Invariant Inner Product in Spaces of

Holomorphic Functions

on Bounded Symmetric Domains

Jonathan Arazy and Harald Upmeier^{1}

Received: March, 24, 1996 Revised: August, 8, 1997

Communicated by Joachim Cuntz

Abstract. We provide new integral formulas for the invariant inner product on spaces of holomorphic functions on bounded symmetric domains of tube type.

1991 Mathematics Subject Classification: Primary 46E20; Secondary 32H10, 32A37, 43A85

0 Introduction

Our main concern in this work is to provide concrete formulas for the invariant inner products and hermitian forms on spaces of holomorphic functions on Cartan domains D of tube type. As will be explained below, the group Aut(D) of all holomorphic automorphisms of D acts transitively. Aut(D) acts projectively on function spaces on D via f ↦ U^{(λ)}(φ)f := (f ∘ φ) (Jφ)^{λ/p}, φ ∈ Aut(D), λ ∈ C, but these actions are not irreducible in general. The inner products we consider are those obtained from the holomorphic discrete series by analytic continuation. The associated Hilbert spaces generalize the weighted Bergman spaces, the Hardy and the Dirichlet space. By “concrete” formulas we mean Besov-type formulas, namely integral formulas involving the functions and some of their derivatives. Possible applications include the study of operators (Toeplitz, Hankel) acting on function spaces and the theory of invariant Banach spaces of analytic functions (where the pairing between an invariant space and its invariant dual is computed via the corresponding invariant inner product).

Our problem is closely related to finding concrete realizations (by means of integral formulas) of the analytic continuation of the Riesz distribution. [Ri], [Go], [FK2], Chapter VII.

1Authors supported

by a grant from the German-Israeli Foundation (GIF), I-415-023.06/95.

Documenta Mathematica 2 (1997) 213–261

In principle, the analytic continuation is obtained from the integral formulas associated with the weighted Bergman spaces (i.e. the holomorphic discrete series) by “partial integration with respect to the radial variables”. This program has been successful in the case of rank 1 (i.e. when D is the open unit ball of C^{d}, see [A3]). The case of rank r > 1 is more difficult, and concrete formulas are known only in special cases, see [A2], [Y4], [Y1], [Y2].

This paper consists of two main parts. In the first part (Sections 2, 3, and 4) we develop in full generality the techniques of [A2], [Y4], and obtain integral formulas for the invariant inner products associated with the so-called Wallach set and pole set. In the second part (section 5) we introduce new techniques (integration on boundary orbits), to obtain new integral formulas for the invariant inner products in the important special cases of Cartan domains of type I and IV. This approach has the potential for further generalizations and applications, including the infinite dimensional setup.

The paper is organized as follows. Section 1 provides background information on Cartan domains, the associated symmetric cones and Siegel domains and the Jordan theoretic approach to the study of bounded symmetric domains [Lo], [FK2], [U2]. We also explain some general facts concerning invariant Hilbert spaces of analytic functions on Cartan domains and the connection to the Riesz distribution. Section 2 is devoted to the study of invariant differential operators on symmetric cones. We study the “shifting operators” introduced by Z. Yan and their derivatives with respect to the “spectral parameter”. Section 3 is devoted to our generalization of Yan’s shifting method, to find explicit integral formulas for the invariant inner products obtained by analytic continuation of the holomorphic discrete series. In section 4 we study the expansion of Yan’s operators, and obtain applications to concrete integral formulas for the invariant inner products. Some of these results were obtained independently by Z. Yan, J. Faraut and A. Korányi, [FK2], [Y4]. We include these results and our proofs, in order to make the paper self contained, and also because in most cases our results go beyond the results in [FK2], [Y4].

In section 5 we propose a new type of integral formulas for the invariant inner products. These formulas involve integration on boundary orbits and the application of the localized versions of the radial derivative associated with the boundary components of Cartan domains. We are able to establish the desired formulas in the important special cases of type I and IV. The techniques established in this section can be used in the study of the remaining cases.

Finally, in the short section 6 we use the quasi-invariant measures on the boundary orbits of the associated symmetric cone in order to obtain integral formulas for some of the invariant inner products in the context of the unbounded realization of the Cartan domains (tube domains). These results are essentially implicitly contained in [VR], where the authors use the Lie-theoretic and Fourier-analytic approach to analysis on symmetric Siegel domains. We use the Jordan-theoretic approach which yields simpler formulation of the results and simpler proofs.

Acknowledgment: We would like to thank Z. Yan, J. Faraut, and A. Korányi for sending us drafts of their work and for many stimulating discussions. We also thank the referee for valuable comments on the first version of the paper.

1 Preliminaries

A Cartan domain D ⊂ C^{d} is an irreducible bounded symmetric domain in its Harish-Chandra realization. Thus D is the open unit ball of a Banach space Z = (C^{d}, ∥ · ∥) which admits the structure of a JB^{∗}-triple, namely there exists a continuous mapping Z × Z × Z ∋ (x, y, z) → {x, y, z} ∈ Z (called the Jordan triple product) which is bilinear and symmetric in x and z, conjugate-linear in y, and so that the operators D(x, x) : Z → Z defined for every x ∈ Z by D(x, x)z := {x, x, z} are hermitian, have positive spectrum, satisfy the ”C^{∗}-axiom” ∥D(x, x)∥ = ∥x∥^{2}, and the operators δ(x) := iD(x, x) are triple derivations, i.e. the Jordan triple identity holds

δ(x){y, z, w} = {δ(x)y, z, w} + {y, δ(x)z, w} + {y, z, δ(x)w}, ∀y, z, w ∈ Z.

The norm ∥ · ∥ is called the spectral norm. We put also D(x, y)z := {x, y, z}. An element v ∈ Z is called a tripotent if {v, v, v} = v. Every tripotent v ∈ Z gives rise to a direct-sum Peirce decomposition

Z = Z_{1}(v) + Z_{\frac{1}{2}} (v) + Z_{0}(v), where Z_{ν}(v) := \{z ∈ Z; D(v, v)z = νz\}, ν = 1, \frac{1}{2}, 0.

The associated Peirce projections are defined for z_{κ} ∈ Z_{κ}(v), κ = 1, \frac{1}{2}, 0, by

P_{ν}(v)(z_{1} + z_{\frac{1}{2}} + z_{0}) = z_{ν}, ν = 1, \frac{1}{2}, 0.

In this paper we are interested in the important special case where Z contains a unitary tripotent e, for which Z = Z_{1}(e). In this case Z has the structure of a JB^{∗}-algebra with respect to the binary product x ∘ y := {x, e, y} and the involution z^{∗} := {e, z, e}, and e is the unit of Z. The binary Jordan product is commutative, but in general non-associative. The triple product is expressed in terms of the binary product and the involution via {x, y, z} = (x∘y^{∗})∘z +(z ∘y^{∗})∘x−(x∘z)∘y^{∗}. In this case the open unit ball D of Z is a Cartan domain of tube-type. This terminology is related to the unbounded realization of D, to be explained later.

