Hardy spaces and analytic continuation of Bergman spaces

W. Bertram, J. Hilgert

Abstract. We introduce a family of weighted Bergman spaces associated to a compactly causal symmetric space and investigate the relation of the corresponding Hardy space with the analytic continuation of this family. For an important class of compactly causal symmetric spaces defined by involutions of Euclidian Jordan algebras we obtain precise results which are in close analogy with the corresponding situation for bounded symmetric domains.

0. Introduction

0.1. A family of weighted Bergman spaces. Let D be a bounded symmetric domain of tube type, realized as a generalized disc in the complexification V_{ℂ} of a Euclidian Jordan algebra V . There is a well known family of Hilbert spaces of holomorphic functions on D, called weighted Bergman spaces and defined by

B_{μ}^{2}(D) := \{f ∈ O(D)| ||f||_{μ}^{2} := c_{μ} ∫_{D} |f(z)|^{2}(Det B(z, \overline{z}))^{μ−1} dz < ∞\}  (0.1)

(cf. [FK94, Ch.XIII]). Here the density appearing in the integral is defined by the Bergman polynomial

B(z, w) := id_{V_{ℂ}} −2z w + P (z)P (w),  (0.2)

where x y and P (x) are related to the triple product T (x, y, z) = x(yz) − y(xz) + (xy)z associated to the Jordan algebra V_{ℂ} via T (x, y, z) = (x y)z and T (x, y, x) = P (x)y, and the constant c_{μ} is chosen such that, if the norm ||1||_{μ} of the constant function 1 is finite, then it is equal to one. If V is simple, then the space B_{μ}(D) is not reduced to zero iff μ > 1 − \frac{r}{2n} (with n = dim V , r the rank of V ; cf. [FK94]; there the parameter is ν = μ\frac{2n}{r}). The space B_{1}(D) is the ordinary Bergman space of D. The harmonic analysis of the weighted Bergman spaces with respect to the group G(D) of holomorphic automorphisms of D (or, more generally, its universal covering G̃(D)) is well developed. These spaces contribute to the holomorphic discrete series of G(D).

In this work we will introduce an additional structure, define a more general class of weighted Bergman spaces and study some natural problems in their harmonic analysis. The additional structure is given by distinguishing an open dense domain

Ξ := {z ∈ D| f_{α}(z) ≠ 0}  (0.3)

in D, where α is an involutive automorphism of V and f_{α} the holomorphic polynomial on V given by

f_{α}(z) := Det(P (z + α(z))).  (0.4)

For example, if V = V_{1} × V_{1} is the direct product of a simple Jordan algebra with itself and α((x, y)) = (y, x), then Ξ is the domain considered by Chadli [Cha97]. Other data lead to the cases considered by Koufany-Ørsted [KØ97], Molchanov [Mo97], and Betten-Ólafsson [BO98]. We now define a family of weighted Bergman spaces depending on two real parameters μ and λ:

B_{μ,λ}^{2}(Ξ) := \{f ∈ O(Ξ)| ||f||_{μ,λ}^{2} := ∫_{Ξ} |f(z)|^{2}|f_{α}(z)|^{−λ}(Det B(z, \overline{z}))^{μ−1} dz < ∞\},  (0.5)

where dz is Lebesgue measure, restricted to the open domain Ξ ⊂ D. For λ = 0, any function from B_{μ,λ}^{2} can be extended to a holomorphic function on D (Prop. 2.1.1), and thus B_{μ,0}^{2}(Ξ) ≅ B_{μ}^{2}(D). In the second parameter, the behavior is “periodic” since clearly for k ∈ ℕ multiplication by f_{α}^{k} yields is an isomorphism of B_{μ,λ}^{2} onto B_{μ,λ+2k}^{2} . The argument remains valid for any rational parameter k = \frac{p}{q} such that there exists a holomorphic function f with f = f_{α}^{k} (i.e. f^{q} = (f_{α})^{p}); we call such parameters admissible.

The weighted Bergman spaces are Hilbert spaces such that for all z ∈ Ξ the point evalutions f ↦ f(z) are continuous. Thus there exists a vector K_{z}^{(μ,λ)} such that f(z) = (f|K_{z}^{(μ,λ)}). The function

K^{(μ,λ)} : Ξ × Ξ → ℂ, (w, z) ↦ K_{z}^{(μ,λ)}(w)

is called the reproducing kernel of B_{μ,λ}^{2}(Ξ). We prove (cf. Cor. 2.1.4):

Theorem 1. If μ > 1 − \frac{r}{2n} and \frac{λ}{2} is an admissible parameter, then the space B_{μ,λ}^{2}(Ξ) is non-trivial, and its reproducing kernel is given by

K^{(μ,λ)}(z, w) = f_{α}(z)^{λ/2}(Det B(z, \overline{w}))^{−μ}\overline{f_{α}(w)}^{λ/2},

where B is given by (0.2) and f_{α} by (0.4).

Next we explain how harmonic analysis enters into this picture.

0.2. Group actions and spectrum of weighted Bergman spaces. As is well known, the group G(D) acts transitively on D. The domain Ξ ⊂ D is stable under the action of the subgroup

G := G(D)^{(−α)_{∗}} := {g ∈ G(D)| (−α)_{∗}g := (−α) ∘ g ∘ (−α) = g}  (0.6)

of G(D), but this action is not transitive. However, the action of G on the open dense subset

X := {u ∈ Σ| f_{α}(u) ≠ 0}  (0.7)

of the Shilov boundary

Σ := \{z ∈ V_{ℂ}| (\overline{z})^{−1} = z\}

of D is transitive. We write X = G/H, where H is the stabilizer of the base point o = ie ∈ X. Then X is a compactly causal symmetric space, and Ξ is the domain in its complexification X_{ℂ} considered in [HOØ91] (cf. [Be98, Th.3.3.5]). The G_{ℂ} -invariant measure dν(z) on X_{ℂ}, restricted to Ξ, is related to the Lebesgue measure dz of V_{ℂ} via

dν(z) = |f_{α}(z)|^{−1} dz  (0.8)

(cf. [Be98, Prop. 2.4.3]). Thus the Bergman space of Ξ

B^{2}(Ξ) := {f ∈ O(Ξ)| ∫_{Ξ} |f(z)|^{2}dν(z) < ∞}

is equal to the space B_{1,1}^{2}(Ξ). The other spaces corresponding to the “diagonal” λ = μ have the following interpretation:

Theorem 2. If \frac{1}{2} is an admissible parameter, then the reproducing kernel of B^{2}(Ξ) is given by the function K^{(1,1)} defined in Theorem 1. If \frac{μ}{2} is an admissible parameter and μ > 1 − \frac{r}{2n}, then the space

B_{μ}^{2}(Ξ) := \{f ∈ O(Ξ)| ∫_{Ξ} |f(z)|^{2}(K^{(1,1)}(z, \overline{z}))^{1−μ}dν(z) < ∞\}

is non-trivial and equal to B_{μ,μ}^{2}(Ξ). Its reproducing kernel is K^{(μ,μ)} = (K^{(1,1)})^{μ}.

The group G acts unitarily on the spaces B_{μ}^{2}(Ξ) via ordinary translation of functions. We are interested in describing the decomposition of B_{μ}^{2}(Ξ) into irreducible representations. There are two approaches to this problem: the first describes the representations appearing in the decomposition as highest weight representations and then determines the corresponding highest weights; this has been worked out for the group cases and μ = 1 by Krötz (cf. [Kr97]). The second approach, which we use in this work, is geometric. We apply an idea that appears in the work of Jakobsen and Vergne [JV79] to the specific geometric situation given by a bounded symmetric domain with an involution α: The space

D^{−} := {z ∈ D| α(z) = −z} ⊂ D

is a bounded symmetric domain in its own right; it is associated to X = G/H in the sense that D^{−} = G/K with K maximal compact in G. It is a special feature of our setup (not predicted by the general theory of the domain Ξ in [HOØ91]) that D^{−} belongs to the boundary of Ξ; this boundary is in a sense “opposite” to the boundary X ⊂ ∂Ξ. Elements of B_{μ,2k}^{2}(Ξ) (k ≥ 0 admissible) have a holomorphic continuation to all of D; thus we can restrict them and all their partial derivatives to D^{−}. These differential restrictions determine the function entirely, and because G acts transitively on D^{−}, it is not too difficult to decompose the space of restrictions thus obtained under the action of G (Theorem 2.3.5). The interest of this technique lies in the fact that it yields not only the spectrum, but also exhibits natural intertwining operators of certain function spaces into the holomorphic discrete series of G.

0.3. Analytic continuation of weighted Bergman spaces and classical Hardy spaces. The reproducing kernel of a Hilbert space is a positive kernel, corresponding to the positivity of the scalar product. Conversely, every positive kernel defines a Hilbert space of functions. For μ > 1 − \frac{r}{2n}, the function K^{(μ,0)}(w, z) given by the formula in Theorem 1 is the reproducing kernel of a weighted Bergman space and is thus positive. The set W ⊂ ℝ of parameters μ for which the formula in Theorem 1 defines a positive kernel K^{(μ,0)} (the so called Wallach set) can be determined explicitly, cf. [FK94, Th.XIII.2.7]:

W = \{0, \frac{n − r}{2(r − 1)n}, …, \frac{1}{2} − \frac{r}{2n}\}∪]\frac{1}{2} − \frac{r}{2n}, ∞[.  (0.9)

It is strictly bigger than the Bergman range μ > 1 − \frac{r}{2n} belonging to the weighted Bergman spaces. The question arises whether the Hilbert spaces B_{μ}(D) belonging to parameters outside the Bergman range have an analytic significance similar to the Bergman spaces. This is in fact true for the parameter μ = \frac{1}{2} : The space B_{\frac{1}{2}} (D) is the Hardy space H^{2}(D) of holomorphic functions f on D having square integrable boundary values in the sense that

||f||_{H^{2}(D)}^{2} := sup_{0<r<1} ∫_{Σ} |f(ru)|^{2} dσ(u)  (0.10)

is finite, where dσ is the normalized U -invariant measure on Σ (cf. [FK94, p.269]). Here U is the stabilizer of 0 in G(D) which is compact and acts transitively on Σ. The space H^{2}(D) has a reproducing kernel, the Cauchy-Szegö kernel S(z, u), defined by the property

f(z) = ∫_{Σ} S(z, u)f(u)dσ(u).  (0.11)

Since H^{2}(D) = B_{\frac{1}{2}} (D), we have the explicit formula S = K^{( \frac{1}{2} ,0)} given by Theorem 1 for the Cauchy-Szegö kernel. Its square is the reproducing kernel of the classical Bergman space B_{1}(D).

0.4. The Hardy space of Ξ and its comparison with the classical one. We are now going to explain to what extent the preceding results can be generalized to the domain Ξ. The Hardy space H^{2}(Ξ) of Ξ is defined as the space of holomorphic functions f on Ξ having square integrable boundary values on X in the sense that

||f||_{H^{2}(Ξ)}^{2} := sup_{γ∈Γ} ∫_{X} |f(γ · x)|^{2} dx < ∞,  (0.12)

where Γ is a certain subsemigroup in the complex group G_{ℂ} such that Ξ = Γ · ie. An explicit formula for the reproducing kernel of H^{2}(Ξ) is known only in some special cases. Our approach to this problem is trying to realize the space H^{2}(Ξ) as the point belonging to the parameter μ = \frac{1}{2} in the analytic continuation of the weighted Bergman spaces B_{μ}^{2}(Ξ). Note that because of Theorem 2 the analytic continuation of the spaces B_{μ}^{2}(Ξ) is easy to describe: These are just the spaces B_{μ,μ}^{2}(Ξ) corresponding to the kernel K^{(μ,μ)} for admissible parameters μ in the Wallach set W. Our main results in this context are:

Theorem 3. Assume that \frac{1}{4} is an admissible parameter. Then we have an inclusion of function spaces

B2_{\frac{1}{2}} (Ξ) ⊂ H^{2}(Ξ).  (0.13)

Theorem 4. Assume that X is irreducible. Then the inclusion (0.13) is an equality if and only if the rank of X is equal to its split rank; this is the case if and only if X admits both compact and non-compact Cartan subspaces. Then the Cauchy-Szegö kernel of H^{2}(Ξ) is given by K^{( \frac{1}{2} , \frac{1}{2} )} , and its square is the Bergman kernel K^{(1,1)} of Ξ.

In the situation of Theorem 4, multiplication by fα^{\frac{1}{4}} is an isomorphism H^{2}(D) = B_{\frac{1}{2}} (D) → B_{\frac{1}{2} , \frac{1}{2}} (Ξ), and in this sense H^{2}(D) is realized as a subspace of H^{2}(Ξ). The question whether in this way the classical and the non-classical Hardy spaces are isomorphic has attracted much interest during the last years, cf. e.g. [BO98], [Cha97], [KØ97], [OØ98]. From our point of view, the problem can the Hardy space be realized as the point with parameter μ = \frac{1}{2} in the analytic continuation of the weighted Bergman spaces? is the mathematically correct formulation of the question are the classical and the non-classical Hardy spaces isomorphic? – the trouble being that “non-canonical” isomorphisms between the two Hardy-spaces may exist, as shows the compact case (Section 3.4). Moreover, our formulation of the problem makes sense for any compactly causal symmetric space, not only for the ones related to Jordan algebras. However, our proof of Theorem 4 uses the Jordan-structure via the explicit formula for the Bergman kernel. It would be interesting to have an abstract proof not involving this formula, either geometric or via highest weight theory. From a geometric point of view it is remarkable that in the cases where (0.13) is an equality, the “causal group” G(D) has a unitary and irreducible representation in H^{2}(Ξ).

Contents of the paper:

1. Geometric and algebraic preliminaries

2. The spectrum of a family of weighted Bergman spaces

3. Comparison of Hardy spaces

4. Open problems

Section 1 contains the classification (Section 1.5) and some preliminary results on involutions of (Euclidian) Jordan algebras which are, however, of interest in its own right (cf. Theorem 1.6.1, Theorem 1.8.2). The full information provided by Chapter 1 is needed only for the proof of Theorem 4; in the other parts of Chapters 2 and 3 only the general facts explained in Sections 1.1 – 1.4 are used. Finally, the approach to Bergman- and Hardy spaces presented in this paper leads naturally to some open problems (Chapter 4) which we will investigate in subsequent work.

We would like to thank M. Chadli and J. Faraut for helpful discussions and the Mathematics Institute of Jussieu, where part of this work was done, for the kind hospitality.

