note that we can choose the positive system Δ^{+} such that β = γ_{1} is the highest root. We write γ̌_{j} ∈ i𝔱 for the coroot of γ_{j}. Denote by 𝔢 the subspace of 𝔱 which is spanned by the iγ̌_{j}. Write Σ: = Δ |_{𝔢}\{0} for the restricted root system. Define Σ_{n}: = Δ_{n} |_{𝔢}, Σ_{c}: = Δ_{c} |_{𝔢}\{0}. Similarly define Σ^{+} , Σ_{n}^{+} , Σ_{c}^{+} . By results of Harish-Chandra and Moore we have:

Σ_{n}^{+} = \{\frac{1}{2}(γ_{i} + γ_{j}): 1 ≤ i, j ≤ r\}

and

Σ_{c}^{+} = \{\frac{1}{2}(γ_{i} − γ_{j}): 1 ≤ i < j ≤ r\} ∪ \{\frac{1}{2}γ_{i}: 1 ≤ i ≤ r\}.

The second term in Σ_{c}^{+} only appears if 𝔰 is not of tube type.

Define the number d: = dim_{ℂ} 𝔰_{ℂ}^{\frac{1}{2} (γ_{i}−γ_{j})} = dim_{ℂ} 𝔰_{ℂ}^{\frac{1}{2} (γ_{i}+γ_{j})} for i ≠ j. This number does not depend on i and j, only on 𝔰. The table is as follows:

\begin{matrix}𝔰 & d \\ 𝔰𝔭(n, ℝ) & 1 \\ 𝔰𝔲(p, q) & 2 \\ 𝔰𝔬(2, n) & n − 2 (n > 2) \\ 𝔰𝔬^{∗}(2n) & 2 \\ 𝔢_{6(−14)} & 3 \\ 𝔢_{(7(−25)} & 4 \end{matrix}

Theorem A.1. (Wallach) (cf. [Wa79]) Let λ_{z}: = zζ for z ∈ ℝ. Then the set of scalar unitary highest weights λ_{z} ∈ i𝔷(𝔲)^{∗} is parametrized by

z ∈ 𝕎 with 𝕎 =] − ∞, −\frac{d(r − 1)}{2}[∪\{−\frac{d(r − 1)}{2}, −\frac{d(r − 2)}{2}, …, 0\}.

The set 𝕎 in Theorem A.1 is called the Wallach set. Note that λ_{z} is regular means precisely that z < −\frac{d(r−1)}{2} .

Lemma A.2. If 𝔰 is of tube type, i.e, if Σ is of type C_{r}, then the following assertions hold: (i) ρ_{n} = (1 + \frac{d(r−1)}{2})ζ. (ii) ⟨ρ, β⟩ = \frac{1}{2}(1 + d(r − 1)). (iii) If λ ∈ i𝔷(𝔲)^{∗} is such that ⟨2λ + ρ, β⟩ < 0, then λ is regular. (iv) The Hardy space weight λ_{h} = −ρ_{n} satisfies the condition ⟨2λ_{h} + ρ, β⟩ < 0. Proof. (i) [ÓØ99, Lemma 2.5]. (ii) Note that ρ = ρ_{n} + ρ_{c}. Thus (ii) is immediate from (i) and the structure of Σ_{c}^{+} . (iii) Write λ = λ_{z}. Then by (ii) the condition ⟨2λ + ρ, β⟩ < 0 means that

z < −\frac{1}{2}(1 + d(r − 1))

and Wallach’s Theorem A.1 implies the assertion. (iv) We obtain from (i) that

⟨2λ_{h} + ρ, β⟩ = ⟨−ρ_{n} + ρ_{c}, γ_{1}⟩ = −\frac{1}{2}(1 + \frac{d(r − 1)}{2}) + ⟨ρ_{c}, γ_{1}⟩.

Now ⟨ρ_{c}, γ_{1}⟩ = \frac{d(r−1)}{4} . Thus

⟨2λ_{h} + ρ, β⟩ = −\frac{1}{2}(1 + \frac{d(r − 1)}{2}) + \frac{d(r − 1)}{4} = −\frac{1}{2} < 0,

as was to be shown.

Proposition A.3. Let (𝔰, 𝔤, 𝔥) be a causally symmetric triple. Then the following assertions hold:

(i) The Hardy space parameter λ_{h} = −ρ_{n} is regular. (ii) The Hardy space parameter λ_{h} = −ρ_{n} satisfies the condition ⟨λ_{h} + ρ_{G}^{c}, β⟩ < 0. Proof. If 𝔰 = 𝔤 ⊕ 𝔤 is the group case, then 𝔤 is of tube type and ρ_{G}^{c} = \frac{1}{2}ρ. Thus (i) and (ii) follow in this case from Lemma A.2. Hence we may assume that 𝔰 is simple. Since 𝔰 is of tube type, (i) follows from Lemma A.2(i) and Theorem A.1.