Let X := {x ∈ Z; x^{∗} = x} be the real part of Z. It is a formally-real (or euclidean) Jordan algebra. Every x ∈ X has a spectral decomposition x = ∑_{j=1}^{r} λ_{j}e_{j}, where {e_{j}}_{j=1}^{r} is a frame of pairwise orthogonal minimal idempotents in X, and {λ_{j}}_{j=1}^{r} are real numbers called the eigenvalues of x. The trace and determinant (or, “norm”) are defined in X via

tr(x) := ∑_{j=1}^{r} λ_{j}, N(x) := ∏_{j=1}^{r} λ_{j}

respectively, and they are polynomials on X. The maximal number r of pairwise orthogonal minimal idempotents in X is called the rank of X. The positive-definite inner product in X, ⟨x, y⟩ = tr(x ∘ y), x, y ∈ X, satisfies

⟨x ∘ y, z⟩ = ⟨x, y ∘ z⟩, x, y, z ∈ X.

Equivalently, the multiplication operators L(x)y := x ∘ y, x, y ∈ X, are self-adjoint. The trace and determinant polynomials, as well as the multiplication operators, have unique extensions to the complexification X^{C} := X + iX = Z. Let

Ω := {x^{2}; x ∈ X, N(x) ≠ 0}.

Then Ω is a symmetric, open convex cone, i.e. Ω is self polar and homogeneous with respect to the group GL(Ω) of linear automorphisms of Ω. We denote the connected component of the identity in GL(Ω) by G(Ω). Define

P (x) := 2L(x)^{2} − L(x^{2}), x ∈ X,  (1.1)

then P (x) ∈ G(Ω) for every x ∈ Ω, and x = P (x^{1/2})e. Thus G(Ω) is transitive on Ω. The map x → P (x) from X into End(X) is called the quadratic representation because of the identity

P (P (x)y) = P (x)P (y)P (x), ∀x, y ∈ X.  (1.2)

The domain T (Ω) := X + iΩ, called the tube over Ω. It is an irreducible symmetric domain which is biholomorphically equivalent to D by means of the Cayley transform c : D → T (Ω), defined by

c(z) := i\frac{e + z}{e − z} , z ∈ Z.

This explains why D is called a tube-type Cartan domain.

Let e_{1}, e_{2}, …, e_{r} be a fixed frame of minimal, pairwise orthogonal idempotents satisfying e_{1} + e_{2} + … + e_{r} = e, where e is the unit of Z. Let

Z = ∑_{1≤i≤j≤r} Z_{i,j}

be the associated joint Peirce decomposition, namely Z_{i,j} := Z_{\frac{1}{2}} (e_{i}) ∩ Z_{\frac{1}{2}} (e_{j}) for 1 ≤ i < j ≤ r and Z_{i,i} := Z_{1}(e_{i}) for 1 ≤ i ≤ r. The characteristic multiplicity is the common dimension a = dim(Z_{i,j}), 1 ≤ i < j ≤ r, and d = r + r(r − 1)a/2. The number p := (r − 1)a + 2 is called the genus of D. It is known that

Det(P (x)) = N(x)^{p}, ∀x ∈ X,

where “Det” is the usual determinant polynomial in End(Z). From this and (1.2) it follows that

N(P (x)y) = N(x)^{2}N(y) ∀x, y ∈ X.  (1.3)

Let u_{j} := e_{1} + e_{2} + … + e_{j} and let Z_{j} := ∑_{1≤i≤k≤j} Z_{i,k} be the JB^{∗}- subalgebra of Z whose unit is u_{j}. Let N_{j} be the determinant polynomials of the Z_{j}, 1 ≤ j ≤ r; they are called the principal minors associated with the frame {e_{j}}_{j=1}^{r}. Notice that Z_{r} = Z and N_{r} = N. For an r-tuple of integers m = (m_{1}, m_{2}, …, m_{r}) write m ≥ 0 if m_{1} ≥ m_{2} ≥ … ≥ m_{r} ≥ 0. Such r-tuples m are called signatures (or, “partitions”). The conical polynomial associated with the signature m is

N_{m}(z) := N_{1}(z)^{m_{1}−m_{2}} N_{2}(z)^{m_{2}−m_{3}} N_{3}(z)^{m_{3}−m_{4}} … N_{r}(z)^{m_{r}}, z ∈ Z.

Notice that N_{m}(∑_{j=1}^{r} t_{j}e_{j}) = ∏_{j=0}^{r} t_{j}^{m_{j}}, thus the conical polynomials are natural generalizations of the monomials. Let Aut(D) be the group of all biholomorphic automorphisms of D, and let G be its connected component of the identity. Let K := {g ∈ G; g(0) = 0} = G ∩ GL(Z) be the maximal compact subgroup of G. For any signature m let P_{m} := span{N_{m} ∘ k; k ∈ K}. Clearly, P_{m} ⊂ P_{ℓ}, where ℓ = |m| = ∑_{j=1}^{r} m_{j} and P_{ℓ} is the space of homogeneous polynomials of degree ℓ. By definition, P_{m} are invariant under the composition with members of K. Let

⟨f, g⟩_{F} := ∂_{f}(g^{♯})(0) = \frac{1}{π^{d}} ∫_{Z} f(z)\overline{g(z)} e^{−|z|^{2}}dm(z)  (1.4)

be the Fock-Fischer inner product on the space P of polynomials, where g^{♯}(z) := \overline{g(z^{∗})}, ∂_{f} = f(\frac{∂}{∂z}), |z| is the unique K-invariant Euclidean norm on Z normalized so that |e_{1}| = 1, and dm(z) is the corresponding Lebesgue volume measure. (Thus ⟨1, 1⟩_{F} = 1). The following result (Peter-Weyl decomposition) is proved in [Sc], see also [U1]. Here the group K acts on functions on D via π(k)f := f ∘ k^{−1}, k ∈ K. Notice that P_{ℓ}, ℓ = 0, 1, 2, … and P are invariant under this action.

Theorem 1.1 (i) The spaces {P_{m}}_{m≥0}, are K-invariant and irreducible. The representations of K on the spaces P_{m} are mutually inequivalent, the P_{m}’s are mutually orthogonal with respect to ⟨·, ·⟩_{F}, and P = ∑_{m≥0} P_{m}. (ii) If H is a Hilbert space of analytic functions on D with a K-invariant inner product in which the polynomials are dense, then H is the orthogonal direct sum H = ∑_{m≥0} ⊕P_{m}. Namely, every f ∈ H is expanded in the norm convergent series f = ∑_{m≥0} f_{m}, with f_{m} ∈ P_{m}, and the spaces P_{m} are mutually orthogonal in H. Moreover, there exist positive numbers {c_{m}}_{m≥0} so that for every f, g ∈ H with expansions f = ∑_{m≥0} f_{m} and g = ∑_{m≥0} g_{m} we have

⟨f, g⟩_{H} = ∑_{m≥0} c_{m} ⟨f_{m}, g_{m}⟩_{F} .