1. Geometric and algebraic preliminaries

1.1. Euclidian Jordan algebras and associated symmetric domains. Let V be a Euclidian Jordan algebra with unit element e and Ω the associated symmetric cone. The cone Ω can be defined as the component of e of the set of invertible elements of V . Our basic reference is [FK94]; we follow the notation introduced there. The tube domain T_{Ω} = V +iΩ ⊂ V_{ℂ} has, via the Cayley transform C : D → T_{Ω} , C(z) = i(e + z)(e − z)^{−1} , a bounded realization D = C^{−1}(T_{Ω}). The Shilov boundary Σ = \{z ∈ V_{ℂ}|\overline{z} = z^{−1}\} of D is, via the Cayley transform, equivalent to the conformal compactification V^{c} of V introduced in [Be96], and the group G(T_{Ω}) of biholomorphic automorphisms of T_{Ω} , acting by birational maps on V , is (up to connected components) the conformal group Co(V) introduced in [Be96]. Its complexification G(T_{Ω})_{ℂ} is the conformal group Co(V_{ℂ}) of V_{ℂ}; it acts by complex birational maps on V_{ℂ}. The stabilizer U of the base point 0 in G(D) is a compact group acting linearly on V_{ℂ}. Its complexification is (up to connected components) the structure group Str(V_{ℂ}) of V_{ℂ}. The group U acts transitively on Σ.

It is well known that the disc D is the connected component of 0 of the set {z ∈ V_{ℂ}| Det B(z, \overline{z}) ≠ 0\}, where B is as in (0.2). The G(D)-invariant measure on D has the density (Det B(z, \overline{z}))^{−1} with respect to the Lebesgue measure of V_{ℂ}, where the determinant is taken over ℂ (cf. [FK94]).

1.2. A decomposition of bounded symmetric domains. Let α be an involution (automorphism of order 2) of the Euclidian Jordan algebra V and denote by the same letter its ℂ-linear extension to V_{ℂ}. Then we have the eigenspace-decompositions V = V^{+} ⊕ V^{−} and V_{ℂ} = V_{ℂ}^{+} ⊕ V_{ℂ}^{−} with respect to α. The intersections of the eigenspaces with the bounded symmetric domain D,

D^{+} := D ∩ V^{+}, D^{−} := D ∩ V^{−},

are easily seen to be bounded symmetric domains (cf. [Be98, Prop.3.1.1]), the former being of tube type (since V^{+} is a Euclidian Jordan algebra), the latter in general not. Their automorphism groups are related to the group G(D) as follows. Let φ_{∗}(g) := φgφ^{−1} be the conjugation by an element φ of Co(V_{ℂ}); then G(D) is stable under the involutions α_{∗} and (−α)_{∗}, and we denote the connected fixed point groups of these involutions by

G^{+} := G(D)_{o}^{α_{∗}}, G^{−} := G(D)_{o}^{(−α)_{∗}}.

It is then easily verified that G^{+} acts transitively on D^{+} and G^{−} acts transitively on D^{−}; in other words, we have surjective homomorphisms

ρ_{±} : G^{±} → G(D^{±}), g ↦ g|_{D^{±}}.

In the “generic case” these homomorphisms are also injective (cf. Remark 1.8.5). The case α = id_{V} is somewhat singular since V^{−} is then reduced to a point. Remark that in this case G^{−} is the compact linear group U defined in Section 1.1, whereas G(D^{−}) should be defined as the trivial group.

1.3. Compactly causal Makarevič spaces. Let, as above, α be an involution of the Euclidian Jordan algebra V and G := G^{−}. Then, according to [Be98, Th.3.3.6], the group G acts transitively on the open dense domain

X = {u ∈ Σ| f_{α}(u) ≠ 0};

of the Shilov boundary Σ. Further, X is a symmetric space with geodesic symmetry −j with respect to the base point ie, where j(z) = z^{−1} is the inverse in the Jordan algebra V_{ℂ}. The corresponding involution of the group G is given by (−j)_{∗}g = (−j)g(−j). We thus can write X = G/H with H open in G^{(−j)_{∗}}.

The space X is a causal symmetric space: The symmetric cone Ω defines on V a flat causal structure (a constant field of cones obtained by translating Ω to each tangent space of V ) which is transported to the Shilov boundary Σ via the Cayley-transform C. The structure on V is invariant under the group Co(V) = G(T_{Ω}), and therefore the structure on Σ is invariant under G(D); in particular, its restriction to X is invariant under G. We describe this causal structure on X in more detail: Let τ be the complex conjugation of V_{ℂ} with respect to V . Then (−jτ)_{∗} is a Cartan involution on G (cf. [Be98, Section 2.2]). Denote by 𝔤 = 𝔨 ⊕ 𝔭 be the corresponding Cartan decomposition of 𝔤. Moreover let 𝔤 = 𝔥 ⊕ 𝔮 be the decomposition associated to the involution (−j)_{∗}. These decompositions can be described as spaces of vector fields. We denote by j_{∗}v the homogeneous quadratic vector field (j_{∗}v)(x) = −P (x)v with v ∈ V and by 𝔰𝔱𝔯(V_{ℂ}) ⊂ 𝔤𝔩(V_{ℂ}) the Lie algebra of the structure group of V_{ℂ}. Then using the fact that 𝔤 is the subalgebra of 𝔠𝔬(V_{ℂ}) fixed under (−α)_{∗} and under (jτ)_{∗}, one gets

\begin{aligned}𝔭 & = \{v + j_{∗}τv| v ∈ V_{ℂ}^{−}\} ≅ V_{ℂ}^{−}, \\ 𝔨 & = iL(V^{+}) ⊕ Der(V)^{α_{∗}}, \\ 𝔮 & = 𝔮_{𝔨} ⊕ 𝔮_{𝔭} = iL(V^{+}) ⊕ \{v + j_{∗}v| v ∈ V^{−}\}, \\ 𝔥 & = 𝔥_{𝔨} ⊕ 𝔥_{𝔭} = Der(V)^{α_{∗}} ⊕ i\{v − j_{∗}v| v ∈ V^{−}\}. \end{aligned}  (1.1)

The cone W in the tangent space T_{o}X ≅ 𝔮 defining the causal structure on X is given by

W := dC(0)^{−1}Ω = −{iL(v^{+}) + (v^{−} + j_{∗}v^{−})| v ∈ Ω, v = v^{+} + v^{−}, v^{±} ∈ V^{±}}.  (1.2)

In fact, the Cayley transformed version of the cone W ⊂ 𝔮 is the cone of vector fields

W′ := C_{∗}W = {v − j_{∗}αv| v ∈ Ω}  (1.3)

in the space of vector fields

𝔮′ := C_{∗}𝔮 = {v − j_{∗}αv| v ∈ V }  (1.4)

where C_{∗} denotes the forward transport of vector fields by the Cayley transform C. This cone corresponds to the causal structure considered in [Be98, Section 3.2.].

In particular,

W ∩ 𝔮_{𝔨} = −i{L(v^{+})| v^{+} ∈ Ω ∩ V^{+}}  (1.5)

is isomorphic to Ω^{+} := Ω ∩ V^{+} . Note that this is just the symmetric cone associated to V^{+} (this holds because Ω^{+} = e^{L(V^{+})} · e). Thus the causal structure we consider has the property that W ∩ 𝔮_{𝔨} ≠ Ø; i.e. X is compactly causal (cf. [HO96, Def. 3.1.8]), and moreover it has the special feature that both W and W ∩ 𝔮_{𝔨} are self-dual (w.r.t. suitable scalar products).

Finally, note that the space X appears as an open orbit in a Shilov boundary. Not all compactly causal symmetric spaces have this property, but “most” of them have (see Section 1.5). These are precisely the Makarevič spaces (cf. [Be96], [Be98]) among the compactly causal symmetric spaces.

1.4. The domain Ξ associated to a compactly causal Makarevič space. The domain Ξ defined by equation (0.3) is related to the holomorphic action of a complex semigroup as follows. If W ⊂ 𝔮 is the Ad(H)-invariant cone given by (1.2), we denote by exp(iW) the image of iW ⊂ i𝔮 ⊂ 𝔤_{ℂ} under the exponential map of G_{ℂ}. Then, according to [Be98, Th.3.3.5],

Ξ = G exp(iW) · ie.

In [HOØ91] the domain Ξ is introduced via an extension of the cone W to an Ad(G)-invariant cone W̃ in 𝔤: by general results on invariant cones (cf. [HO96]) there exists a G-invariant open convex cone W̃ ⊆ 𝔤 which satisfies (−j)_{∗}(W̃) = −W̃ and W̃ ∩ 𝔮 = W. Then (cf. [HOØ91, Lemma 3.1 and its proof]) G exp(iW̃) is a subsemigroup of G_{ℂ} such that Ξ = G exp(iW̃) · ie. It should be noted here that the extension W̃ of W with the given properties is in general not unique. Our results will not depend on the extension W̃ chosen. However, there is one “biggest” extension which is rather canonical in the given geometric context and which we will fix in the sequel.

Proposition 1.4.1. Let S(D) := {g ∈ Co(V_{ℂ})| g(D) ⊂ D} be the compression semigroup of D and let

Γ := G_{ℂ} ∩ S(D) = {g ∈ G_{ℂ}| g(D) ⊂ D}.

Then Γ is a semigroup with Ξ = Γ · ie, and it contains G exp(iW). It is of the form Γ = G exp(iW̃), where W̃ is a regular Ad(G)-invariant cone in 𝔤 with (−j)_{∗}(W̃) = −W̃ and W̃ ∩ 𝔮 = W.

Proof. From [Be98, Cor.3.3.3] it follows via Cayley-transform that G exp(iW) ⊂ Γ. Therefore [Be98, Th.3.3.5] implies that Ξ ⊂ Γ · ie. The other inclusion follows from the fact that Γ ⊂ (G_{ℂ} ∩ S(D)) and thus Γ · ie ⊂ (X_{ℂ} ∩ D) which is, according to [Be98, Th. 2.1.3 (iv)] equal to Ξ.

The remaining statements are best proved using the tube realization T_{Ω} of D. Then the compression semigroup S(T_{Ω}) is of the form S(T_{Ω}) = G(T_{Ω}) exp(iC_{max}) with the maximal invariant cone C_{max} ⊂ 𝔤(T_{Ω}) given by

C_{max} = {X ∈ 𝔤(T_{Ω})| (∀v ∈ V) X(v) ∈ Ω}

([Cha97, Th.1.1]). From Equations (1.3) and (1.4) we deduce that

W′ = 𝔮′ ∩ C_{max}.

Moreover, the Cayley transformed version of the involution (−j)^{∗} is (− id)_{∗}, and the description of C_{max} given above shows that (− id)_{∗}C_{max} = −C_{max}. It follows that the cone W̃ := C_{max} ∩ 𝔤 has all the desired properties. (Note, however, that we cannot conclude that this cone is maximal in 𝔤.)

1.5. Classification. In this section we classify simple pairs (V, α); this means that V is a Euclidian Jordan algebra having no proper ideal invariant under the involution α. The spaces V and V^{+} are described as Jordan algebras; in particular, Sym and Herm denote Jordan algebras of symmetric, respectively Hermitian, matrices with their natural Jordan product X · Y = \frac{1}{2}(AB + BA), and ℝ^{p} × ℝ^{q} is the space ℝ^{p+q} with the Jordan product x · y = b(x, e)y + b(y, e)x − b(x, y)e, where b is a form of signature (p, q) and e such that b(e, e) = 1. The spaces V^{−} are described as Jordan triple-systems (JTS): they inherit from V the Jordan triple product (x, y, z) ↦ x(yz) − y(xz) + (xy)z which defines V^{−} as non-degenerate JTS in the sense of [Sa80,I.6]. In particular, Asym and Aherm denote JTS of skew-symmetric, resp. skew-Hermitian matrices and Mat(p × q; 𝔽) denotes the JTS of p × q -matrices over 𝔽. By Hermitian we mean, if there is no other specification: Hermitian with respect to the canonical involution of the base field (cf. [Be96]). In the case 𝔽 is the skew field ℍ of quaternions, we consider also its involution φ: 1 ↦ 1, i ↦ −i, j ↦ j, k ↦ −k. Then the Jordan algebra Herm(m, φ, ℍ) is isomorphic to the Jordan triple system Aherm(m, ℍ), cf. [Be96, 1.2.1]. Further, we consider the real form V_{α} := V^{+} + iV^{−} of V_{ℂ} (the corresponding conjugation is τα, where τ, as before, denotes conjugation with respect to V .) Finally, J = ({0 \atop −I} {I \atop 0}) and I_{p,q} = ({I_{p} \atop 0} {0 \atop −I_{q}} ), where I_{n}, or simply I if the size is clear from the context, is the identity matrix of size n × n.

1.5.1 Table of simple pairs (V, α). (Cf. [Ma73] and [Be96].)

I) V = Herm(m, ℂ), V_{ℂ} = Mat(m × m, ℂ), Co(V)_{0} ≅ SU(m, m), \frac{n}{r} = m.

1) α(Z) = I_{p,q}ZI_{p,q} (p + q = m), X ≅ U(p, q), V^{+} = Herm(p, ℂ) ⊕ Herm(q, ℂ), V^{−} ≅ Mat(p × q; ℂ), V_{α} ≅ Herm(I_{p,q}; ℂ) f_{α}(Z) = Det(Z + I_{p,q}ZI_{p,q})^{2m}.

2) α(Z) = Z^{t}, X ≅ SO^{∗}(2m)/ SO(m, ℂ), V^{+} = Sym(m, ℝ),V^{−} = i Asym(m, ℝ), V_{α} = Mat(m × m, ℝ), f_{α}(Z) = Det(Z + Z^{t})^{2m}.

3) α(Z) = JZ^{t}J^{−1} (m = 2k even), X ≅ Sp(2k, ℝ)/ Sp(k, ℂ), V^{+} ≅ Herm(k, ℍ), V^{−} ≅ Aherm(k, ℍ), V_{α} ≅ Mat(k × k, ℍ), f_{α}(Z) = Det(Z − JZ^{t}J)^{4k}.

II) V = Sym(m, ℝ), V_{ℂ} = Sym(m, ℂ), Co(V)_{0} ≅ Sp(m, ℝ), \frac{n}{r} = \frac{m+1}{2} .

1) α(Z) = I_{p,q}ZI_{p,q} (p + q = m), X ≅ U(p, q)/ O(p, q), V^{+} = Sym(p, ℝ) ⊕ Sym(q, ℝ), V^{−} ≅ Mat(p × q; ℝ), V_{α} ≅ Sym(I_{p,q}; ℝ), f_{α}(Z) = Det(Z + I_{p,q}ZI_{p,q})^{m+1} .

2) α(Z) = JZJ^{−1} (m = 2k even), X ≅ Sp(k, ℝ), V^{+} ≅ Herm(k, ℂ), V^{−} ≅ Sym(k, ℂ), V_{α} ≅ Herm(k, φ, ℍ) ≅ Aherm(k, ℍ), f_{α}(Z) = Det(Z − JZJ)^{2k+1} .

III) V = Herm(m, ℍ), V_{ℂ} = Sym(J, ℂ), Co(V)_{0} ≅ SO^{∗}(4m), \frac{n}{r} = 2m − 1.

1) α(Z) = I_{p,q}ZI_{p,q} (p + q = m), X ≅ U(2p, 2q)/ Sp(p, q), V^{+} ≅ Herm(p, ℍ) ⊕ Herm(q, ℍ), V^{−} ≅ Mat(p × q; ℍ), V_{α} ≅ Herm(I_{p,q}; ℍ), f_{α}(Z) = Det(Z + I_{p,q}ZI_{p,q})^{2m−1} .