It remains to check that (ii) is satisfied for the pairs

(𝔰, 𝔤) = (𝔰𝔭(2n, ℝ), 𝔰𝔭(n, ℝ)), (𝔰𝔲(2n, 2n), 𝔰𝔭(n, ℂ)), and (𝔰𝔬(2, n), 𝔰𝔬(1, n)) .

Let us start with (𝔰𝔭(2n, ℝ), 𝔰𝔭(n, ℝ)). In this case it is easy to see that ⟨ρ_{G}^{c}, β⟩ = \frac{1}{2}⟨ρ, β⟩. Thus we obtain

⟨λ_{h} + ρ_{G}^{c}, β⟩ = \frac{1}{2}⟨2λ_{h} + ρ, β⟩ < 0

by Lemma A.2 (iv).

Next consider the case (𝔰𝔲(2n, 2n), 𝔰𝔭(n, ℂ)). Note that the restricted root system of 𝔰𝔭(n, ℂ) is of type C_{n} with with root space dimension 2 for all roots. Thus we obtain from Lemma A.2(ii) that

⟨ρ_{G}^{c}, β⟩ = \frac{1}{2}(\frac{1}{2}(1 + 2(n − 1))) = −\frac{1}{4} + \frac{n}{2}.

On the other hand we have

⟨λ_{h}, β⟩ = −\frac{1}{2}(1 + 1 · \frac{2n − 1}{2}) = −(\frac{1}{4} + \frac{n}{2}).

Thus ⟨λ_{h} + ρ_{G}^{c}, β⟩ = −\frac{1}{2} < 0.

Finally consider the pair (𝔰𝔬(2, n), 𝔰𝔬(1, n)). Here one has

⟨λ_{h} + ρ_{g}^{c}, β⟩ = ⟨λ_{h}, β⟩ + ⟨ρ_{g}^{c}, β⟩ = −\frac{n}{4} + \frac{n − 1}{4} = −\frac{1}{4} < 0,

concluding the proof of the proposition.

Appendix B: Structure theory for totally symmetric triples

In this subsection we will develop some structure theory for causally symmetric triples (𝔰, 𝔤, 𝔥). The results will be useful for certain 𝔰𝔲(1, 1)-reductions used throughout the paper. Also we will give a structural description of the maximal parabolic subgroup P of S. We will state all results without proof since they entirely consist of minor variations of results already in the literature (cf. [HiÓl96, Ch. 1, Ch. 4, App. A]).

As in Section 2 let 𝔱 ⊆ 𝔲 denote a compact τ -stable Cartan subalgebra of 𝔰. Set 𝔢: = 𝔱 ∩ 𝔮_{∗} and write Δ_{𝔢} = Δ(𝔰_{ℂ}, 𝔢_{ℂ}) for the restricted root system with respect to 𝔢_{ℂ}. We can choose Δ^{+} (without changing Δ_{n}^{+} ) such that Δ_{𝔢}^{+}: = Δ^{+} |_{𝔢}\{0} is a positive system for Δ_{𝔢}. Further we set Δ_{𝔢,n}^{+}: = Δ_{n}^{+} |_{𝔢}. Note that Δ_{𝔢} is a root system of type C_{r}. Write γ_{1}, …, γ_{r} for a maximal set of long positive strongly orthogonal roots in Δ_{𝔢}. Define H_{j} ∈ i𝔢 by γ_{k}(H_{j}) = 2δ_{kj}. Note that

X_{0} = −i\frac{1}{2}(H_{1} + … + H_{r}).

We can choose E_{±j} ∈ 𝔰_{ℂ}^{±γ_{j}} such that

(B.1)  \overline{E_{j}} = E_{−j}, τ(E_{j}) = E_{−j}, θ(E_{j}) = −E_{−j} and [E_{j}, E_{−j}] = H_{j}.

Note that the elements {H_{j}, E_{j}, E_{−j}} form an 𝔰𝔲(1, 1)-triple in 𝔰_{ℂ}. Set

Z_{j}: = −i(E_{j} − E_{−j})  and  Y_{j} := E_{j} + E_{−j}

and note that Z_{j} ∈ 𝔮_{∗} ∩ 𝔭_{∗} and Y_{j} ∈ 𝔤 ∩ 𝔭_{∗} = 𝔭.

If we set X_{j}: = −iH_{j} then the elements {X_{j}, Y_{j}, Z_{j}} form an 𝔰𝔲(1, 1)-triple in 𝔰 with relations

(B.2)  [X_{j}, Y_{j}] = Z_{j}, [X_{j}, Z_{j}] = −Y_{j}, [Y_{j}, Z_{j}] = X_{j}.

Our choice of the maximal abelian subspace 𝔞 ⊆ 𝔭 then is

(B.3)  𝔞: = ⨁_{j=1}^{r} ℝY_{j}.