For every signature m let K_{m}(z, w) be the reproducing kernel of P_{m} with respect to (1.4). Clearly, the reproducing kernel of the Fock-Fischer space F (the completion of P with respect to ⟨·, ·⟩_{F}) is

F (z, w) := ∑_{m} K_{m}(z, w) = e^{⟨z,w⟩}.

An important property of the norm polynomial N is its transformation rule under the group K

N(k(z)) = χ(k)N(z), k ∈ K, z ∈ Z  (1.5)

where χ : K → T := {λ ∈ C; |λ| = 1} is a character. In fact, χ(k) = N(k(e)) = Det(k)^{2/p} ∀k ∈ K. Notice that (1.5) implies that P_{(m,m,...,m)} = C N^{m} for m = 0, 1, 2, . . ..

The subgroup L of K defined via

L := {k ∈ K; k(e) = 1}  (1.6)

plays an important role in the theory.

Lemma 1.1 For every signature m ≥ 0 the function

ϕ_{m}(z) := ∫_{L} N_{m}(ℓ(z))dℓ  (1.7)

is the unique spherical (i.e., L-invariant) polynomial in P_{m} satisfying ϕ_{m}(e) = 1.

For example, ϕ_{(m,m,...,m)} = N^{m} by (1.5). The L-invariant real polynomial on X

h(x) = h(x, x) := N(e − x^{2})

admits a unique K-invariant, hermitian extension h(z, w) to all of Z. Thus, h(k(z), k(w)) = h(z, w) for all z, w ∈ Z and k ∈ K, h(z, w) is holomorphic in z and anti-holomorphic in w, and h(z, w) = \overline{h(w, z)}, [FK1]. The transformation rule of h(z, w) under Aut(D) is

h(φ(z), φ(w)) = Jφ(z)^{\frac{1}{p}} h(z, w) \overline{Jφ(w)}^{\frac{1}{p}} , φ ∈ Aut(D), z, w ∈ D,  (1.8)

where Jφ(z) := Det(φ′(z)) is the complex Jacobian of φ, and p is the genus of D. For s = (s_{1}, s_{2}, …, s_{r}) ∈ C^{r} one defines the conical function N_{s} on Ω via

N_{s}(x) := N_{1}^{s_{1}−s_{2}}(x) N_{2}^{s_{2}−s_{3}}(x) N_{3}^{s_{3}−s_{4}}(x) . . . · N_{r}^{s_{r}}(x), x ∈ Ω,

which generalize the conical polynomials N_{m}. In what follows use the following notation:

λ_{j} := (j − 1)\frac{a}{2}, 1 ≤ j ≤ r.

The Gindikin - Koecher Gamma function is defined for s = (s_{1}, s_{2}, …, s_{r}) ∈ C^{r} with ℜ(s_{j}) > λ_{j}, 1 ≤ j ≤ r, via

Γ_{Ω}(s) := ∫_{Ω} e^{−tr(x)}N_{s}(x)dμ_{Ω}(x).

Here tr(x) = ⟨x, e⟩ is the Jordan-theoretic trace of x, and

dμ_{Ω}(x) := N(x)^{− \frac{d}{r}} dx

is the (unique, up to a multiplicative constant) G(Ω)-invariant measure on Ω. The following formula [Gi] reduces the computation of Γ_{Ω}(s) to that of ordinary Gamma functions:

Γ_{Ω}(s) = (2π)^{(d−r)/2} ∏_{1≤j≤r} Γ(s_{j} − λ_{j}),  (1.9)

and provides a meromorphic continuation of Γ_{Ω} to all of C^{r}. In particular, Γ_{Ω}(λ) := Γ_{Ω}(λ, λ, …, λ) is given by

Γ_{Ω}(λ) = ∫_{Ω} e^{−tr(x)} N(x)^{λ} dμ_{Ω}(x) = (2π)^{(d−r)/2} ∏_{1≤j≤r} Γ(λ − λ_{j}),

and it is an entire meromorphic function. The pole set of Γ_{Ω}(λ) is precisely

P(D) := ∪_{1≤j≤r}(λ_{j} − N) = {λ_{j} − n; 1 ≤ j ≤ r, n ∈ N}.  (1.10)

For λ ∈ C and a signature m = (m_{1}, m_{2}, …, m_{r}) one defines

(λ)_{m} := \frac{Γ_{Ω}(m + λ)}{Γ_{Ω}(λ)} = ∏_{j=1}^{r}(λ − λ_{j})_{m_{j}} = ∏_{j=1}^{r} ∏_{n=0}^{m_{j}−1}(n + λ − λ_{j}),

where m + λ := (m_{1} + λ, m_{2} + λ, …, m_{r} + λ).

We recall two important formulas for integration in polar coordinates [FK2], Chapters VI and IX. The first formula uses the fact that K · Ω = Z, namely the fact that every z ∈ Z can be written (not uniquely) in the form z = k(x), where x ∈ Ω and k ∈ K. This is the first (or “conical”) type of polar decomposition of x, and it generalizes the polar decomposition of matrices. This leads to the formula

∫_{Z} f(z)dm(z) = \frac{π^{d}}{Γ_{Ω}(\frac{d}{r})} ∫_{Ω} (∫_{K} f(k(x^{\frac{1}{2}} )) dk) dx  (1.11)

which holds for every f ∈ L^{1}(Z, m). Next, fix a frame e_{1}, …, e_{r}, and define

R := span_{R}{e_{j}}_{j=1}^{r} and R_{+} := {∑_{j=1}^{r} t_{j}e_{j}; t_{1} > t_{2} > … > t_{r} > 0}

and

R_{+}^{r} := {t = (t_{1}, … t_{r}); t_{1} > t_{2} > … > t_{r} > 0}.