2) α(Z) = φ(Z) (i.e., (φ(Z))_{ij} = (φ(Z_{ij}) with φ as above), X ≅ SO^{∗}(2m), V^{+} ≅ Herm(m, ℂ), V^{−} ≅ Asym(m, ℂ), V_{α} ≅ Sym(F, ℝ), f_{α}(Z) = Det(Z + Z^{t})^{2m−1}.

IV) V = ℝ × ℝ^{n−1} , V_{ℂ} = ℂ^{n}, Co(V)_{0} ≅ SO(2, n), \frac{n}{r} = \frac{n}{2} .

1) α = id_{V} , X ≅ S^{n} × S^{1} = (SO(n) × S^{1})/ SO(n − 1), f_{α}(z) = (z_{1} + … + z_{n})^{2n}.

2) α = I_{p,n−p}, p > 1, X ≅ (SO(p) × SO(2, n − p)/(SO(p − 1) × SO(1, n − p)), V^{+} ≅ ℝ × ℝ^{p−1} , V^{−} ≅ ℝ^{n−p}, V_{α} ≅ ℝ^{p−1} × ℝ^{n−p+1} , f_{α}(z) = (z_{1} + … + z_{p} − z_{p+1} − … z_{n})^{2n}.

3) α = I_{1,n−1} , X ≅ SO(2, n − 1)/ SO(1, n − 1), V^{+} ≅ ℝ, V^{−} ≅ ℝ^{n−1} , V_{α} ≅ ℝ^{n}, f_{α}(z) = (z_{1} − … − z_{n})^{2n}.

V) V = Herm(3, 𝕆), V_{ℂ} = Herm(3, 𝕆_{ℂ}), Str(V) ≅ E_{6(−25)} × ℝ^{∗}, Co(V)_{0} ≅ E_{7(−25)} , \frac{n}{r} = 9.

1) α = id_{V} , X ≅ (E_{6(−14)} × U(1))/F_{4(−20)} , f_{α}(Z) = Δ(Z)^{18} (where Δ is the Jordan determinant) (The Peirce-involution α(Z) = I_{1,2}ZI_{1,2} leads to the same space as α = id_{V} .)

2) α(Z) = Z̃ (the non-trivial involution of 𝕆), X ≅ SU(6, 2)/ Sp(3, 1), V^{+} = Herm(3, ℍ), V_{ℂ}^{−} ≅ Mat(6 × 2; ℂ), V_{α} = Herm(3, 𝕆_{s}),

f_{α}(Z) = Δ(Z + Z̃)^{18}

CT) (Cayley type): V = V_{1} × V_{1} , where V_{1} is one of the above Euclidian Jordan algebras, Co(V)_{0} = Co(V_{1})_{0}×Co(V_{1})_{0} , P ((x, y)) = ({P_{1}(x) \atop 0} {0 \atop P_{1}(y)}), where P_{1} is the quadratic representation of V_{1} . α((x, y)) = (y, x), V^{+} ≅ V^{−} ≅ V_{1} , V_{α} ≅ (V_{1})_{ℂ}, f_{α}((z, w)) = Det P ((z + w, z + w)) = Det P_{1}(z + w)^{2} , and X ≅ Co(V_{1})/ Str(V_{1}). We obtain the following cases:

1) X ≅ SU(n, n)/(Sl(n, ℂ) × ℝ),

2) X ≅ Sp(n, ℝ)/ Gl(n, ℝ),

3) X ≅ SO^{∗}(4n)/(Sl(n, ℍ) × ℝ),

4) X ≅ SO(2, n)/(SO(1, n − 1) × ℝ),

5) X ≅ E_{7(−25)}/(E_{6(−26)} × ℝ).

Remark 1.5.2. The following compactly causal symmetric spaces are not Makarevič spaces: the simple parts of the reductive spaces in the above list (e.g. SU(p, q) is the simple part of U(p, q)), the spaces SO(2, p + q)/(SO(1, p) × SO(1, q)) with min(p, q) > 1, the group case SO(2, n) and some exceptional spaces (cf. [HO96, p.89]). Conversely, the symmetric spaces X from the above list which are not irreducible do not appear in the classification [HO96, p.89] although their causal structure cannot be reduced to direct products (i.e. they are reducible as symmetric spaces, but irreducible as causal symmetric spaces; in [HO96] this distinction is not made, and a causal symmetric space is called irreducible if it is so as a symmetric space). These are the spaces I. – V.1 and IV.2. The latter case plays a rather exceptional role.

Remark 1.5.3. In all cases, the Jordan algebra V^{+} is either simple or a direct sum of two simple ideals. The latter happens precisely in the cases where α is a (non-trivial) Peircereflection, i.e. α = P (w) for some w ∈ V (w ≠ e) with w^{2} = e (cf. [Hw69, Lemma 3]). These are the cases I.1, II.1, III.1, IV.2 (p = 2) and V.1 (always α ≠ id_{V} ).

Remark 1.5.4. The Jordan triple system (JTS) V^{−} is simple in all cases, but V_{ℂ}^{−} may be a direct sum of two simple JTS’s. This happens if and only if V^{−} is in fact a complex JTS, and this corresponds exactly to the cases where X is of group type, i.e.

(i) X = (G × G)/dia(G × G) with G = U(p, q) (I.1), (ii) G = Sp(k, ℝ) (II.2) or (iii) G = SO^{∗}(2m) (III.2). In these cases, D^{−} is a direct product of two isomorphic bounded symmetric domains.

Remark 1.5.5. (Admissible parameters) Recall that a rational number k = \frac{p}{q} is called an admissible parameter if there exists a holomorphic function f on Ξ with f = f_{α}^{k}. Such a function is again constant in V^{−} -direction and is therefore determined by its restriction to V^{+} . Since the restriction f_{α}|_{Ω_{+}} generates a one-dimensional space of functions under the action of Str(V^{+}), so will do the restriction f|_{Ω_{+}} . If V^{+} is simple, it follows that f|_{Ω_{+}} is a power of the Jordan determinant Δ_{+} of V^{+} , i.e.

f(x) = (Δ_{+}(x + αx))^{l}

with some integer l (cf. [FK94, Th. XII.2.2]). Thus the lowest admissible parameter k is given by the condition f_{α}(x)^{k} = Δ_{+}(x + αx). Since f_{α} is homogeneous of degree 2n and Δ_{+} is homogeneous of degree r_{+} (the rank of V^{+} ), we get k = \frac{r_{+}}{2n} for the lowest admissible parameter if V^{+} is simple. (In the Cayley case this reduces to k = \frac{r_{+}}{4n_{+}} . Note that \frac{2n_{+}}{r_{+}} is always an integer, cf. Section 1.7.) If V^{+} is not simple (Peirce involutions), we write V^{+} as a sum of two simple algebras and apply the above argument to each term. Again we obtain that k = \frac{r_{+}}{2n} is the lowest admissible parameter. (Note that here r_{+} = r.) We get the following list of admissible parameters in the cases corresponding to Table 1.5.1:

I.1.  I.2.  I.3.  II.1. II.2. III.1. III.2. IV.1. IV.2. IV.3. V.1. V.2.
\frac{ℤ}{2m}  \frac{ℤ}{2m}  \frac{ℤ}{8k}  \frac{ℤ}{m+1}  \frac{ℤ}{4k+2}  \frac{ℤ}{2m−1} \frac{ℤ}{4m−2} \frac{ℤ}{n}  \frac{ℤ}{n}  \frac{ℤ}{2n}  \frac{ℤ}{18}  \frac{ℤ}{18}

Later on it will be important to know whether \frac{1}{2} and \frac{1}{4} are admissible parameters. This can be read off the preceding table.

1.6. Split and non-split involutions. In this section we prove in a classification free way that the involutions α given in the preceding section fall into two classes, called split and non-split.

Recall that the rank of a Euclidian Jordan algebra is the number of elements of a complete system of primitive orthogonal idempotents c_{1}, …, c_{r}, also called a Jordan frame. The space R := ⊕_{i=1}^{r}ℝc_{i} is a maximal associative subalgebra of V , and the space L(R) is a Cartan subspace for the Cartan decomposition 𝔰𝔱𝔯(V) = Der(V) ⊕ L(V), i.e. it is maximal abelian in L(V). We can always find a Cartan subspace which is stable under a given involution, and correspondingly there is always a Jordan frame such that α(R) = R for a given involution α of V . Then we write R = R^{+} ⊕ R^{−} for the corresponding decomposition of R into eigenspaces of α.

Theorem 1.6.1. Let V be a Euclidian Jordan algebra and α an involution such that V has no proper α-stable ideal. Then one can find a Jordan frame in V such that:

(i) α(R) = R (ii) The rank of the Euclidian Jordan algebra V^{+} is equal to dim R^{+} . (iii) Either R = R^{+} or dim R^{+} = dim R^{−}.

Proof. Let e_{1}, …, e_{r_{+}} be a Jordan frame for the Euclidian Jordan algebra V^{+} . Since e_{i} is an idempotent also in V , the eigenspace

W_{i} := V (e_{i}, 1) := {x ∈ V | L(e_{i})x = x}

is a subalgebra of V with unit element e_{i} ([FK94, Prop. IV.1.1]), and it inherits from V a positive associative bilinear form, i.e. it is Euclidian. We claim that either W_{i} = ℝe_{i} or W_{i} has rank two. In fact, W_{i} is α-stable; let W_{i} = W_{i}^{+} ⊕ W_{i}^{−} be the corresponding decomposition. Since e_{i} is primitive in V^{+} , it follows from the spectral theorem ([FK94, Th.III.1.1]) that V^{+}(e_{i}, 1) = W_{i}^{+} is one-dimensional. By the argument given before stating the theorem, we know that there exists a frame w_{1}, …, w_{l} in W_{i} such that the vector space spanned by the w_{j} ’s is α-stable. It follows that α permutes the elements of this frame. If dim W_{i} > 1, then e_{i} = ∑_{j} w_{j} is not a primitive idempotent, and since W_{i}^{+} = ℝe_{i}, it follows that no w_{j} is fixed under α. On the other hand, w_{j} + α(w_{j}) is an idempotent fixed under α. It follows that already w_{1}, α(w_{1}) is a frame of W_{i}, and hence the rank of W_{i} is two. We define f_{i} := w_{1} .

Next we are going to show that f_{i} is actually primitive in V . To this end, let e_{i} = ∑_{j} λ_{j}c_{j} be the spectral decomposition of e_{i} in V with respect to a Jordan frame c_{1}, …, c_{r} of V ([FK94, Th.III.1.2.]). Since e_{i} is idempotent, it follows that λ_{j}^{2} = λ_{j}, whence λ_{j} = 1 or λ_{j} = 0. Because e_{i}c_{j} = λ_{j}c_{j}, this implies that those c_{j} actually appearing in the decomposition e_{i} = ∑_{j} λ_{j}c_{j} belong to W_{i}. But then they are also primitive idempotents in W_{i}, and since the rank of W_{i} is at most two, there are at most two non-vanishing terms in the decomposition of e_{i}. If there is just one term, then e_{i} is actually primitive in V . Otherwise we have e_{i} = f_{i} + α(f_{i}) with f_{i} as above, and we conclude that f_{i} is also primitive in V .

Now we can prove that

{f_{i}, αf_{i}, e_{j}| 1 ≤ i, j ≤ r_{+}, rk W_{i} = 2, rk W_{j} = 1}

is a frame in V : it is clear from the construction that this is a complete system of idempotents which are primitive by what we have proved above. It only remains to be shown that they are orthogonal. But since the e_{j} are orthogonal, we have for i ≠ j W_{i} = V (e_{i}, 1), W_{j} ⊂ V (e_{i}, 0), and since V (e_{1}, 1)V (e_{j}, 0) = 0 ([FK94, Prop.IV.1.1]), it follows that W_{i}W_{j} = 0. We know already that f_{i} and α(f_{i}) are orthogonal (if they are distinct). Together this implies that the idempotents in the system in question are orthogonal. It is clear that the vector space R spanned by this system is α-stable, and by construction dim R^{+} is the rank of V^{+} . Thus we have established properties (i) and (ii).

In order to prove (iii) we assume first that V^{+} is simple. Then Aut(V^{+})_{o} acts transitively on the set of primitive idempotents ([FK94, Cor.IV.2.7]). Since Aut(V)_{o}^{α_{∗}} → Aut(V^{+})_{o}, g ↦ g|_{V^{+}} is surjective (cf. Prop.1.8.1), it follows that any two of the e_{i} are conjugate under Aut(V)^{α_{∗}}, and therefore the corresponding W_{i} are conjugate under this group. In particular, they have all the same rank. Thus in the frame of V constructed above, either all idempotents are of the form e_{j} or all appear in pairs f_{i}, α(f_{i}), and this implies (iii).

If V^{+} is not simple, then by a result of Helwig ([Hw69, Lemma 3]), α is a Peirce-reflection P (w) with respect to an idempotent c of V , w = 2c − e. Then V^{+} = V (c, 1) ⊕ V (c, 0) (cf. [FK94, p.65]), and there is a frame of V contained in V^{+} , i.e. R = R^{+} , and (iii) holds.

Definition 1.6.2. Let α be an involution of V and R a frame having properties (i) – (iii) of the preceding theorem. Then, if R = R^{+} we say that α is split (w.r.t. R), and if dim R^{+} = dim R^{−} we say that α is non-split (w.r.t. R).

Remark 1.6.3. We specify which of the involutions from Table 1.5.1 are split and which are non-split. In the matrix cases we choose Jordan frames c_{1}, …, c_{r} such that R = ∑_{i=1}^{r} ℝc_{i} is the respective space of diagonal matrices. In the case of rank 2 (Type IV) we choose c_{1} and c_{2} such that they span the same space as the first two canonical base vectors of ℝ^{n}. The involutions α from Table 1.5.1 are either split or non-split for these Jordan frames. We distinguish further the cases that both V and V^{+} are simple and that one of these algebras is not simple. Thus the involutions α are partitioned into four classes. Using the notation from Table 1.5.1, this partition looks as follows:

V and V^{+} simple  V or V^{+} not simple
α split  I.2, III.2, IV.2, V.2  Peirce involutions
and α = id_{V}
α non-split  I.3, II.2, IV.3  Cayley-type

Next we draw some consequences concerning the geometry of the associated compactly causal space X. Recall that the rank of a symmetric space X is the dimension of a maximal abelian subspace of 𝔮 consisting of semisimple elements. Since our space X is compactly causal, [HO96, Prop. 3.1.11 (3)] together with c-duality shows that there is a compact Cartan subspace, i.e. a maximal abelian subspace contained in 𝔮_{𝔨} = 𝔮 ∩ 𝔨. The split rank of X is defined to be the dimension of a maximal abelian subspace of 𝔮_{𝔭}.

Proposition 1.6.4.

(i) The rank of X is equal to the rank of the Jordan algebra V^{+} , and the space 𝔞 := iL(R^{+}) is a compact Cartan subspace of 𝔮.