Further

(B.4)  𝔟: = ⨁_{j=1}^{r} ℝZ_{j}

defines a maximal abelian subspace in 𝔮_{∗} ∩ 𝔭_{∗}. Define elements

Y_{0}: = \frac{1}{2}(Y_{1} + … + Y_{r}) and Z_{0}: = \frac{1}{2}(Z_{1} + … + Z_{r}).

Note that Spec(ad Y_{0}) = Spec(ad Z_{0}) = {−1, 0, 1}. Next we define a Cayley transform

C: 𝔰_{ℂ} → 𝔰_{ℂ}, X ↦ e^{i \frac{π}{2} ad Y_{0}}(X).

Notice the relations

C(H_{j}) = Z_{j}, C(Z_{j}) = −H_{j}, C(Y_{j}) = Y_{j}.

Since Spec(ad Y_{0}) = {−1, 0, 1}, the prescription

σ(X) = (e^{iπ ad Y_{0}} ∘ θ)(X) (X ∈ 𝔰)

defines an involution on 𝔰. Note that σ commutes with τ by construction. Write 𝔰 = 𝔥^{⋆} ⊕ 𝔮^{⋆} for the σ -eigenspace decomposition of 𝔰 and 𝔤 = 𝔥 ⊕ 𝔮 for the σ -eigenspace decomposition of 𝔤, i.e., 𝔥 = 𝔥^{⋆} ∩ 𝔤 and 𝔮 = 𝔤 ∩ 𝔮^{⋆}. Then

(B.5)  𝔞 ⊆ 𝔭 ∩ 𝔮  and  𝔟 ⊆ 𝔮_{∗} ∩ 𝔮^{⋆} ∩ 𝔭_{∗}.

As in Section 2 we let S_{ℂ} be a simply connected Lie group with Lie algebra 𝔰_{ℂ} and S, G, U, K the analytic subgroups of S_{ℂ} corresponding to 𝔰, 𝔤, 𝔲 and 𝔨. Our final goal is to describe the maximal parabolic subgroup P = C(U_{ℂ}P^{−}) ∩ S of S. For that write Σ_{𝔟} = Σ(𝔰, 𝔟) for the restricted root system of 𝔰 with respect to 𝔟. Note that Σ_{𝔟} = C^{t}(Δ_{𝔢}) and so Σ_{𝔟} is of type C_{r}. Define Σ_{𝔟,n}^{+}: = C^{t}(Δ_{𝔢,n}^{+}) and write

(B.6)  𝔫_{𝔟}^{±}: = ⨁_{α∈±Σ_{𝔟,n}^{+}} 𝔰^{α}.

Note that 𝔫_{𝔟}^{±} are both abelian and real forms of C(𝔭^{±}). Write N_{𝔟}^{±} for the analytic subgroups of S corresponding to 𝔫_{𝔟}^{±}. Write L: = C(U_{ℂ}) ∩ S. Then L is the Levi factor of P and we have

(B.7)  P = L ⋊ N_{𝔟}^{−}.

We will show that L = H^{⋆}: = S^{σ}. First we describe 𝔩: = Lie(𝔩). For that define subalgebras

𝔰(0): = 𝔷_{𝔰}(Y_{0}) and 𝔤(0) = 𝔷_{𝔤}(Y_{0}).

Then 𝔰(0) = 𝔰^{σθ} and 𝔤(0) = 𝔤^{σθ} and therefore

(B.8)  𝔤(0) = (𝔥 ∩ 𝔨) ⊕ (𝔭 ∩ 𝔮) and 𝔰(0) = (𝔥^{⋆} ∩ 𝔲) ⊕ (𝔮^{⋆} ∩ 𝔭_{∗}).

Thus we obtain

\begin{aligned}𝔩 & = C(𝔲_{ℂ}) ∩ 𝔰 = (𝔲_{ℂ} ∩ 𝔰(0)) ⊕ (C(𝔲_{ℂ}) ∩ (𝔥^{⋆} ∩ 𝔭_{∗} ⊕ 𝔮^{⋆} ∩ 𝔲)) \\ & = (𝔥^{⋆} ∩ 𝔲) ⊕ (𝔥^{⋆} ∩ 𝔭_{∗}) = 𝔥^{⋆}. \end{aligned}

Hence L = H^{⋆}. Summarizing our discussion we have shown:

Lemma B.1. Let P = C(U_{ℂ}P^{−}) ∩ S be the stabilizer of x_{1} ∈ ∂_{s}D in S. Then the Levidecomposition of the maximal parabolic subgroup P is given by

P = H^{⋆} ⋊ N_{𝔟}^{−}.

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Simon Gindikin Department of Mathematics Rutgers University New Brunswick, NJ 08903 USA

gindikin@math.rutgers.edu

Gestur Ólafsson Louisiana State University Department of Mathematics Baton Rouge, LA 70803 USA

olafsson@math.lsu.edu

Bernhard Krötz

The Ohio State University Department of Mathematics 231 West 18th Avenue Columbus, OH 43210–1174 USA

                              kroetz@math.ohio-state.edu