Then Z = K · R, namely every z ∈ Z has a representation z = k(∑_{j=1}^{r} t_{j}e_{j}) for some (again, not unique) ∑_{j=1}^{r} t_{j}e_{j} ∈ R and k ∈ K. This representation is the second type of polar decomposition of z. Moreover, m(Z \ K · R_{+}) = 0, namely up to a subset of measure zero, every z ∈ Z has a representation z = k(∑_{j=1}^{r} t_{j}^{1/2}e_{j}) with t_{1} > t_{2} > … > t_{r} > 0. This leads to the formula

∫_{Z} f(z)dm(z) = c_{0} ∫_{R_{+}^{r}} (∫_{K} f(k(∑_{j=1}^{r} tj^{\frac{1}{2}} e_{j})) dk) ∏_{1≤i<j≤r}(t_{i} − t_{j})^{a} dt_{1} dt_{2} … dt_{r},

(1.12)

which holds for every f ∈ L^{1}(Z, m). The constant c_{0} will be determined as a byproduct of our work in section 5 below. For convenience, we can write (1.12) in the form

∫_{Z} f(z)dm(z) = c_{0} ∫_{R_{+}^{r}} f^{#}(t) w(t)^{a} dt,  (1.13)

where

f^{#}(t) := ∫_{K} f(k(∑_{j=1}^{r} tj^{\frac{1}{2}} e_{j})) dk, t = (t_{1}, t_{2}, …, t_{r}) ∈ R_{+}^{r}

is the radial part of F and

w(t) := ∏_{1≤i<j≤r}(t_{i} − t_{j}), t = (t_{1}, t_{2}, …, t_{r}) ∈ R_{+}^{r}  (1.14)

is the Vandermonde polynomial.

By [Hu], [Be], [La1], [FK1], we have the binomial formula:

Theorem 1.2 For λ ∈ C we have

N(e − x)^{−λ} = ∑_{m≥0}(λ)_{m} \frac{ϕ_{m}(x)}{∥ϕ_{m}∥_{F}^{2}} , ∀x ∈ Ω ∩ (e − Ω),  (1.15)

and

h(z, w)^{−λ} = ∑_{m≥0}(λ)_{m} K_{m}(z, w), ∀z, w ∈ D.  (1.16)

The two series converge absolutely, (1.15) converges uniformly on compact subsets of (λ, x) ∈ C × (Ω ∩ (e − Ω)), and (1.16) converges uniformly on compact subsets of (λ, z, w) ∈ C × D × D.

In particular, it follows that for fixed z, w ∈ D, the function λ → h(z, w)^{−λ} is analytic in all of C (this can be proved also by showing that h(z, w) ≠ 0 for z, w ∈ D).

The Wallach set of D, denoted by W(D), is the set of all λ ∈ C for which the function (z, w) → h(z, w)^{−λ} is non-negative definite in D × D, namely

∑_{i,j} a_{i}\overline{a}_{j} h(z_{i}, z_{j})^{−λ} ≥ 0

for all finite sequences {a_{j}} ⊆ C and {z_{j}} ⊆ D. For λ ∈ W(D) let H_{λ} be the completion of the linear span of the functions {h(·, w)^{−λ}; w ∈ D} with respect to the inner product ⟨·, ·⟩_{λ} determined by

⟨h(·, w)^{−λ}, h(·, z)^{−λ}⟩_{λ} = h(z, w)^{−λ}, z, w ∈ D.

Since h(z, w)^{−λ} is continuous in D × D, it is the reproducing kernel of H_{λ}. The transformation rule (1.8) implies that ⟨·, ·⟩_{λ} is K-invariant, namely ⟨f ∘ k, g ∘ k⟩_{λ} = ⟨f, g⟩_{λ} for all f, g ∈ H_{λ} and k ∈ K. Thus, by Theorems 1.1 and 1.2, for every f, g ∈ H_{λ} with Peter-Weyl expansions f = ∑_{m≥0} f_{m}, g = ∑_{m≥0} g_{m}, we have

⟨f, g⟩_{λ} = ∑_{m≥0} \frac{⟨f_{m}, g_{m}⟩_{F}}{(λ)_{m}} .  (1.17)

This formula defines λ ↦ ⟨f, g⟩_{λ} as a meromorphic function in all of C, whose poles are contained in the pole set P(D) of Γ_{Ω}, see (1.10) and (1.16). Of course, for λ ∈ C \ W(D) (1.17) is not an inner product, but merely a sesqui-linear form; it is hermitian precisely when λ ∈ R.

Using (1.16) and (1.17) one obtains a complete description of the Wallach set W(D) and the Hilbert spaces H_{λ} for λ ∈ W(D).

Theorem 1.3 (i) The Wallach set is given by W(D) = W_{d}(D) ∪ W_{c}(D) where W_{d}(D) := \{λ_{j} = (j − 1)\frac{a}{2}; 1 ≤ j ≤ r\} is the discrete part, and W_{c}(D) := (λ_{r}, ∞) is the continuous part.

(ii) For λ ∈ W_{c}(D) the polynomials are dense in H_{λ} and H_{λ} = ∑_{m≥0} ⊕P_{m} as in Theorem 1.1;

(iii) For 1 ≤ j ≤ r, let S_{0}(λ_{j}) := {m ≥ 0; m_{j} = m_{j+1} = … = m_{r} = 0}. Then H_{λ_{j}} = ∑_{m∈S_{0}(λ_{j})} P_{m} and

h(z, w)^{−λ_{j}} = ∑_{m∈S_{0}(λ_{j})}(λ_{j})_{m} K_{m}(z, w), z, w ∈ D.

For λ ∈ C, φ ∈ G and a functions f on D, we define

U^{(λ)}(φ)f := (f ∘ φ) · (Jφ)^{\frac{λ}{p}}

Then, U^{(λ)}(id_{D}) = I and for φ, ψ ∈ G we have

U^{(λ)}(φ ∘ ψ) = c_{λ}(φ, ψ) U^{(λ)}(ψ) U^{(λ)}(φ),

where c_{λ}(φ, ψ) is a unimodular scalar which transforms as a cocycle (projective representation of G). In particular, U^{(λ)}(φ^{−1}) = U^{(λ)}(φ)^{−1}.

Using (1.8) we see that

Jφ(z)^{\frac{λ}{p}} h(φ(z), φ(w))^{−λ} \overline{Jφ(w)}^{\frac{λ}{p}} = h(z, w)^{−λ}, ∀z, w ∈ D, ∀φ ∈ G.

From this it follows that the hermitian forms ⟨·, ·⟩_{λ} given by (1.17) are U^{(λ)}-invariant:

⟨ U^{(λ)}(φ)f , U^{(λ)}(φ)g ⟩_{λ} = ⟨f, g⟩_{λ}, ∀f, g ∈ H_{λ}, ∀φ ∈ G.

In particular, for λ ∈ W(D) the inner products ⟨·, ·⟩_{λ} are U^{(λ)}-invariant and U^{(λ)}(φ), φ ∈ G, are unitary operators on H_{λ}.

There are other spaces of analytic functions on D which carry U^{(λ)}-invariant hermitian forms, some of which are non-negative. For any signature m and λ ∈ C let q(λ, m) := deg_{λ}(·)_{m} be the multiplicity of λ as a zero of the polynomial ξ ↦ (ξ)_{m}. Notice that 0 ≤ q(λ, m) ≤ r for all λ and m. Let

q(λ) := max{q(λ, m); m ≥ 0}.  (1.18)

Let

P^{(λ)} := span{U^{(λ)}(φ)f ; f polynomial , φ ∈ G}

For 0 ≤ j ≤ q(λ) set

S_{j}(λ) := {m ≥ 0; q(λ, m) ≤ j} M_{j}^{(λ)} := {f ∈ P^{(λ)}; f = ∑_{m∈S_{j}(λ)} f_{m}, f_{m} ∈ P_{m}}.