(ii) If W ⊂ 𝔮 is the invariant cone defined in Section 1.3 and e_{1}, …, e_{r^{+}} the Jordan frame in V^{+} corresponding to R^{+} and 𝔞 is as in Part (i), then

W ∩ 𝔞 = iL({∑ t_{i}e_{i}| t_{i} < 0}).

Proof. (i) For any Euclidian Jordan algebra, L(R) is a maximal abelian subspace in L(V). Since by part (ii) of Theorem 1.6.1 L(R^{+}) is a maximial abelian subspace of L(V^{+}), it follows that iL(R^{+}) is maximal abelian in iL(V^{+}) = 𝔮_{𝔨}.

(ii) We have seen in Section 1.3 that W ∩ 𝔮_{𝔨} = −iL(Ω^{+}). Since Ω^{+} is the cone of squares in V^{+} , we have Ω^{+} ∩ R^{+} = {∑ t_{i}e_{i}| t_{i} > 0}, and the claim follows directly from the definition of 𝔞 in part (i).

Proposition 1.6.5. Assume that X is irreducible. Then the following are equivalent: (i) α is non-split. (ii) The rank of X is equal to the split rank of X (i.e. X admits both compact and non-compact

Cartan subspaces).

Proof. If α is non-split, then 𝔞_{1} := {v + j_{∗}v| v ∈ R^{−}} is a Cartan subspace contained in 𝔮_{𝔭} having the same dimension as 𝔞. If α is split, then one checks, using the classification, that the rank of 𝔮_{𝔭} is always strictly lower than the rank of V^{+} . (According to Remark 1.6.3, these are the cases I.2 and III.2 from Table 1.5.1; note that in case IV.2 V^{+} and V^{−} have both rank equal to 2, but this case is excluded since in case IV.2 X is not simple. In Jordan theoretic terms, the rank of 𝔮_{𝔭} is equal to the rank of the Jordan triple system V^{−} whose isomorphism class is indicated in Table 1.5.1.)

1.7. Peirce decomposition and root structure. Recall that the space L(R) defined by a Jordan frame c_{1}, …, c_{r} is maximal abelian in L(V). The weight decomposition of V under the action of the abelian Lie algebra L(R) is given by the Peirce decomposition (cf. [FK94, Th.IV.2.1])

V = ⨁_{i=1}^{r} ℝc_{i} ⊕ ⨁_{i<j} V_{ij},

where V_{ij} for i ≠ j is the intersection of the \frac{1}{2} -eigenspaces of L(c_{i}) and L(c_{j}). In other words, if we denote by c_{1}^{∗}, …, c_{r}^{∗} the basis of L(R)^{∗} dual to L(c_{1}), …, L(c_{r}), then V_{ij} is the weight space for the weight \frac{1}{2}(c_{i}^{∗} +c_{j}^{∗}). For a simple algebra V , all the spaces V_{ij} (i ≠ j) have the same dimension called the genus of V and denoted by d. We then have the relation n = r + d\frac{r(r−1)}{2} .

1.7.1. Action of an involution on the Peirce spaces. Assume that α is an involution with α(R) = R. We analyze the behavior of the spaces V_{ij} under α in the cases corresponding to Remark 1.6.3:

– Split case: It is immediate that all Peirce spaces V_{ij} are stable under α. We assume first that V^{+} is simple and consider the matrix cases (I.2, III.2, V.2). Then there is a structure of Euclidian Hurwitz algebra on V_{ij} ([FK94,Prop.V.3.4]), and α acts on V_{ij} by a conjugation of ℂ, ℍ or 𝕆 according to the cases d = 2, 4 or 8. The Peirce spaces for V^{+} are the α-fixed parts of the Peirce spaces of V , having half the dimension of the latter. In the case of rank 2 there remains only Type IV.2 in which X is not irreducible and which we will not consider.

If V^{+} is not simple, then α = P (w) is a Peirce reflection (cf. Remark 1.5.3) with w = c_{1} + … + c_{p} (1 ≤ p < r), and then α acts trivially on V_{ij} for 1 ≤ i, j ≤ p and for p < i, j ≤ r and by −1 on the other V_{ij} ’s.

– Non-split case: The rank of V is even, r = 2r_{+} , and we order the idempotents such that c_{r_{+}+j} = α(c_{j}). Then the idempotents e_{j} of V^{+} are e_{j} = c_{j} + αc_{j}, j = 1, …, r_{+} . In the matrix cases (I.3 and II.2) the Peirce-structure of the Euclidean Jordan algebra V^{+} w.r.t. the e_{i} looks like the Peirce-structure of V , but with d replaced by d_{+} = 2d. Let for i, j = 1, …, r_{+}

U_{ij} := V (L(e_{i}), \frac{1}{2}) ∩ V (L(e_{j}), \frac{1}{2})

be the intersection of the \frac{1}{2} -eigenspaces of L(e_{i}) and L(e_{j}) in V . Then U_{ij} ∩ V^{+} for i < j are the Peirce-spaces of V^{+} . On the other hand, we get from the definition of the Peirce-spaces V_{ij} of V that

U_{ij} = V_{ij} ⊕ V_{i+r_{+},j+r_{+}} ⊕ (V_{i,j+r_{+}} + V_{j,i+r_{+}}),

and thus V = R⊕⨁_{i≤j} U_{ij}. Since we know already that dim(U_{ij} ∩V^{+}) = d_{+} = 2d, we conclude that the dimension of U_{ij} ∩ V^{−} is 2d if i < j and d if i = j. In case of rank 2 there remains only Case IV.3: there is just one Peirce-space in V ; it belongs to V^{−}.

Finally, if V is not simple, then we are in the Cayley-type case. Here V^{+} ≅ V^{−} ≅ V_{1} , and the decomposition of V^{+} and V^{−} w.r.t. the idempotents of V^{+} coincides with the usual Peirce-decomposition in V_{1} .

1.7.2. Root structure. The complexification of the Lie algebra 𝔤 is given by

𝔤_{ℂ} = 𝔠𝔬(V_{ℂ})^{(−α)_{∗}} = V_{ℂ}^{−} ⊕ 𝔰𝔱𝔯(V_{ℂ})^{α_{∗}} ⊕ j_{∗}V_{ℂ}^{−}.

Since 𝔞 = L(R^{+}) is a compact Cartan subspace in 𝔮 (Prop. 1.6.4), the adjoint action of 𝔞_{ℂ} on the Lie algebra 𝔤_{ℂ} is diagonalizable. Let Δ := Δ(𝔤_{ℂ}, 𝔞_{ℂ}) be the root system of 𝔤_{ℂ} with respect to 𝔞_{ℂ} = L(R^{+})_{ℂ}. Since the action of 𝔞_{ℂ} commutes with − id_{V_{ℂ}}, the root spaces are contained in the homogeneous parts of the graded Lie algebra 𝔤_{ℂ}, and the root system Δ decomposes as

Δ = Δ_{1} ∪ Δ_{0} ∪ Δ_{−1}, Δ_{1} = Δ(V_{ℂ}^{−}, 𝔞_{ℂ}), Δ_{0} = Δ(𝔰𝔱𝔯(V_{ℂ})^{α_{∗}}, 𝔞_{ℂ}), Δ_{−1} = −Δ_{1}.

The roots from Δ_{0} are called compact and the roots from Δ_{1} ∪ Δ_{−1} non-compact. All roots are real-valued on the real form 𝔞 = L(R^{+}) of 𝔞_{ℂ}. A root is called positive if it is positive on the positive Weyl chamber \{∑_{i} t_{i}e_{i}| 0 ≤ t_{1} ≤ … ≤ t_{r}\}. As usual, ρ = \frac{1}{2} ∑_{α∈Δ^{+}} m_{α}α is the half-sum of the positive roots, weighted by the dimension m_{α} of the root spaces.

In Theorem 1.8.3 we will prove that, if X is irreducible and not of type IV.3, then 𝔰𝔱𝔯(V_{ℂ})^{α_{∗}} and 𝔰𝔱𝔯(V_{ℂ}^{+}) are canonically isomorphic. This permits to describe Δ_{k}, k = −1, 0, 1, in terms of the Peirce decompositions of V^{+} and V . The details are given below. For simplicity of notation, we identify 𝔞_{ℂ} with R_{ℂ}^{+} . Let e_{1}, …, e_{r_{+}} be the Jordan frame belonging to R^{+} and e_{1}^{∗}, …, e_{r_{+}}^{∗} be its dual basis, considered as basis of 𝔞_{ℂ}^{∗}. Let V be of dimension n, rank r and with genus d and denote by a lower index + the corresponding constants for V^{+} .

– Case A: X = G/H is simple and not of Type IV.3. In this case V^{+} is a simple Jordan algebra, and Theorem 1.8.3 shows that 𝔰𝔱𝔯(V)^{α_{∗}} ≅ 𝔰𝔱𝔯(V^{+}). Thus, according to [FK94, p.212],

Δ_{0} = Δ(𝔰𝔱𝔯(V_{ℂ}^{+}), 𝔞_{ℂ}) = \{\frac{1}{2}(e_{j}^{∗} − e_{k}^{∗})| j ≠ k; j, k = 1, …, r_{+}\},

ρ_{0} = ρ(Δ_{0}) = \frac{d_{+}}{4} ∑_{j<k}(e_{k}^{∗} − e_{j}^{∗}) = \frac{d_{+}}{4} ∑_{j}(2j − r − 1)e_{j}^{∗}.

The root system Δ_{1} := Δ(V^{−}, 𝔞) can be determined from the Peirce decompositions of V and V^{+} : V^{−} is the sum of the weight spaces of 𝔞 not appearing in V^{+} . According to Section 1.7.1 we obtain:

(i) Case A.1: α split, i.e. r_{+} = r: Note that for type IV such an involution does not exist (since we exclude case IV.2 in which X is not irreducible). Thus we are in the matrix cases where α acts by a conjugation on the Euclidian Hurwitz algebra isomorphic to the V_{ij} ’s. Thus

Δ_{1} = Δ(V_{ℂ}^{−}, 𝔞_{ℂ}) = \{\frac{1}{2}(e_{j}^{∗} + e_{k}^{∗})| j, k = 1, …, r_{+}, j < k\},

and the corresponding root space dimensions are d_{+} = \frac{d}{2} with d = 2 in Case I.2, d = 4 in Case III.2 and d = 8 in Case V.2. Therefore

ρ = ρ_{0} + ρ_{1} = \frac{d}{8}(∑_{j<k}(e_{k}^{∗} − e_{j}^{∗}) + ∑_{j<k}(e_{k}^{∗} + e_{j}^{∗})) = \frac{d}{4} ∑_{j}(j − 1)e_{j}^{∗}.

(ii) Case A.2: α non-split, i.e. r_{+} = \frac{r}{2} : There are r_{+} one dimensional weight spaces in V^{−} corresponding to R^{−}. In the matrix cases (I.3 with d = 2, II.2 with d = 1) there are in addition weight spaces for the weight \frac{1}{2}(e_{j}^{∗} + e_{k}^{∗}) (j < k) with dimension 2d and for the weight e_{k}^{∗} with dimension d. Thus

Δ_{1} = \{e_{j}^{∗}, \frac{1}{2}(e_{j}^{∗} + e_{k}^{∗})| j, k = 1, …, r_{+}; j < k\}

ρ = ρ_{0} + ρ_{1} = \frac{d}{2} ∑_{j<k}(e_{k}^{∗} − e_{j}^{∗}) + \frac{1 + d}{2} ∑_{j} e_{j}^{∗} + \frac{d}{2} ∑_{j<k}(e_{j}^{∗} + e_{k}^{∗}) = \frac{1 + d}{2} ∑_{j} e_{j}^{∗} + d ∑_{j}(j − 1)e_{j}^{∗}.

For the Cayley-type we have d_{+} = d, n_{+} = \frac{n}{2} , and the decomposition of V^{+} ≅ V^{−} with respect to 𝔞 is the usual Peirce decomposition of V^{+} with d_{+} = d:

Δ_{1} = \{e_{j}^{∗}, \frac{1}{2}(e_{j}^{∗} + e_{k}^{∗})| j, k = 1, …, r_{+}; j < k\},

ρ = \frac{1}{2} ∑_{j} e_{j}^{∗} + \frac{d}{4} ∑_{k<j}(e_{k}^{∗} − e_{j}^{∗} + e_{k}^{∗} + e_{j}^{∗}) = \frac{1}{2} ∑_{j} e_{j}^{∗} + \frac{d}{2} ∑_{j}(j − 1)e_{j}^{∗}.

– Case B: Case IV.3 (X ≅ SO(2, n − 1)/ SO(1, n − 1)). We have r_{+} = \frac{r}{2} = 1; thus e_{1} = c_{1} + c_{2} is the unit element e and R^{+} = ℝe, n_{+} = 1. Clearly Δ_{1} = {e^{∗}} with weight space V^{−}. Since L(R^{+}) = ℝ id_{V} , we have Δ_{0} = Δ(𝔰𝔱𝔯(V_{ℂ})^{α_{∗}}, L(R^{+})_{ℂ}) = Ø, and thus

ρ = \frac{n − 1}{2}e^{∗}.

– Case C: X is not irreducible. If we discard the case IV.2, we may assume that α is a Peirce involution: α = P (w), w = c_{1}+…+c_{p}, r_{1} = p, r_{−1} = q = r−p, d_{1} = d_{−1} = d, n_{1} = p+d\frac{p(p−1)}{2} , n_{−1} = q + d\frac{q(q−1)}{2} .

Δ_{0} = Δ(𝔰𝔱𝔯(V^{+}), 𝔞) = \{\frac{1}{2}(c_{j}^{∗} − c_{k}^{∗})|j ≠ k, 1 ≤ j, k ≤ q, or q < j, k ≤ r\},

Δ_{1} = \{\frac{1}{2}(e_{j}^{∗} + e_{k}^{∗})|1 ≤ j ≤ p < k ≤ r\},

ρ = \frac{d}{4}(∑_{j=1}^{p}(2j − p − 1)e_{j}^{∗} + ∑_{j=p+1}^{r}(2j − q − 1)e_{j+p}^{∗} + ∑_{1≤j≤p<k≤r}(e_{j}^{∗} + e_{k}^{∗}) = \frac{d}{4} ∑_{j=1}^{r}(2j − 2p − r − 1)e_{j}^{∗}.

1.8. The behavior of the structure group under involutions. In this subsection α is an involution of an arbitrary semisimple Jordan algebra V . According to [Sa80, I.7.3], the structure algebra 𝔰𝔱𝔯(V^{−}) of the non-degenerate JTS V^{−} is equal to the inner structure algebra V^{−} _{−}V^{−} generated by v _{−}w (v, w ∈ V^{−}); here v _{−}w is the restriction to V^{−} of v w = L(vw) + [L(v), L(w)]. Similarly, 𝔰𝔱𝔯(V^{+}) = V^{+} _{+}V^{+} . In this subsection we show (in a classification free way) that “generically” 𝔰𝔱𝔯(V^{+}) and 𝔰𝔱𝔯(V^{−}) can be identified. See [Hw69] for similar results.