(1.19)

The following result is established in [FK1], see also [A1], [O].

Theorem 1.4 Let λ ∈ C and let 0 ≤ j ≤ q(λ).

(i) The spaces M_{j}^{(λ)}, 0 ≤ j ≤ q(λ), are U^{(λ)}-invariant,

M_{0}^{(λ)} ⊂ M_{1}^{(λ)} ⊂ M_{2}^{(λ)} ⊂ … ⊂ M_{q(λ)}^{(λ)} = P^{(λ)},  (1.20)

and every non-zero U^{(λ)}-invariant subspace of P^{(λ)} is one of the spaces M_{j}^{(λ)}, 0 ≤ j ≤ q(λ).

(ii) The quotients M_{j}^{(λ)}/M_{j−1}^{(λ)}, 1 ≤ j ≤ q(λ), are U^{(λ)}-irreducible.

(iii) The sesqui-linear forms ⟨·, ·⟩_{λ,j} on M_{j}^{(λ)}, 1 ≤ j ≤ q(λ), defined for f, g ∈ M_{j}^{(λ)}

by

⟨f, g⟩_{λ,j} := lim_{ξ→λ} (ξ − λ)^{j} ⟨f, g⟩_{ξ}

are U^{(λ)}-invariant and {f ∈ M_{j}^{(λ)}; ⟨f, g⟩_{λ,j} = 0, ∀g ∈ M_{j}^{(λ)}} = M_{j−1}^{(λ)}.

(iv) For f, g ∈ M_{j}^{(λ)} with Peter-Weyl expansions f = ∑_{m} f_{m} and g = ∑_{m} g_{m},

we have

⟨f, g⟩_{λ,j} = ∑_{m∈S_{j}(λ)\S_{j−1}(λ)} \frac{⟨f_{m}, g_{m}⟩_{F}}{(λ)_{m,j}}

where

(λ)_{m,j} := lim_{ξ→λ} \frac{(ξ)_{m}}{(ξ − λ)^{j}} = \frac{1}{j!} (\frac{d}{dξ})^{j} (ξ)_{m}_{|ξ=λ}.  (1.21)

(v) The forms ⟨·, ·⟩_{λ,j} are hermitian if and only if λ ∈ R.

(vi) The quotient M_{j}^{(λ)}/M_{j−1}^{(λ)} is unitarizable (namely, ⟨·, ·⟩_{λ,j} is either positive definite or negative definite on M_{j}^{(λ)}/M_{j−1}^{(λ)}) if and only if either: λ ∈ W(D) and j = 0, or: λ ∈ P(D), j = q(λ), and λ_{r} − λ ∈ N.

The sequence (1.20) is called the composition series of P^{(λ)}.

Definition 1.1 H_{λ,j} = H_{λ,j}(D) is the completion of M_{j}^{(λ)}/M_{j−1}^{(λ)} with respect to ⟨·, ·⟩_{λ,j}.

Observe that H_{λ,0} = H_{λ} for λ ∈ W(D). Also, q(λ) > 0 if and only if λ ∈ P(D).

The main objective of this work is to provide natural integral formulas for the U^{(λ)}-invariant hermitian forms ⟨·, ·⟩_{λ,j}, with special emphasis on the case where the forms are definite, namely the case where H_{λ,j} is a U^{(λ)}-invariant Hilbert space. These integral formulas provide a characterization of the membership in the spaces H_{λ,j} in terms of finiteness of some weighted L^{2}-norms of the functions or of some of their derivatives. We discuss now some examples which motivate our study.

The weighted Bergman spaces: It is known [FK1] that for λ ∈ R the integral c(λ)^{−1} := ∫_{D} h(z, z)^{λ−p}dm(z) is finite if and only if λ > p − 1, and in this case

c(λ) = \frac{Γ_{Ω}(λ)}{π^{d} Γ_{Ω}(λ − \frac{d}{r})} .  (1.22)

For λ > p − 1 we consider the probability measure

dμ_{λ}(z) := c(λ) h(z, z)^{λ−p} dm(z)  (1.23)

on D. The weighted Bergman space L_{a}^{2}(D, μ_{λ}) consists of all analytic functions in L^{2}(D, μ_{λ}). Using (1.8) one obtains the transformation rule of μ_{λ} under composition with φ ∈ G:

dμ_{λ}(φ(z)) = |Jφ(z)|^{\frac{2λ}{p}} dμ_{λ}(z).

(The same argument yields the invariance of the measure dμ_{0}(z) := h(z, z)^{−p}dm(z)). From this it follows that the operators U^{(λ)}(φ) are isometries of L^{2}(D, μ_{λ}) which leave L_{a}^{2}(D, μ_{λ}) invariant. It is easy to verify that point evaluations are continuous linear functionals on L_{a}^{2}(D, μ_{λ}) and that the reproducing kernel of L_{a}^{2}(D, μ_{λ}) is h(z, w)^{−λ}. (For w = 0 this is trivial, and the general case follows by invariance.) It follows that H_{λ} = L_{a}^{2}(D, μ_{λ}).

The Hardy space: The Shilov boundary S of a general Cartan domain D is the set of all maximal tripotents in Z. S is invariant and irreducible under both of G and K. Let σ be the unique K-invariant probability measure on S, defined via

∫_{S} f(ξ) dσ(ξ) := ∫_{K} f(k(e)) dk.

The Hardy space H^{2}(S) is the space of all analytic functions f on D for which

∥f∥_{H^{2}(S)}^{2} := lim_{ρ→1−} ∫_{S} |f(ρξ)|^{2} dσ(ξ)

is finite. The polynomials are dense in H^{2}(S) and every f ∈ H^{2}(S) has radial limits f̃(ξ) := lim_{ρ→1−} f(ρξ) at σ-almost every ξ ∈ S. Moreover, for f ∈ H^{2}(S), ∥f∥_{H^{2}(S)} = ∥f̃∥_{L^{2}(S,σ)}. This identifies H^{2}(S) as the closed subspace of L^{2}(S, σ) consisting of those functions f ∈ L^{2}(S, σ) which extend analytically to D by means of the Poisson integral. Again, the point evaluations f ↦ f(z), z ∈ D, are continuous linear functionals on H^{2}(S). The corresponding reproducing kernel is called the Szegö kernel and is given (as a function on S) by S_{z}(ξ) = S(ξ, z) := h(ξ, z)^{−d/r}. See [Hu], [FK1]. This non-trivial fact implies that H_{d/r} = H^{2}(S). The transformation rule of the measure σ under the automorphisms φ ∈ G is

dσ(φ(ξ)) = |Jφ(ξ)| dσ(ξ).

Hence, U^{(d/r)}(φ)f = (f ∘ φ) (Jφ)^{1/2}, φ ∈ G, are isometries of L^{2}(S, σ) which leave H^{2}(S) invariant.

The Dirichlet space: The classical Dirichlet space B_{2} consists of those analytic functions f on the open unit disk D ⊂ C for which the Dirichlet integral

∥f∥_{B_{2}}^{2} := ∫_{D} |f′(z)|^{2} dA(z)  (1.24)

is finite. Here dA(z) := \frac{1}{π} dx dy. Clearly, B_{2} is a Hilbert space modulo constant functions, and ∥f ∘ φ∥_{B_{2}} = ∥f∥_{B_{2}} for every f ∈ B_{2} and φ ∈ Aut(D). Thus, B_{2} is U^{(0)}-invariant. The composition series corresponding to λ = λ_{1} = 0 is C1 = M_{0}^{(0)} ⊂ M_{1}^{(0)} = P^{(0)}. Hence B_{2} = H_{0,1}(D). The inner product in B_{2} can be computed also via integration on the boundary T := ∂D (which coincides with the Shilov boundary in this simple case):

⟨f, g⟩_{B_{2}} = \frac{1}{2π} ∫_{T} ξf′(ξ) \overline{g(ξ)} |dξ|.  (1.25)

Motivated by this example we call the spaces H_{0,q(0)} for a general Cartan domain D the (generalized) Dirichlet space of D. The paper [A2] provides integral formulas generalizing (1.24) and (1.25) for the norms in H_{λ,q(λ)} for λ ∈ W_{d}(D), in the context of a Cartan domain of tube type (in [A1] these formulas are extended to all λ ∈ P(D)). Formula (1.24) says that f ∈ B_{2} = H_{0,1} if and only if f′ ∈ H_{2}. Namely, differentiation “shifts” the space corresponding to λ = 0 to the one corresponding to λ = 2. This shifting technique is developed in [Y3] in order to get integral formulas for the inner products in certain spaces H_{λ} with λ ≤ p − 1. The general idea is to obtain such integral formulas via “partial integration in the radial directions”, see [Ri], [Go], and [FK2], Chapter VII. (For the open unit ball of C^{d}, the simplest (i.e. rank-one) nontube Cartan domain, cf. [A3], [Pel]).

Finally, we describe the relationship between the invariant inner product and the Riesz distribution. The Riesz distribution was introduced in [Ri] for the Lorentz cone, i.e. the symmetric cone associated with the Cartan domain of type IV (the “Lie ball”). It was studied in [Go] for the cone of symmetric, positive definite real matrices (associated with the Cartan domain of type III) and for a general symmetric cone in [FK2], chapter VII. Let Ω be the symmetric cone associated with the Cartan domain of tube type D. For α ∈ C with ℜα > (r − 1)\frac{a}{2} let R_{α} be the linear functional on the Schwartz space S(X) of X defined via

R_{α}(f) := \frac{1}{Γ_{Ω}(α)} ∫_{Ω} f(x)N(x)^{α− \frac{d}{r}} dx.

Then R_{α} is a tempered distribution satisfying ∂_{N}R_{α} = R_{α−1}, R_{α}⋆R_{β} = R_{α+β}, R_{0} = δ, i.e. R_{1} is the fundamental solution for the “wave operator” ∂_{N} := N(\frac{∂}{∂x}). These formulas permit analytic continuation of α ↦ R_{α} to an entire meromorphic function. It is very interesting to find the explicit description of the action of R_{α} for general α, but this is still open. What is known is that the Riesz distribution R_{α} is represented by a positive measure if and only if α ∈ W (D).

Writing the inner products ⟨·, ·⟩_{λ} in conical polar coordinates (1.11), we get for λ > p − 1

⟨f, g⟩_{λ} = \frac{Γ_{Ω}(λ)}{Γ_{Ω}(\frac{d}{r}) Γ_{Ω}(λ − \frac{d}{r})} ∫_{Ω∩(e−Ω)}(f\overline{g})^{}˜ (x) N(e − x)^{λ−p} dx, ∀f, g ∈ H_{λ}(D),

where (fḡ)^{}˜ (x) := ∫_{K} f(k(x^{\frac{1}{2}} )) \overline{g(k(x^{\frac{1}{2}} ))} dk. Thus

⟨f, g⟩_{λ} = \frac{Γ_{Ω}(λ)}{Γ_{Ω}(\frac{d}{r})} (R_{λ− \frac{d}{r}} ⋆ (fḡ)^{}˜ ) (e),

where the convolution of functions u and v on Ω is

(u ⋆ v)(x) := ∫_{Ω∩(x−Ω)} u(y) v(x − y) dy.

Also, the inner product ⟨·, ·⟩_{λ}, λ > p − 1, in the context of the tube domain T (Ω) := X + iΩ (holomorphically equivalent to D) is

⟨f, g⟩_{λ} := c(λ) ∫_{Ω} (∫_{X} f(x + iy) \overline{g(x + iy)} dx) N(2y)^{λ−p} dy.

See section 6 for the details. Thus

⟨f, g⟩_{λ} = π^{−d} 2^{λ−p} Γ_{Ω}(λ) R_{λ− \frac{d}{r}} ((f ḡ)^{♭}) ,

where (f ḡ)^{♭})(y) := ∫_{X} f(x + iy) \overline{g(x + iy)} dx, y ∈ Ω.

In view of these formulas the problem of obtaining an explicit description of the analytic continuation of the maps λ ↦ ⟨f, g⟩_{λ} is equivalent to the problem of determining the analytic continuation of the maps λ ↦ R_{λ− \frac{d}{r}} (u).

2 G(Ω)-invariant differential operators

Let Ω be the symmetric cone associated with the Cartan domain of tube type D, i.e. the interior of the cone of squares in the Euclidean Jordan algebra X. In this section we study G(Ω)-invariant differential operators that will be used later for the invariant inner products. The ring Diff(Ω)^{G(Ω)} of G(Ω)-invariant differential operators is a (commutative) polynomial ring C[X_{1}, X_{2}, …, X_{r}], [He], [FK2]. By [FK2], Proposition IX.1.1, Ω is a set of uniqueness for analytic functions on Z (namely, if an analytic function on Z vanishes identically on Ω, it vanishes identically on Z). Similarly, Ω ∩ D = Ω ∩ (e − Ω) is a set of uniqueness for analytic functions on D. Thus, if f, g and q are polynomials on Z so that ∂_{f}(g)(x) = f(\frac{d}{dx})g(x) = q(x) for every x ∈ Ω, then ∂_{f}(g)(z) = f(\frac{∂}{∂z})g(z) = q(z) for every z ∈ Z. We begin with the following known result [FK2], Proposition VII.1.6.