Proposition 1.8.1. Let V be a semisimple Jordan algebra and α an involution of V . Then the following extension homomorphisms i_{±} and restriction homomorphisms r_{±} are well defined:

i_{±} : 𝔰𝔱𝔯(V^{±}) = V^{±} _{±}V^{±} → 𝔰𝔱𝔯(V)^{α_{∗}}, v _{±}w ↦ v w
r_{±} : 𝔰𝔱𝔯(V)^{α_{∗}} → 𝔰𝔱𝔯(V^{±}), X ↦ X|_{V^{±}}.

They satisfy r_{±} ∘ i_{±} = id_{𝔰𝔱𝔯(V^{±})} . In particular, i_{±} is injective and r_{±} is surjective.

Proof. Let us show that the formula for i_{±} indeed yields a well defined map. In fact, the inclusion V^{±} ⊂ V is a homomorphism of non-degenerate JTS, and by a result of Neher (cf. [Sa80, p.39/40]), such a homomorphism uniquely extends to a homomorphism of the associated graded Lie algebras which by restriction to 𝔰𝔱𝔯(V^{±}) precisely yields i_{±}. Now, if v, w ∈ V^{±}, α ∘ (v w) ∘ α = (αv) (αw) = v w, and therefore the image of i_{±} is contained in 𝔰𝔱𝔯(V)^{α_{∗}}.

We now show that the restriction maps are well defined: it is clear that V^{+} and V^{−} are stable under Str(V)^{α_{∗}} and under 𝔰𝔱𝔯(V)^{α_{∗}}. Further, the restriction of X ∈ 𝔰𝔱𝔯(V)^{α_{∗}} to the subspaces V^{±} belongs to the respective structure algebras: this follows easily from the decomposition 𝔰𝔱𝔯(V)^{α_{∗}} = Der(V)^{α_{∗}} ⊕ L(V^{+}). Finally, it is clear that r_{±} ∘ i_{±} = id_{𝔰𝔱𝔯(V^{±})} .

We say that V^{+} and V^{−} commute iff [i_{−}(𝔰𝔱𝔯(V^{−})), i_{+}(𝔰𝔱𝔯(V^{+}))] = 0.

Theorem 1.8.2. Assume that V^{−} is simple and that V^{+} and V^{−} don’t commute. (i) If V^{+} is simple, then r_{−} is injective and i_{−} is surjective. (ii) If V^{+} is not simple, then α is a Peirce reflection P (w) and r_{−} has a one dimensional

kernel given by ℝL(w); then r_{+}(ℝL(w)) is in the center of 𝔰𝔱𝔯(V^{+}). (iii) r_{+} is injective and i_{+} is surjective.

Proof. By Proposition 1.8.1, i_{±}∘r_{±} is a projection onto the image of i_{±}; therefore 𝔰𝔱𝔯(V)^{α_{∗}} = im i_{+} ⊕ ker r_{+} = im i_{−} ⊕ ker r_{−}. We will now determine the kernel of the homomorphism

r_{±} ∘ i_{∓} : 𝔰𝔱𝔯(V^{∓}) → 𝔰𝔱𝔯(V^{±}), v _{∓}w ↦ (v w)|_{V^{±}}.

Assume first that both V^{−} and V^{+} are simple. Then it is known that the center of the respective structure algebras is one dimensional, given by scalar multiples of the identity, and that their derived algebras are simple. Therefore r_{+} ∘ i_{−} and r_{−} ∘ i_{+} must be inverse isomorphisms of the derived algebras. In fact, if the intersection of the derived algebras with the respective kernels were not zero, V^{+} and V^{−} would commute. In particular, we see that i_{∓} ∘ r_{∓} ∘ i_{±} = i_{±} on the derived algebra.

We now prove that r_{±} ∘ i_{∓}(id_{V^{∓}}) = id_{V^{±}}. First, i_{+}(id_{V^{+}}) = i_{+}(L(e)|_{V^{+}}) = id_{V} , therefore r_{−} ∘ i_{+}(id_{V^{+}}) = id_{V^{−}}. According to [Sa80, I.6.1], we can write id_{V^{−}} = ∑_{i} c_{i} _{−}c′_{i} where (c_{i}) and (c′_{i}) are dual basis of V^{−} with respect to the trace form (v, w) ↦ Tr(v _{−}w), and similarly for V^{+} with a basis (d_{i}) of V^{+} . Because V^{+} and V^{−} are simple, the respective trace forms are multiples of the respective restrictions of the trace form of V . Hence (\frac{dim V^{+}}{dim V} d′_{i}, \frac{dim V^{−}}{dim V} c′_{j}) is the dual basis of the basis (d_{i}, c_{j}) of V . One deduces that

i_{+}(id_{V^{+}}) + i_{−}(id_{V^{−}}) = ∑_{i} c_{i} c′_{i} + ∑_{i} d_{i} d′_{i} = 2 id_{V} .

Since i_{+}(id_{V^{+}}) = id_{V} , we get i_{−}(id_{V^{−}}) = id_{V} . Thus r_{±} ∘ i_{∓} is an isomorphism on the center and, by what we have seen above, on all of 𝔰𝔱𝔯(V^{±}).

We have proved that i_{+} and i_{−} have the same image. This implies that their complements are equal: ker r_{+} = ker r_{−}. But clearly ker r_{+} ∩ ker r_{−} = 0, and hence r_{±} and i_{±} are isomorphisms.

It remains to consider the case when V^{+} is not simple. According to Remark 1.5.3, α is then a Peirce reflection P (w). By standard facts on the Peirce decomposition (cf. [FK94, p.65]), V^{+} is then the 0-eigenspace of L(w) and V^{−} is the direct sum of the +1-eigenspace V_{1} and the −1-eigenspace V_{−1} of L(w); both are ideals in V^{+} . We first prove that the image of i_{−} is contained in the image of i_{+} . In fact, since V^{−} is assumed to be simple, the arguments given above still go through. We conclude that ker r_{+} ⊂ ker r_{−}, and hence ker r_{+} = 0 since ker r_{−} ∩ ker r_{+} = 0. Thus i_{+}(𝔰𝔱𝔯(V^{+})) = 𝔰𝔱𝔯(V)^{α_{∗}}, and it is then immediately verified that ker r_{−} = ℝL(w).

Proposition 1.8.3. If V is Euclidian and the space X associated to α is irreducible, then r_{−} : 𝔰𝔱𝔯(V)^{α_{∗}} → 𝔰𝔱𝔯(V^{−}) is bijective.

Proof. Using the classification of irreducible spaces from Table 1.5.1, we see that in all cases with the exception of IV.3 V^{+} and V^{−} don’t commute, and in these cases Theorem 1.8.2 implies the claim. In the remaining case IV.3 we have V^{+} = ℝ and thus 𝔰𝔱𝔯(V^{+}) ≅ ℝ, and the claim can be proved by similar arguments as in the proof of Theorem 1.8.2.

Remark 1.8.4. In case IV.2 the statement of Theorem 1.8.2 is false: indeed, V^{+} and V^{−} do commute, and 𝔰𝔱𝔯(V)^{α_{∗}} contains a direct product of the simple parts of the non-isomorphic structure groups of V^{+} and V^{−}.

Remark 1.8.5. The surjective homomorphisms ρ_{±} : G^{±} → G(D^{±}) (cf. Section 1.2) have discrete kernel if and only if r_{±} : 𝔰𝔱𝔯(V)^{α_{∗}} → 𝔰𝔱𝔯(V^{±}) are bijective. In fact, the kernel of the derived homomorphism of Lie algebras ρ̇_{±} : 𝔤_{ℂ}^{±} → 𝔤(D^{±})_{ℂ} is contained in the homogeneous part of degree 1 in 𝔤_{ℂ}^{±} which is just 𝔰𝔱𝔯(V_{ℂ})^{α_{∗}}. Thus Theorem 1.8.1 yields a criterion when ρ_{±} is essentially injective.

2. The spectrum of a family of weighted Bergman spaces

In this section we fix an involution α, the corresponding domain Ξ ⊂ D and the compactly causal space X = G/H.

2.1. The spaces B_{λ,μ}^{2}(Ξ) (proof of Theorem 1). We define the weighted Bergman spaces B_{μ}^{2}(D) and B_{λ,μ}^{2}(Ξ) by Equations (0.1) and (0.5).

Proposition 2.1.1. A function f ∈ B_{μ,0}^{2}(Ξ) extends to a holomorphic function on D.

Proof. This follows from [Pe96, Th.1.2] since D \ Ξ is an analytic (even algebraic) set in D.

The map assigning to f ∈ B_{μ,0}^{2}(Ξ) its holomorphic extension onto D is a unitary bijection of B_{μ,0}^{2}(Ξ) onto B_{μ}^{2}(D) which we will consider as an identification.

Corollary 2.1.2. If B_{μ,0}(Ξ) is not reduced to zero, then its reproducing kernel is given by

K^{(μ,0)}(z, w) = (Det B(z, \overline{w}))^{−μ}

with B given by Equation (0.2).

Proof. The preceding proposition implies that B_{μ,0}^{2}(Ξ) and B_{μ}^{2}(D) have the same reproducing kernel. Now the claim follows from [FK94, Prop. XIII.1.4].

Proposition 2.1.3. For all admissible parameters k,

A_{μ,λ,2k} : B_{μ,λ}^{2}(Ξ) → B_{μ,λ+2k}^{2}(Ξ), f ↦ (f_{α})^{k} · f

is an isometry onto.

Proof. This is immediate from the definition of the spaces B_{μ,λ}^{2} (Equation (0.5)).

Corollary 2.1.4. If B_{μ,0}^{2}(Ξ) is not reduced to zero and k is an admissible parameter, then the reproducing kernel of B_{μ,2k}^{2}(Ξ) is given by

K^{(μ,2k)}(z, w) = f_{α}(z)^{k}(Det B(z, \overline{w}))^{−μ}\overline{f_{α}(w)}^{k}.

Proof. We drop the index μ and denote by K^{(λ)}(z, w) the reproducing kernel corresponding to λ. Then

f(z) = (f|K_{z}^{(0)})_{0} = (f_{α}^{k}f|f_{α}^{k}K_{z}^{(0)})_{2k},
(f_{α}^{k}f)(z) = (f_{α}^{k}f|f_{α}^{k}K_{z}^{(0)}\overline{f_{α}(z)}^{k})_{2k}.

Therefore

K^{(2k)}(z, w) = f_{α}(z)^{k}K^{(0)}(z, w)\overline{f_{α}(w)}^{k}

with K^{(0)} from Corollary 2.1.2.

This completes the proof of Theorem 1. It is immediate from the definition of a positive kernel (cf. [FK94, p.171]) that, for k admissible, K^{(μ,2k)} is positive if and only if K^{(μ,0)} is. This in turn is the case if and only if μ belongs to the Wallach set W given by Equation (0.9) (see [FK94, Th.XIII.2.7]). We will denote by B_{μ,2k}(Ξ) the corresponding “abstract” reproducing kernel space. Then A_{μ,2k} : B_{μ,0}(Ξ) → B_{μ,2k}(Ξ) as in Proposition 2.1.3 is an isomorphism of reproducing kernel spaces.

2.2. Group and semigroup actions. The following lemma describes how the group of holomorphic automorphisms of a circled complex domain acts on spaces of holomorphic functions.

Lemma 2.2.1. Let M = G(M)/U ⊂ E be a circled bounded symmetric domain in a finite dimensional complex vector space E, where G(M) = Aut(M)_{o} und U is the stabilizer of the base point 0_{E}. If (ρ, V_{ρ}) is a finite dimensional representation of U_{ℂ}, then the formula

(π_{ρ}(g)f)(z) := ρ(dg^{−1}(z))^{−1} · f(g^{−1}z)

defines a linear action of G(M) on the space O(M, V_{ρ}) of holomorphic V_{ρ} -valued functions on M. The same formula defines a linear action of the semigroup S(M)^{−1} , where

S(M) := {g ∈ G(M)_{ℂ}| g(M) ⊂ M}.

Proof. It is known that, whenever dg^{−1}(z) exists for g ∈ G(M)_{ℂ} and z ∈ M, it is an element of U_{ℂ} (cf. [Lo77, 8.15] or [Sa80, Lemma II.5.3]). Thus the term on the right hand side is well defined. Using the chain rule one easily verifies that π_{ρ}(gh) = π_{ρ}(g)π_{ρ}(h).

Remark 2.2.2. The representation π_{ρ} is equivalent to the subspace of the induced representation Ind_{U}^{G(M)}ρ given by holomorphic sections of the associated bundle over G(M)/U = M with fiber V_{ρ} (cf. [FO95, p.25]).

Remark 2.2.3. We want to apply the preceding lemma to the representation ρ_{q}(u) = (Det u)^{q} for q a rational number, the case q = \frac{1}{2} being of particular interest. If V is a simple Jordan algebra, then ρ_{q} is a well defined representation of U if and only if q is a multiple of \frac{r}{n}. (In fact, the equation Δ(gz) = Det(g)^{\frac{r}{n}} Δ(z) for all z ∈ V_{ℂ}, g ∈ Str(V_{ℂ}) shows that g ↦ Det(g)^{\frac{r}{n}} is a well defined character of Str(V_{ℂ}). The converse is deduced from the fact that every relative invariant, i.e. a function on Ω transforming according to a character, is a power of the Jordan determinant Δ; cf. Remark 1.5.5.) In order to be able to apply a version of Lemma 2.2.1, we construct explicitly an m-fold covering group of G(M) (cf. [KØ97, Section 5, “Principle 1”] for more details).

Since M is simply connected, for any g ∈ G(M) the nowhere vanishing polynomial χ_{g}(z) = det(dg^{−1}(z))^{−1} has m holomorphic m-th roots. Thus the first projection of

G̃(M)_{m} := {(g, F) ∈ G(M) × O(M)| ∀z ∈ M : F (z)^{m} = χ_{g}(z)}

defines an m-fold covering p_{m} : G̃(M)_{m} → G(M). If, in this definition, we replace the term F (z)^{m} by e^{F (z)} , then we obtain a covering p_{∞} : G̃(M)_{∞} → G(M) of infinite order. We assume from now on that m ∈ ℕ ∪ {∞}. The following product defines a group structure on G̃(M)_{m}:

(g, F) · (g′, F′) = (gg′, F′′), F′′(z) = F (g′z)F′(z).

Since a root of χ_{g} is determined by its value at the base point 0, G̃(M)_{m} may topologically also be described as

G̃(M)_{m} = {(g, z) ∈ G(M) × ℂ | z^{m} = χ_{g}(0)}.

If Ũ_{m} = p_{m}^{−1}(U), then the second projection Ũ_{m} → ℂ, (u, z) ↦ z is a character whose m-th power is the character Det; therefore we denote it by Det^{1/m}.

All definitions go through for G(M) replaced by the semigroup S(M); we thus define m-th order covering semigroups S̃(M)_{m}.