Lemma 2.1 For every s = (s_{1}, s_{2}, …, s_{r}) ∈ C^{r} and ℓ ∈ N, we have

N^{ℓ}(\frac{d}{dx}) N_{s}(x) = μ_{s}(ℓ) N_{s−ℓ}(x), ∀x ∈ Ω,

where

μ_{s}(ℓ) := \frac{(\frac{d}{r})_{s}}{(\frac{d}{r})_{s−ℓ}} = \frac{Γ_{Ω}(s + \frac{d}{r})}{Γ_{Ω}(s + \frac{d}{r} − ℓ)} = ∏_{j=1}^{r} ∏_{ν=0}^{ℓ−1}(s_{j} − ν + (r − j)\frac{a}{2}),

and

Γ_{Ω}(s) N(\frac{d}{dx}) N_{s}(x^{−1}) = (−1)^{r} Γ_{Ω}(s + 1) N_{s+1}(x^{−1}).

Let N_{j}^{∗} be the norm polynomial of the JB^{∗}-subalgebra V_{j} := ∑_{r−j+1≤j≤k≤r} Z_{i,k}, where Z_{i,k} are the Peirce subspaces of Z associated with the fixed frame {e_{j}}_{j=1}^{r}. For every s = (s_{1}, …, s_{r}) ∈ C^{r} let

N_{s}^{∗}(x) := N_{1}^{∗}(x)^{s_{1}−s_{2}} N_{2}^{∗}(x)^{s_{2}−s_{3}} … N_{r}^{∗}(x)^{s_{r}}, x ∈ Ω,

and

s^{∗} := (s_{r}, s_{r−1}, s_{r−2}, …, s_{1}).

Then we have N_{s}(x^{−1}) = N_{−s^{∗}}^{∗}(x) for x ∈ Ω, [FK2],Proposition VII.1.5.

Definition 2.1 For ℓ ∈ N and λ ∈ C let D_{ℓ}(λ) be the operator on C^{∞}(Ω) defined by

D_{ℓ}(λ) = N^{\frac{d}{r} −λ}(x) N^{ℓ}(\frac{d}{dx}) N^{ℓ+λ− \frac{d}{r}} (x).  (2.1)

In the special case of the Cartan domain of type II the operators D_{1}(λ) have been considered by Selberg (see [T], p.208). The operators D_{ℓ}(λ) were studied in full generality in [Y3], see also [FK2], Chapter XIV. Notice that by Lemma 2.1 we have

D_{ℓ}(λ) N_{s} = \frac{Γ_{Ω}(s + λ + ℓ)}{Γ_{Ω}(s + λ)} N_{s}.  (2.2)

In section 4 below we will extend D_{ℓ}(λ) to a polynomial differential operator on Z, i.e. D_{ℓ}(λ) = Q_{ℓ,λ}(z, \frac{∂}{∂z}) for some polynomial Q_{ℓ,λ}.

Lemma 2.2 The operator D_{ℓ}(λ) is K-invariant, i.e.

D_{ℓ}(λ)(f ∘ k) = (D_{ℓ}(λ)f) ∘ k ∀f ∈ C^{∞}(Ω), ∀k ∈ K.

Proof: We have N(kz) = χ(k)N(z) for every z ∈ Z. Since the operator ∂_{N} = N(\frac{∂}{∂z}) is the adjoint of the operator of multiplication by N with respect to the inner product ⟨·, ·⟩_{F}, K-invariance of ⟨·, ·⟩_{F} implies ∂_{N}(f ∘ k) = χ(k)(∂_{N}f) ∘ k. It follows that

D_{ℓ}(λ)(f(kz)) = \overline{χ(k)}^{ℓ+λ− \frac{d}{r}} N(z)^{\frac{d}{r} −λ} N^{ℓ}(\frac{∂}{∂z}) (N^{ℓ+λ− \frac{d}{r}} (kz)f(kz))

= \overline{χ(k)}^{ℓ+λ− \frac{d}{r}} N(z)^{\frac{d}{r} −λ} χ(k)^{ℓ} (N^{ℓ}(\frac{∂}{∂z})(N^{ℓ+λ− \frac{d}{r}} f)) (kz)

= N^{\frac{d}{r} −λ}(kz) (N^{ℓ}(\frac{∂}{∂z})(N^{ℓ+λ− \frac{d}{r}} f)) (kz) = (D_{ℓ}(λ)f)(kz).

Using (2.2) and the fact that Ω ∩ D = Ω ∩ (e − Ω) is a set of uniqueness for analytic functions on D, we obtain the following result.

Corollary 2.1 The spaces P_{m} are eigenspaces of D_{ℓ}(λ) with eigenvalues

μ_{ℓ,m}(λ) := \frac{Γ_{Ω}(m + λ + ℓ)}{Γ_{Ω}(m + λ)}.  (2.3)

Thus for every analytic function f on D with Peter-Weyl expansion f = ∑_{m≥0} f_{m},

D_{ℓ}(λ)f = ∑_{m≥0} \frac{Γ_{Ω}(m + λ + ℓ)}{Γ_{Ω}(m + λ)} f_{m} = (λ)_{(ℓ,ℓ,...,ℓ)} ∑_{m≥0} \frac{(λ + ℓ)_{m}}{(λ)_{m}} f_{m}.  (2.4)

Indeed, for every signature m and every k ∈ K,

D_{ℓ}(λ)(N_{m} ∘ k) = (D_{ℓ}(λ)N_{m}) ∘ k = \frac{Γ_{Ω}(m + λ + ℓ)}{Γ_{Ω}(m + λ)} N_{m} ∘ k.

Since P_{m} = span{N_{m} ∘ k; k ∈ K}, (2.4) follows from the continuity of D_{ℓ}(λ) with respect to the topology of compact convergence on D.

Corollary 2.2 Let λ ∈ C \ P(D), ℓ ∈ N, and w ∈ D. Then

D_{ℓ}(λ) h(·, w)^{−λ} = (λ)_{(ℓ,ℓ,...,ℓ)} h(·, w)^{−(λ+ℓ)}.  (2.5)

Proof: Using (1.16) and Corollary 2.2, we get

D_{ℓ}(λ) h(·, w)^{−λ} = ∑_{m≥0}(λ)_{m} D_{ℓ}(λ) K_{m}(·, w)

= (λ)_{(ℓ,...,ℓ)} ∑_{m≥0}(λ)_{m} \frac{(λ + ℓ)_{m}}{(λ)_{m}} K_{m}(·, w)

= (λ)_{(ℓ,...,ℓ)} ∑_{m≥0}(λ + ℓ)_{m} K_{m}(·, w) = (λ)_{(ℓ,...,ℓ)} h(·, w)^{−(λ+ℓ)}.