Now we return to the setup from Section 2.1 with M := D. Note that G ⊂ G(D) and G exp(iW) ⊂ S(D), where W ⊂ 𝔮 is the cone defined in Section 1.3. Note that s = g exp(iX) ↦ s^{∗} = exp(iX)g^{−1} defines an anti-holomorphic involution on S(D) and similarly for the covering semigroups. Recall the concept of a Hermitian representation of an involutive semigroup S on a pre-Hilbert space ℌ^{0} from [Ne98, Def. II.3.3]: The vector space B_{0}(ℌ^{0}) of linear operators A: ℌ^{0} → ℌ^{0} for which a formal adjoint exists is an involutive semigroup, and a Hermitian representation of S on ℌ^{0} is a semigroup homomorphism π: S → B_{0}(ℌ^{0}) preserving the involutions, i.e. π(s^{∗}) = π(s)^{∗}.

Proposition 2.2.4. The formula

(π_{μ}(g)f)(z) := (Det dg^{−1}(z))^{μ}f(g^{−1}z)

defines a unitary representation of G(D)_{∞} and a Hermitian holomorphic contractive representation of the semigroup S(D)_{∞}^{−1} in B_{μ}^{2}(D). Proof. Because of the construction of the covering (semi-)group and of Lemma 2.2.1, π_{μ} is an action on the space of holomorphic functions on D. We have to show that it preserves (resp. contracts) the norm of B_{μ}^{2}(D).

The relation

B(g · z, \overline{g · w}) = dg(z)B(z, \overline{w})(dg(w))^{∗}

for g ∈ G(D) (cf. [Be98, 1.3.(9)] or [Sa80,p.65]) yields

K_{μ}(g · z, g · w) = (Det dg(z))^{−μ}K_{μ}(z, w)\overline{Det(dg(w))}^{−μ}  (2.1)

which is equivalent to

π_{μ}(g) · K_{w} = \overline{Det(dg(w))}^{−μ}K_{g·w}.  (2.2)

Since B_{μ}^{2}(D) is spanned by the K_{w}, the space is stable under π_{μ}. Further, (2.1) and (2.2) together imply that

(π_{μ}(g)K_{w}|π_{μ}(g)K_{z})_{μ} = K(z, w) = (K_{w}|K_{z}),

and therefore π_{μ} is a unitary representation of G(D)_{∞}.

Analytic extension of (2.1) in the g -variable yields

(Det ds(z)^{−1})^{−μ}K_{μ}(s · z, w) = K_{μ}(z, s^{∗} · w)\overline{Det(ds^{∗}(w)^{−1})}^{−μ}.  (2.1′)

But then [Ne98, Prop.II.4.3] implies that S(D)^{−1} acts on B_{μ}^{2}(D) via a Hermitian representation and the action of S(D)_{∞}^{−1} clearly is holomorphic.

It remains to show that any g ∈ S(D)_{∞}^{−1} acts by a contraction. We prove this first for μ = 1: then B_{μ} = B_{μ}^{2} is the classical Bergman space, and a change of variables yields immediately

||π_{1}(g)f||_{1}^{2} = ∫_{g^{−1}D} |f(z)|^{2} dz ≤ ∫_{D} |f(z)|^{2} dz = ||f||_{1}^{2}

since g^{−1}D ⊂ D. From this we deduce the result for general μ using Th.II.4.4 of [Ne98] which states that, in the situation we consider, for all g ∈ S(D)_{∞}^{−1} ,

||π_{μ}(g)||^{2} = sup\{| Det Dg^{−1}(z)|^{2μ} \frac{K_{μ}(gz, gz)}{K_{μ}(z, z)} | z ∈ D, K_{μ}(z, z) > 0\}.

In fact, we have just seen that for μ = 1 the right hand side is bounded by 1, and since K_{μ} = K_{1}^{μ}, it is bounded by 1 for all positive μ.

Note that if μ is rational, the previous proposition holds with G(D)_{∞} replaced by some G(D)_{m} with finite m, and if μ = \frac{1}{2} one may choose m = 2. In the following we assume that m ∈ ℕ ∪ {∞} is suitably chosen. Now let k be an admissible parameter (cf. Section 2.1). Then we can push forward the unitary action of G(D)_{m} by the isomorphism A_{μ,0,2k} and obtain a unitary action π_{μ,2k} of G(D)_{m} on B_{μ,2k}^{2}(Ξ) by

(π_{μ,2k}(g)f)(z) = f_{α}(z)^{k}(Det dg^{−1}(z))^{μ}(f_{α}^{−k} · f)(g^{−1}z).  (2.3)

The same formula describes the pushforward of the holomorphic contractive action of the semigroup S(D)_{m}^{−1} . Let G_{m} ⊂ G(D)_{m} be the preimage of G ⊂ G(D) under the canonical projection G(D)_{m} → G(D) and S_{m} ⊂ S(D)_{m} the preimage of S := G_{ℂ} ∩ S(D) ⊂ S(D).

Proposition 2.2.5. Let k be an admissible parameter.

(i) For all g ∈ G_{m},

(π_{μ,2k}(g)f)(z) = (Det dg^{−1}(z))^{μ−2k}f(g^{−1}z).

In particular,

(π_{2k,2k}(g)f)(z) = f(g^{−1}z).

By holomorphic extension, the same formulas hold for g ∈ S_{m}. (ii) The kernel K^{(2k,2k)} is a G_{m} -invariant function: for all g ∈ G_{m} and z, w ∈ D,

K^{(2k,2k)}(gz, gw) = K^{(2k,2k)}(z, w).

Proof. (i) We prove first the following transformation law: for all g ∈ G and x, y where the following expressions are defined, we have

P (g(x) + αg(y)) = dg(x)P (x + αy)jαdg(y)^{−1}jα.

In fact, this follows from the corresponding transformation property of B_{α} ([Be98, 1.3.(9)]) by a calculation similar to [Be98, 2.1.2]. We observe further that jαdg(y)^{−1}jα is just the transposed of dg(y) with respect to the form Tr L(xαy) (cf. [Be98]) and therefore has the same determinant as dg(y); we deduce that for g ∈ G_{m},

f_{α}(g^{−1}z) = (Det dg^{−1}(z))^{2}f_{α}(z).

Now the claim follows using this transformation law in the expression for π_{μ,2k}.

(ii) This can be verified directly from the transformation properties of f_{α} just proved, or it can be seen as an easy consequence of (i).

2.3. The differential restriction operators. If α = id_{V} , then G is the compact linear group U, and the decomposition of B_{μ}^{2}(D) with respect to this group is described in [FK94, Th.XIII.2.7]. We want to obtain a similar decomposition of this space under the action of G (resp. G_{m}) for general α. Recall that D^{−} = D ∩ V_{ℂ}^{−} is a bounded symmetric domain on which G acts transitively, and D^{−} = G/K (Section 1.2). The basic idea, due to Jakobsen and Vergne ([JV79]), is to decompose B_{μ}^{2}(D) as a G-module by restricting functions and their transversal derivatives to D^{−}. For a smooth function f: E ⊃ Y → F defined on a domain Y in a vector space E with values in a vector space F, we denote by df: Y → Hom(E, F) its ordinary total differential, and by d^{k}f: Y → Hom(S^{k}E, F) its k -th total differential; here S^{k}E is the k -th symmetric power of E. If furthermore f is holomorphic and scalar-valued, then d^{k}f is a holomorphic function with values in (S^{k}E)^{∗}; we write d^{k}f ∈ O(Y, (S^{k}E)^{∗}).

Let f be a holomorphic function D → F. The restriction of the linear function d^{k}f(z): S^{k}V_{ℂ} → F to the subspace S^{k}V_{ℂ}^{+} will be called the V^{+} -transversal k-th derivative of f at z and is denoted by d_{+}^{k}f(z). We now define the restriction operators

R^{(0)} :O(D, F) → O(D^{−}, F), f ↦ f|_{D^{−}},
R^{(1)} : ker(R^{(0)}) → O(D^{−}, Hom(V_{ℂ}^{+}, F)), f ↦ d_{+}f|_{D^{−}},
⋮
R^{(k)} : ker(R^{(k−1)}) → O(D^{−}, Hom(S^{k}V_{ℂ}^{+}, F)), f ↦ d_{+}^{k}f|_{D^{−}}.

Proposition 2.3.1. If F is a K_{m} -module, then the maps

R^{(k)} : O(D, F) ⊃ ker(R^{(k−1)}) → O(D^{−}, Hom(S^{k}V_{ℂ}^{+}, F)), f ↦ d_{+}^{k}f|_{D^{−}}

are G_{m} -equivariant with respect to the actions of G_{m} defined by Lemma 2.2.1. Proof. The claim is verified by a direct computation: for k = 0 it is immediate. For k = 1 we differentiate

d(g · f)(z) = d(ρ(dg^{−1}(·)^{−1}) · (f ∘ g^{−1}))(z)
= d(ρ(dg^{−1}(·)^{−1}))(z) · f(g^{−1}(z)) + ρ(dg^{−1}(z))^{−1} · (df(g^{−1}(z)) ∘ dg^{−1}(z)).

We evaluate for f ∈ ker R^{(0)} and z ∈ D^{−}: then f(g^{−1}(z)) = 0 since g preserves D^{−}; therefore the first term vanishes, and the remaining term (considered as operator on V_{ℂ}^{+} ) is just (g·df)(z). For k > 1 we differentiate the above expression k −1 times, using product and Leibnitz rule, and obtain a sum of multilinear terms containing differentials of f of order 0, 1, …, k. The terms containing differentials of f of order less than k vanish when we evaluate for f ∈ ker R^{(k−1)} and z ∈ D^{−}, and only one term remains, namely

ρ(dg^{−1}(z))^{−1} · (d^{k}f(g^{−1}(z)) ∘ (S^{k}dg^{−1}(z))),

and this is precisely (g · d^{k}f)(z) (when restricted to S^{k}V_{ℂ}^{+} ).

Now let H ⊂ O(D, F) be a G-invariant Hilbert subspace (i.e. a subspace with a structure of Hilbert space for which the point evaluations are continuous und such that G acts unitarily in H). Let R_{H}^{(k)} be the restriction of R^{(k)} to H.

Lemma 2.3.2. For all k ≥ 0, the kernels ker R_{H}^{(k)} are closed subspaces of H, and the restriction operators R_{H}^{(k)} are continuous.

Proof. It follows from the Cauchy inequalities (cf. [Chab90, p.30]) that the maps

d^{k}: O(D, F) → O(D, Hom(S^{k}V_{ℂ}, F)), f ↦ d^{k}f

are continuous with respect to the topologies of uniform convergence on compact sets. Therefore these maps and hence the differential restriction operators are continuous on H, and their kernels are closed.

For k > 1 we let H_{k} be the orthocomplement of ker R_{H}^{(k)} in ker R_{H}^{(k−1)} , and let H_{0} be the orthocomplement of ker R_{H}^{(0)} in H. Since G acts unitarily, it follows from Proposition 2.3.1 that H_{k} is a G-submodule of H. Therefore

R_{H}^{(k)} : H_{k} → O(D^{−}, Hom(S^{k}V_{ℂ}^{+}, F))

is a G-isomorphism onto the image of R_{H}^{(k)} .

Proposition 2.3.3. We have the following orthogonal G-invariant decomposition:

H = ⨁_{k=1}^{∞} H_{k}.

Proof. It only remains to show that the sum ⊕_{k}H_{k} is dense in H: in fact, if f ∈ (⨁_{k} H_{k})^{⊥}, then f|_{D^{−}} = 0 since f ∈ H_{0}^{⊥} = ker R_{μ}^{(0)} . Further, all partial derivatives of f in V_{ℂ}^{+} -direction vanish on D^{−}, and therefore the Taylor series of f at any point of D^{−} is zero, implying that f = 0 by analyticity.

We transport the Hilbert structure of H_{k} via the isomorphism R_{H}^{(k)} onto its image. This defines a Hilbert subspace (denoted by RH_{k}) of O(D^{−}, Hom(S^{k}V_{ℂ}^{+}, F)); in fact, the point evaluations of this space are continuous since they are given by differentials on D which have the continuity property described in Lemma 2.3.2.

Since G acts transitively on D^{−}, we can apply a theorem due to S. Kobayashi (cf.

[Ko68] for the square integrable version and [BH97, Th.2.5] for the abstract kernel version) in order to decompose the space RH_{k} into irreducible submodules. In our situation the theorem implies: a G-invariant Hilbert subspace ℌ of O(D^{−}, E) is irreducible under G if and only if ℌ(0) := {f(0)|f ∈ ℌ} is irreducible under K = G∩U, where U is the maximal compact subgroup of G(D) defined in Section 1.1. Put in another way: given an irreducible K -module E_{0} , there is at most one G-invariant Hilbert subspace ℌ ⊂ O(D^{−}, U) with ℌ(0) = E_{0} . If it exists, we denote it by H(D^{−}, E_{0}); by Kobayashi’s theorem, it is irreducible. (Using the mean value property of holomorphic functions, one can show that H(D^{−}, E_{0}) is the G-module generated by all constant functions D^{−} → E_{0} ; cf. [Ach96, Section 2].)

Theorem 2.3.4. Let H ⊂ O(D, F) be a G-invariant Hilbert subspace. For k = 0, 1, 2, … let

RH_{k}(0) = ⨁_{i} W_{k,i}

be the decomposition into irreducible K -modules. Then H(D^{−}, W_{k,i}) is irreducible, and

H ≅ ⨁_{i,k} H(D^{−}, W_{k,i})

is the decomposition into irreducible G-modules.

Proof. Since by Proposition 2.3.3 H ≅ ⨁_{k} RH_{k}, we have to decompose RH_{k} into irreducible G-modules. If RH_{k} = ⨁_{j} E_{j} is this decomposition, then one direction of Kobayashi’s theorem tells us that RH_{k}(0) = ⨁_{j} E_{j}(0) is a decomposition into irreducible K -modules, and by the unicity statement we have E_{j} = H(D^{−}, E_{j}(0)). This implies the claim.

In the case where F is one dimensional and H = B_{μ}(D), we have the following, more precise, result:

Theorem 2.3.5. Let ℂ_{μ} be the space ℂ with the action of K_{∞} by (Det k)^{−μ}, let Pol(V_{ℂ}^{+}, ℂ_{μ}) be the space of holomorphic polynomials on V_{ℂ}^{+} with the action of K_{∞} by k · p = (Det k)^{−μ} · (p ∘ k^{−1}), and let

Pol(V_{ℂ}^{+}, ℂ_{μ}) = ⨁_{j∈J} W_{j}

be its decomposition into irreducible K_{∞} -modules. Then for μ > \frac{1}{2} − \frac{r}{2n}

B_{μ}(D) = ⨁_{j∈J} H(D^{−}, W_{j})

is the decomposition into irreducible G-modules.

Proof. We apply the preceding theorem for F = ℂ_{μ} and H = B_{μ}(D). By [FK94, Th.XIII.2.7], B_{μ}(D) (μ > \frac{1}{2} − \frac{r}{2n}) contains all holomorphic polynomials; one deduces that

RH_{k}(0) = Hom(S^{k}V_{ℂ}^{+}, ℂ_{μ}).

In fact, if q: D^{−} → (S^{k}V_{ℂ}^{+})^{∗}, z ↦ q_{0} is a constant function, then one may find a holomorphic polynomial p such that d_{+}^{k}p(z) = q(z) = q_{0} and d_{+}^{k−1}p(z) = 0 for all z ∈ D^{−}, namely p(z) = \frac{1}{k!}q_{0}(\frac{z+αz}{2} ).) Now we can apply the preceding theorem, using the natural identification

Pol(V_{ℂ}^{+}, F) = ⊕_{k=0}^{∞} Hom(S^{k}V_{ℂ}^{+}, F).

2.4. Spectrum of B_{μ}(D) in terms of highest weights. We can make Theorem 2.3.5 still more specific by describing the highest weights of the representations appearing there. Note that K is a compact real form of the group K_{ℂ} = Str(V_{ℂ})^{α_{∗}}. The homomorphism

(K_{ℂ})_{o} = Str(V_{ℂ})_{o}^{α_{∗}} → Str(V_{ℂ}^{+})_{o}, g ↦ g|_{V^{+}}  (2.4)

is surjective since the map r_{+} considered in Proposition 1.8.1 is surjective. Thus the decomposition of Pol(V_{ℂ}^{+}, ℂ_{μ}) under K_{ℂ} (or under one of its covering groups) can be treated as a problem in the Jordan algebra V^{+} .

Now let V_{0} be any Euclidian Jordan algebra (later we take V_{0} = V^{+} ). If V_{0} is simple, then according to [FK94, Th.XI.2.4] the space Pol(V_{0}) of polynomials on V_{0} decomposes under the standard action of Str(V_{0})_{o} by g · p = p ∘ g^{−1} as

Pol(V_{0}) = ⨁_{m≥0} P_{m}.

Here m = (m_{1}, …, m_{r}) is a multi-integer (r the rank of V_{0} ) with m_{1} ≥ … ≥ m_{r} ≥ 0, and P_{m} is a (Aut(V_{0})_{o} -spherical) irreducible module of polynomials having highest weight (cf. [FK94, p.227])

− ∑_{i=1}^{r} m_{i}c_{i}^{∗},

with respect to the Cartan subspace 𝔞 = ⊕_{i=1}^{r}ℝL(c_{i}) of L(V_{0}) given by a Jordan frame c_{1}, …, c_{r} and to the order induced by the ordered basis (L(c_{1}), …, L(c_{r})) of 𝔞. We denote the dual basis of 𝔞^{∗} by (c_{1}^{∗}, …, c_{r}^{∗}).

The module P_{(1,...,1)} is one dimensional and generated by the Jordan determinant Δ of V_{0} . We have the relations Δ(x)^{\frac{2n}{r}} = Det P (x) and Det P (gx) = (Det g)^{2} Det P (x), Δ(gx) = (Det g)^{\frac{r}{n}} Δ(x) for g ∈ Str(V_{0}), where P is the quadratic representation of V_{0} . The function Δ is positive on the symmetric cone Ω of V_{0} , and therefore we can define Δ^{t} as a positive function on Ω for all t ∈ ℝ. In the following, we will often identify a real parameter t with (t, …, t) ∈ ℝ^{r}. If m is a multi-integer, we let

P_{m+t} = {Δ^{t} · p | p ∈ P_{m}};

for general t ∈ ℝ, this is a space of functions on the symmetric cone Ω, and it is a Str(V_{0})_{o} - module of highest weight

− ∑_{i}(m_{i} + t)c_{i}^{∗}.  (HW)

For t ∈ ℕ it is a space of polynomials coinciding with the space of polynomials defined previously.

Theorem 2.4.1. Assume α be such that X = G/H is irreducible. Then for all μ ∈ ℝ we have the following decomposition into irreducible K_{∞} -modules:

Pol(V^{+}, ℂ_{μ}) ≅ ⨁_{m≥0} P_{m+μ \frac{n}{r_{+}}} ,

(where notation is as above with V_{0} = V^{+} and r_{+} = rk V^{+} , n = dim V ). For μ > \frac{1}{2} − \frac{r}{2n}, we have the following decomposition into irreducible G_{∞} -modules:

B_{μ}(D) ≅ ⨁_{m≥0} H(D^{−}, P_{m+μ \frac{n}{r_{+}}} ).

The highest weight of the G_{∞} -module H(D^{−}, P_{m+μ \frac{n}{r_{+}}} ) is the same as the one of the K_{∞} -module P_{m+μ \frac{n}{r_{+}}} , namely − ∑_{i}(m_{i} + μ\frac{n}{r_{+}} ).

Proof. We denote by Det the holomorphic character

Det : K_{ℂ} → ℂ^{∗}, g ↦ Det g.

We also consider the characters

Det_{±} : K_{ℂ} → ℂ^{∗}, g ↦ Det(g|_{V_{ℂ}^{±}});

then Det = Det_{+} · Det_{−}. Now K_{ℂ} is connected with one dimensional center; in fact, 𝔨_{ℂ} is the complexification of 𝔰𝔱𝔯(V)^{α_{∗}} which is (according to Proposition 1.8.3) isomorphic to 𝔰𝔱𝔯(V^{−}); moreover V^{−} is simple (cf. loc.cit.), whence 𝔰𝔱𝔯(V^{−}) has a one dimensional center. Thus the center of 𝔨_{ℂ} consists only of the multiples of the identity which then acts by the same scalar on

V^{+} and V^{−}. Therefore Det = Det+^{\frac{n}{n_{+}}} = (Det_{+}^{− \frac{r_{+}}{n_{+}}} )^{− \frac{n}{r_{+}}} so that we can calculate,

Pol(V^{+}, ℂ_{μ}) = Pol(V^{+}) ⊗ ℂ_{μ}

≅ ⨁_{m≥0} P_{m} ⊗ Det^{−μ}

= ⨁_{m≥0} P_{m} ⊗ (Det_{+}^{− \frac{r_{+}}{n_{+}}} )^{−μ \frac{n}{r_{+}}}

= ⨁_{m≥0} P_{m+μ \frac{n}{r^{+}}}

is the decomposition under Str(V^{+})_{o}, under Str(V)_{o}^{α_{∗}} (because of the surjectivity of (2.4)), and thus also under K_{ℂ} and K.

The statement about the decomposition of B_{μ}(D) now follows from Theorem 2.3.5, and the formula for the highest weight w.r.t. K_{∞} is an immediate consequence of formula (HW) above. The remaining statement about the corresponding highest weights w.r.t. G_{∞} is a general fact on holomorphically induced representations, using the fact that 𝔞_{ℂ} is a Cartan subspace both in 𝔨_{ℂ} and in 𝔤_{ℂ} (cf. [Ach96,Th.3.7]).

Remark 2.4.2.

(i) Note that \frac{n}{r_{+}} = \frac{2n}{r} if α is split and \frac{n}{r_{+}} = \frac{n}{r} if α is non-split.

(ii) If α is a Peirce involution, then V^{+} = V_{1} ⊕ V_{−1} is a direct sum of two simple algebras, and the preceding theorem cannot be applied. In fact, for the explicit decomposition of Pol(V_{1} ⊕V_{−1}) ≅ Pol(V_{1})⊗Pol(V_{−1}) in terms of the ones of Pol(V_{1}) and Pol(V_{−1}) one needs a Clebsch-Gordan formula. Then one could apply similar arguments as in the preceding proof.

3. Comparison of Hardy spaces.

3.1. The Hardy space associated to Ξ. We keep the notation and hypotheses from Section 2. Following [HOØ91], we define a Hardy space H^{2}(Ξ) associated to the compactly causal Makarevič space X as the space of holomorphic functions f on Ξ having finite Hardy space norm

||f||_{H^{2}(Ξ)}^{2} := sup_{γ∈Γ^{−1}} ||γ · f||_{L^{2}(X)}^{2},  (3.1)

where Γ = G_{ℂ} ∩ S(D) ⊆ G_{ℂ} is the semigroup introduced in Proposition 1.4.1; it acts on the space of holomorphic functions f: Ξ → ℂ by γ · f := f ∘ γ^{−1} . Since γ · f is holomorphic on a neighborhood of Ξ, it is continuous on X, and

||γ · f||_{L^{2}(X)}^{2} := ∫_{X} |f(γ^{−1}x)|^{2}dx

is defined. We recall the following general results on H^{2}(Ξ) from [HOØ91]:

Theorem 3.1.2.

(i) Equation (3.1) defines a Hilbert space norm on H^{2}(Ξ). (ii) The semigroup Γ^{−1} acts by contractions on H^{2}(Ξ) via (γ, f) ↦ γ · f.

(iii) There is a G-equivariant isometry b = b_{Ξ} : H^{2}(Ξ) → L^{2}(X) defined by b_{Ξ}(f) = lim(γ_{j} · f) for any sequence γ_{j} ∈ Γ^{−1} converging to 1 ∈ G. Here the action of G on L^{2}(X) is the left regular one.

(iv) Set W_{0} := −W̃ and suppose that X ∈ iW_{0} . Then X acts on b_{Ξ}(H^{2}(Ξ)) via the regular representation as a negative operator. Conversely, if H ⊆ L^{2}(X) is a closed subspace on which all the X ∈ iW_{0} act as negative operators, i.e. H is W_{0} -dissipative, then H ⊆ b_{Ξ}(H^{2}(Ξ)).

(v) The Hilbert space H^{2}(Ξ) admits a reproducing kernel K: Ξ × Ξ → ℂ which is holomorphic in the first and antiholomorphic in the second variable.

Since the Hardy space H^{2}(Ξ) is defined by integration over X, we will need the following result about the G-invariant measure on X. Let us denote by dσ the U -invariant normalized measure on Σ.

Proposition 3.1.3. The density of the G-invariant measure on the open dense set X ⊂ Σ with respect to dσ is given by the function

|f_{α}(u)|^{− \frac{1}{2}} , u ∈ X.

Proof. We use the corresponding result for the Cayley transformed realization: by [Be98, Cor.2.4.2], the invariant measure on X^{(−α)} = C(X) is given by | Det B_{−α}(x, x)|^{− \frac{1}{2}} dλ (where dλ is Lebesgue measure on V ) and the invariant measure on V^{c} = X^{(− id_{V} )} by | Det B_{− id}(x, x)|^{− \frac{1}{2}} dλ. In both formulas, Det A denotes the determinant of an endomorphism A of V ; it is the same as the determinant of its ℂ-linear extension A_{ℂ} taken over ℂ (since both have the same matrix). By Cayley transform, the density we are looking for is given by

\frac{| Det B_{−α}(Cx, Cx)|^{− \frac{1}{2}}}{| Det B_{− id}(Cx, Cx)|^{− \frac{1}{2}}}

The calculation given in the proof of [Be98, Th.2.1.1] together with the fact that | Det P (u)| = 1 for all u ∈ Σ shows that the quotient equals

| Det P (z + αz)|^{− \frac{1}{2}} , z = C(x) ∈ C(X).

3.2. Embedding of H^{2}(D) into H^{2}(Ξ) (proof of Theorem 3). Recall the definition of the Hardy space H^{2}(D) and its reproducing kernel from Equations (0.10) and (0.11). We are now ready to give a precise formulation for Theorem 3.

Theorem 3.2.1. Assume that \frac{1}{4} is an admissible parameter, i.e. f_{α} has a holomorphic fourth root.

(i) We have a norm preserving inclusion of function spaces

B_{\frac{1}{2} , \frac{1}{2}} (Ξ) ⊂ H^{2}(Ξ).

(ii) The map

H^{2}(D) = B2_{\frac{1}{2}} (D) → B_{\frac{1}{2} , \frac{1}{2}}^{2} (Ξ) ⊂ H^{2}(Ξ), f ↦ fα^{\frac{1}{4}} · f

is an isometric and G-equivariant imbedding.

Proof. (i) Proposition 2.2.4 shows that Γ^{−1} acts by contractions in B2_{\frac{1}{2}} (D) and thus, by Gisomorphy, also in the space B_{\frac{1}{2} , \frac{1}{2}}^{2} (Ξ). Recall further that the classical Hardy space H^{2}(D) also admits an isometric embedding b_{D}: H^{2}(D) → L^{2}(Σ), f ↦ lim_{t→1} t^{−1} id ·f (cf. [FK94, p.270]). Using this and Proposition 3.1.3, we have for all f ∈ B_{\frac{1}{2} , \frac{1}{2}}^{2} (Ξ)

||f||_{B_{\frac{1}{2} , \frac{1}{2}} (Ξ)} = sup_{γ∈Γ^{−1}} ||γ · f||_{B_{\frac{1}{2} , \frac{1}{2}} (Ξ)}

= sup_{γ∈Γ^{−1}} ||f_{α}^{− \frac{1}{4}} (γ · f)||_{B_{\frac{1}{2} ,0}(Ξ)}

= sup_{γ∈Γ^{−1}} ||f_{α}^{− \frac{1}{4}} (γ · f)||_{H^{2}(D)}

= lim_{t→1}(sup_{γ∈Γ^{−1}} ||t^{−1} id ·(f_{α}^{− \frac{1}{4}} (γ · f))||_{H^{2}(D)})

= sup_{γ∈Γ^{−1}} ||f_{α}^{− \frac{1}{4}} (γ · f)||_{L^{2}(Σ)}

= sup_{γ∈Γ^{−1}} ||γ · f||_{L^{2}(X)} = ||f||_{H^{2}(Ξ)}.

Thus proves part (i). Part (ii) follows by combining part (i) and Proposition 2.1.3.

3.3. Criterion for equality (proof of Theorem 4). Recall the description of the spectrum of H^{2}(Ξ) in terms of highest weight representations from [HOØ91, Th.3.3 and Th.3.4] and [OØ91, Th. 5.2].

Theorem 3.3.1. Suppose that X is irreducible.

(i) H^{2}(Ξ) is a multiplicity free direct sum of unitary highest weight modules. (ii) The highest weight representations occurring in H^{2}(Ξ) admit H -invariant distribution

vectors so that the minimal K -types are K ∩ H -spherical and the corresponding highest weights can be viewed as elements of 𝔞^{∗}.

(iii) An element of λ ∈ i𝔞^{∗} occurs as a highest weight of an irreducible subrepresentation of H^{2}(Ξ) if and only the following conditions are satisfied:

(1) The K -module with highest weight λ is K ∩ H -spherical.

(2) λ(iX) ≤ 0 for all X ∈ W ∩ 𝔞 = −W_{0} ∩ 𝔞.

(3) (λ + ρ | β) < 0 for all β ∈ Δ_{1} .

The generalization of the Cartan-Helgason theorem given by Takeuchi in [Ta73, Th.2.4] allows a characterization of the property of the K -module with highest weight λ to be K ∩ H - spherical. The integrality condition on λ given in that theorem refers to the lattice

L := {Z ∈ 𝔞| exp(Z)ie = ie}

If e_{1}, …, e_{r_{+}} is a Jordan frame in V^{+} , then

L = 2πi ∑_{j=1}^{r_{+}} ℤL(e_{j});

this is immediate from the equation exp(L(∑_{j} a_{j}e_{j}))ie = i ∑_{j} e^{a_{j}}e_{j}. If now λ = ∑_{j=1}^{r_{+}} m_{j}e_{j}^{∗}, the integrality condition from Takeuchi’s Cartan-Helgason theorem simply is

m_{j} ∈ ℤ, ∀j = 1, …, r_{+}.

Of course the corresponding dominance condition depends on the choices of Δ_{0}^{+} made in Section 1.7.

Theorem 3.3.2. Assume that X is irreducible and that \frac{1}{4} is an admissible parameter.

(i) If α is non-split, then B_{\frac{1}{2} , \frac{1}{2}} (Ξ) = H^{2}(Ξ). These are the cases I.2 (Sp(2k, ℝ)/ Sp(k, ℂ)), II.2 (Sp(k, ℝ)), IV.3 (Hyperboloids) and Cayley type from the classification in Table 1.5.1.

(ii) If α is split, then B_{\frac{1}{2} , \frac{1}{2}} (Ξ) is a proper subspace of H^{2}(Ξ), and the quotient H^{2}(Ξ)/B_{\frac{1}{2} , \frac{1}{2}} (Ξ) decomposes as a direct sum of irreducible G-modules with highest weights λ = − ∑_{j=1}^{r} m_{j}e_{j}^{∗} as follows:

m_{1} ≥ … ≥ m_{r}, m_{i} ∈ ℤ^{+}, \frac{n}{2r} > m_{r} > \frac{n}{2r} − \frac{d + 1}{2}.

These are the cases I.2 (SO^{∗}(2m)/ SO(m, ℂ); d = 2), III.2 (SO^{∗}(2m); d = 4) and V.2

(d = 8).

Proof. We have to specify the conditions of Theorem 3.3.1 (iii) in terms of the Jordan data from Section 1.7. If we let λ = − ∑_{i=1}^{r_{+}} m_{i}e_{i}^{∗}, then the description of Δ_{0}^{+} shows that (1) can be written

m_{1} ≥ … ≥ m_{r}; m_{i} ∈ ℤ.

The cone admissibility condition (2) can be written, since 𝔞 ∩ W = − ∑_{i=1}^{r_{+}} ℝ^{+}e_{i}

λ = − ∑_{i=1}^{r_{+}} m_{i}e_{i}^{∗}, ∀i : m_{i} > 0.

The conditions (1) and (2) together have been written earlier in the form m ≥ 0. The Harish-Chandra condition (3) has to be made explicit by using the structural data from Section 1.7.

– α split. The condition (ρ − m|β) < 0 for all β ∈ Δ_{1} is equivalent to

m_{j} + m_{k} > ρ_{j} + ρ_{k} ∀j ≠ k,

where ρ = ∑_{j} ρ_{j}e_{j}^{∗}. Since m_{j} + m_{k} ≥ 2m_{r} and ρ_{j} + ρ_{k} ≤ ρ_{r} + ρ_{r−1} = \frac{d}{4}(2r − 3), this condition is equivalent to

m_{r} > \frac{d}{2}(2r − 3) = \frac{n}{2r} − \frac{d + 1}{2}.

– α non-split. The condition (ρ − m|β) < 0 forall β ∈ Δ_{1} is equivalent to

m_{j} > ρ_{j} ∀j.

In the Cayley-type case the Harish-Chandra condition is equivalent to

m_{r_{+}} > ρ_{r_{+}} = \frac{1}{2} + \frac{d}{2}(r_{+} − 1) = \frac{n_{+}}{r_{+}} − \frac{1}{2} = \frac{n}{r} − \frac{1}{2}.

Since \frac{n}{r} is either an integer or a half-integer, this is equivalent to m_{r_{+}} ≥ \frac{n}{r}.

In the matrix cases the Harish-Chandra condition is equivalent to

m_{r_{+}} > ρ_{r_{+}} = d(r_{+} − 1) + \frac{d + 1}{2} = \frac{d(r − 1)}{2} + \frac{1}{2} = \frac{n}{r} − \frac{1}{2}.

As above, this is seen to be equivalent to m_{r_{+}} ≥ \frac{n}{r}.

In the case of Type IV.3 we have ρ = \frac{n−1}{2}e^{∗}; thus the Harish-Chandra condition is equivalent to

n_{r_{+}} > ρ_{r_{+}} = \frac{n − 1}{2} = \frac{n}{r} − \frac{1}{2},

and again this is equivalent to m_{r_{+}} ≥ \frac{n}{r}.

Now we can compare conditions (1) – (3) of Theorem 3.3.1 (iii) with the description of the spectrum of B_{\frac{1}{2} , \frac{1}{2}} (Ξ) ≅ B_{\frac{1}{2}} (D) (Theorem 2.4.2): the spectrum of B_{\frac{1}{2}} (D) is given by the condition m ≥ 0 and m_{r} ≥ \frac{n}{2r_{+}} . In the non-split case the last condition is equivalent to m_{r} ≥ \frac{n}{r}. Since the Harish-Chandra condition was also equivalent to this condition, the spectra of B_{\frac{1}{2} , \frac{1}{2}} (Ξ) and H^{2}(Ξ) are the same, and since according to Th. 3.3.1 (i) we have multiplicity one in H^{2}(Ξ), both spaces are actually equal. In the split case the spectrum of B_{\frac{1}{2}} (D) is given by the conditions m ≥ 0 and m_{r} ≥ \frac{n}{2r}. Since in these cases d ≥ 2, this does not coincide with the conditions (1)

– (3) given above, the difference of both conditions being precisely the set of parameters given in the theorem.

Remark 3.3.3. If α is a Peirce involution, it is more difficult to get precise results. However, going through the arguments leading to Theorem 3.3.2, one gets rough estimates indicating that in these cases again B_{\frac{1}{2} , \frac{1}{2}} (D) is a proper subspace of H^{2}(Ξ).

3.4. The compact case. We consider the case α = id_{V} : X = Σ is the Shilov boundary of D, and Ξ = {z ∈ D| det z ≠ 0} = D ∩ Ω_{ℂ}. Note that Theorem 3.3.2 does not apply in this case since X is not irreducible. The following result, due to Lassalle ([La85, Th. 8]), replaces Theorem 3.3.2 in the compact case.

Proposition 3.4.1. H^{2}(Ξ) = H^{2}(D) Proof. “⊃”: Clearly a function holomorphic on D is holomorphic on Ξ, and an element of H^{2}(D) satisfies also the Hardy condition of H^{2}(Ξ) because according to Proposition 2.2.4, S(D)^{−1} acts by contractions on H^{2}(D).

“⊂”: It suffices to show that elements of H^{2}(Ξ) are actually holomorphic on D for then the Hardy condition of H^{2}(Ξ) clearly implies the one of H^{2}(D). (The supremum is taken over the semigroup ]0, 1[⊂ Γ(C) = {g ∈ Str(V_{ℂ})| g(Ξ) ⊂ Ξ}.) According to [FK94, Ch.XII], every holomorphic function f on Ξ can be developped in a Laurent series

f(z) = ∑_{m∈ℤ_{+}^{r}} a_{m}p_{m}(z),

where p_{m} belongs to the space P_{m} defined in [FK94], p. 245. The elements of this space are certain rational functions which are homogeous of degree ∑_{i=1}^{r} m_{i}. Since

sup_{0<r<1} ∫_{Σ} |p_{m}(ru)|^{2} dσ(u) = (sup_{0<r<1} r^{∑ m_{i}}) ∫_{Σ} |p_{m}(ru)|^{2}

is finite if and only if ∑ m_{i} ≥ 0, only terms satisfying this condition appear in the Laurent series of f which is thus actually a Taylor series converging on D.

Corollary 3.4.2. The reproducing kernel of H^{2}(Ξ) is given by K(z, w) = Det B(z, \overline{w})^{− \frac{1}{2}} .

Corollary 3.4.3. In the compact case, the inclusion B_{\frac{1}{2}} (Ξ) ⊂ H^{2}(Ξ) is strict, and the Bergman kernel associated to Ξ is not the square of the Cauchy kernel. Proof. The first statement follows from the second, and this is proved by remarking that the Bergman kernel for Ξ is given by

K^{(1,1)}(z, w) = Det P (z)^{\frac{1}{2}} Det B(z, \overline{w})^{−1} Det P (\overline{w})^{\frac{1}{2}}

which is not the square of K.

The results of the last corollary make appear the equality in Proposition 3.4.1 rather “noncanonical”. It is difficult to give a reasonable interpretation of this equality. It can be proved that for any α the inclusion H^{2}(Ξ) ⊂ H^{2}(D) holds; thus the space H^{2}(D) appears as an “upper bound” which is reached in the compact case.

4. Open problems.

At this stage there are two classes of open problems:

Problem 4.1. A more conceptual definition of the spaces B_{μ,λ}(Ξ) has to be made in the framework of Hilbert spaces of sections of vector bundles; then the spaces B_{μ,2k}(Ξ) for various admissible parameters k are all interpreted as spaces of sections of the same line bundle. The groundwork for such a theory is laid in the note [BH97]. However, the definition of Hardy spaces as spaces of sections of line bundles (of half-forms, to be more precise) is more delicate than the definition of Bergman spaces of sections. For example, topological problems come in, related to the problem of existence of holomorphic fourth roots of f_{α}. But only in this way it is possible to get an interpretation of the “inclusion” of the classical Hardy space in the non-classical one as a natural inclusion of invariantly defined Hardy spaces, and at the same time to make definitions independent of the special realization chosen. The desire for having such an invariant description arises already in the case of the classical Hardy spaces: in fact, the classical Hardy space H^{2}(D) and the Hardy space H^{2}(T_{Ω}) of the tube T_{Ω} equivalent to D ([FK94, p.178]) are isomorphic, but this is not at all clear from the definitions (it can be deduced by comparing the reproducing kernels). Therefore one would like to understand H^{2}(D) and H^{2}(T_{Ω}) as different aspects of one invariantly defined object. Then, as shows the present work, the invariantly defined Hardy space of Ξ sometimes is equal to the one of D, but sometimes it is strictly bigger. One would like to have a geometric understanding of Theorem 4 in this context.

Problem 4.2. One wants to find a formula for the Cauchy-Szegö kernel in the remaining cases, where one does not have equality of classical and non-commutative Hardy spaces. There is one case in which this is easy, namely the compact case; cf. Section 3.4. In the intermediate cases it seems to be rather difficult to find a “simple” formula for the Cauchy-Szegö kernel (cf. [KØ96] for the case X = U(1, 1)). Moreover, there are a few compactly causal spaces which cannot be realized using Jordan algebras (cf. Remark 1.5.2), and for these literally nothing seems to be known.

References

[Ach96] Achab, D., “Représentations C -dissipatives irréductibles”, Journées ‘Programme de Gelfand-Gindikin’, Paris 1996.

[Be96] Bertram, W., “On some Causal and Conformal Groups”, J. Lie Theory 6 (1996), 215 – 244.

[Be98] Bertram, W., “Algebraic Structures of Makarevič Spaces I ”, to appear in Transformation Groups, 1998.

[BH97] Bertram, W. and J. Hilgert, “Reproducing Kernels on Vector Bundles”, Preprint, Clausthal 1997.

[BO98] Betten, F. and G. Ólafsson, “Causal compactifications and Hardy spaces for spaces of Hermitian type”, submitted.

[Cha97] Chadli, M.,“Espace de Hardy d’un espace symétrique de type Cayley”, preprint, Paris 1997.

[Chab90] Chabat, B., Introduction à l’analyse complexe. Vol. 2, Edition MIR, Moscow, [FK94] Faraut, J., and A. Korányi, Analysis on symmetric cones, Clarendon Press, Oxford, 1994.

[FO95] Faraut, J., and G. Ólafsson, “Causal semi-simple symmetric spaces, the geometry and harmonic analysis”, in: Semigroups in Algebra, Geometry and Analysis, Eds. Hofmann, Lawson, Vinberg; de Gruyter, Berlin 1995..

[Hw69] Helwig, K.H., “Involutionen von Jordan-Algebren”, Man. Math. 1 (1969), 211 – 229.

[HO96] Hilgert, J. and G. Ólafsson, Causal Symmetric Spaces - Geometry and Harmonic Analysis, Perspectives in Mathematics Vol. 18, Academic Press, San Diego 1996.

[HOØ91] Hilgert, J., G. Ólafsson and B. Ørsted, “Hardy spaces on affine symmetric spaces”, J. reine angew. Math. 415 (1991), 189 – 218.

[JV79] Jakobsen, H.P. and M. Vergne, “Restriction and Expansion of Holomorphic Representations”, J.of Functional Analysis 34 (1979), 29 – 53.

[Ko68] Kobayashi, S., “Irreducibility of certain unitary representations”, J. Math. Soc. Japan 20 (1968), 638 – 642.

[KØ96] Koufany, K. and B. Ørsted, “Function spaces on the Olshanski semi-group and the Gelfand-Gindikin program”, Ann. Inst. Fourier 46 (1996), 689 – 722.

[KØ97] Koufany, K. and B. Ørsted, “Hardy spaces on two-sheeted covering semigroups”, J. of Lie theory 7 (1997), 245 – 267.

[Kr97] Krötz, B., “On Hardy and Bergman spaces on complex Olshanskii semigroups”, submitted.

[La85] Lassalle, M., “L’espace de Hardy d’un domaine de Reinhard généralisé”, J. Funct. An. 60 (1985), 309 – 340.

[Lo77] Loos, O., Bounded symmetric domains and Jordan pairs, Lecture Notes, Irvine 1977.

[Ma73] Makarevič, B.O., “Open symmetric orbits of reductive groups in symmetric R-spaces”, Math. USSR Sbornik 20 (1973), 406 – 418.

[Mo97] Molchanov, V., “Holomorphic Disrete Series for Hyperboloids of Hermitian Type”, J. Funct. Anal. 147 (1997), 26 – 50.

[Ne98]  Neeb, K.H., Holomorphy and Convexity in Lie Theory, de Gruyter, to appear.

[OØ91] Ólafsson, G. and B. Ørsted, “The holomorphic discrete series of an affine symmetric space and representations with reproducing kernel”, Trans. AMS 326 (1991), 385 – 405.

[OØ98] Ólafsson, G. and B. Ørsted, “Causal compactifications and Hardy Spaces”, Trans. AMS, to appear.

[Pe96] Pevsner, M., “Espace de Bergman d’un semi-groupe complexe”, Comptes Rendus Ac. Sci. Paris 322, série 1 (1996), 635 – 640.

[Sa80] Satake, I., Algebraic structures of symmetric domains, Iwanami Shoten, Princeton 1980.

[Ta73] Takeuchi, M., “Polynomial representations associated with symmetric bounded domains”, Osaka J. Math. 10 (1973), 441 – 475.

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