Notice that the assumption that λ is not in P(D) is used in the above proof to ensure that (λ)_{m} ≠ 0 for every m ≥ 0. This is due to the fact that the zero set of the polynomial (λ)_{m} is

Z((·)_{m}) = ∪_{j=1}^{r} {λ_{j} − k; k = 0, 1, …, m_{j} − 1},  (2.6)

while P(D) = ∪_{j=1}^{r} (λ_{j} − N) = ∪_{m≥0} Z((·)_{m}). Similarly, for each m ≥ 0 the zero set of the polynomial defined by (2.3) is given by

Z(μ_{ℓ,m}(·)) = ∪_{j=1}^{r} {λ_{j} − k; m_{j} ≤ k ≤ m_{j} + ℓ − 1}.  (2.7)

The multiplicities of the zeros are equal to the number of their appearances on the right hand side of (2.7).

Corollary 2.3 Let λ ∈ C, ℓ ∈ N be so that {m ≥ 0; (λ)_{m} = 0} ⊆ {m ≥ 0; (λ + ℓ)_{m} = 0}. Then (2.5) holds.

Proof: Notice first that (λ)_{(ℓ,ℓ,...,ℓ)}(λ + ℓ)_{m} = (λ)_{m+ℓ} for all λ ∈ C, ℓ ∈ N, and m ≥ 0. Hence, using the fact that {m; (λ + ℓ)_{m} ≠ 0} ⊆ {m; (λ)_{m} ≠ 0}, we get for every w ∈ D

D_{ℓ}(λ)h(·, w)^{−λ} = D_{ℓ}(λ) ∑_{(λ)_{m}≠0}(λ)_{m}K_{m}(·, w)
= (λ)_{(ℓ,ℓ,...,ℓ)} ∑_{(λ)_{m}≠0}(λ + ℓ)_{m}K_{m}(·, w)

= (λ)_{(ℓ,...,ℓ)} ∑_{(λ+ℓ)_{m}≠0}(λ + ℓ)_{m}K_{m}(·, w)
= (λ)_{(ℓ,...,ℓ)}h(·, w)^{−(λ+ℓ)}.

For λ ∈ P(D) let q = q(λ) be as in (1.18), and for 0 ≤ j ≤ q consider S_{j}(λ) and M_{j}^{(λ)} as in (1.19).

Lemma 2.3 Let λ, and q = q(λ) be as above, and choose an integer ℓ so that λ + ℓ ≥ \frac{d}{r} = λ_{r} + 1. Then

(i) deg_{λ}((·)_{(ℓ,ℓ,...,ℓ)}) = q.

(ii) For every j = 0, 1, 2, …, q and every m ∈ S_{j}(λ) \ S_{j−1}(λ), deg_{λ}(μ_{ℓ,m}) = q − j.

(iii) If 0 ≤ j ≤ q and m ∈ S_{j−1}(λ), then deg_{λ}(μ_{ℓ,m}) ≥ q − j + 1.

Proof: Using (2.6) it is clear that

q(λ, m) = q ⇔ λ_{j} − m_{j} + 1 ≤ λ ∀j ⇔ λ_{r} − m_{r} + 1 ≤ λ.

Since λ_{r} +1 ≥ λ+ℓ, we see that m = (ℓ, ℓ, …, ℓ) satisfies the above condition, namely deg_{λ}((·)_{(ℓ,...,ℓ)}) = q(λ, (ℓ, …, ℓ)) = q. This establishes (i). Next, m ∈ S_{j}^{(λ)} \ S_{j−1}^{(λ)} is equivalent to q(λ, m) = j. By the argument given above, q(λ, m + ℓ) = q. Since deg_{λ}(f/g) = deg_{λ}(f) − deg_{λ}(g), we get

deg_{λ}(μ_{ℓ,m}) = deg_{λ} (\frac{(·)_{m+ℓ}}{(·)_{m}} ) =

= deg_{λ}((·)_{m+ℓ}) − deg_{λ}((·)_{m}) = q(λ, m + ℓ) − q(λ, m) = q − j.

This yields (ii). Finally, (iii) follows by similar computations.

Let λ ∈ P(D), ℓ ∈ N, and q = q(λ) as above. For every m ≥ 0 and ν ∈ N we define

μ_{ℓ,m}^{ν}(λ) := \frac{1}{ν!}(\frac{∂}{∂ξ})^{ν}μ_{ℓ,m}(ξ)_{|ξ=λ}.

Using Lemma 2.3 (ii), we have

Corollary 2.4 (i) If m ∈ S_{j}(λ) \ S_{j−1}(λ) then

μ_{ℓ,m}^{q−j}(λ) = ∏_{i=1}^{r} ∏^{′}_{k=m_{i}}^{m_{i}+ℓ−1} (λ + k − λ_{i}),

where the product ^{∏′}_{k=m_{j}}^{m_{j}+ℓ−1} ranges over all non-zero terms. In particular, μ_{ℓ,m}^{q−j}(λ) ≠ 0.

(ii) If m ∈ S_{j−1}(λ) then μ_{ℓ,m}^{q−j}(λ) = 0.

Definition 2.2 For λ ∈ C and ν, ℓ ∈ N let D_{ℓ}^{ν}(λ) be the operator on C^{∞}(D) defined by

D_{ℓ}^{ν}(λ)f := \frac{1}{ν!}(\frac{∂}{∂ξ})^{ν}(D_{ℓ}(ξ)f)_{|ξ=λ}.  (2.8)

Notice that if f = ∑_{m≥0} f_{m} is analytic in D, then D_{ℓ}^{ν}(λ)f := ∑_{m≥0} μ_{ℓ,m}^{ν}(λ) f_{m}.

By [FK2], Chapter VI the group G(Ω) admits an Iwasawa decomposition G(Ω) = NAL, where L is the group defined via (1.6), and NA is a maximal solvable subgroup of G(Ω) (called the triangular subgroup with respect to the frame {e_{i}}_{i=1}^{r}) which acts simply transitively on Ω and for which all the conical functions N_{s}, s ∈ C^{r}, are eigenfunctions:

N_{s}(τ(x)) = N_{s}(τ(e)) N_{s}(x), ∀τ ∈ NA, ∀x ∈ Ω.  (2.9)

Lemma 2.4 The operators D_{ℓ}(λ) are G(Ω)-invariant, i.e. D_{ℓ}(λ)(f ∘ φ) = (D_{ℓ}(λ)f) ∘ φ, ∀f ∈ C^{∞}(Ω), ∀φ ∈ G(Ω).

Proof: By the L-invariance of D_{ℓ}(λ) (see Lemma 2.2) it is enough to verify the NA-invariance of D_{ℓ}(λ) for functions f of the form f = N_{s} ∘ ℓ for some s ∈ C^{r} and ℓ ∈ L. Let τ ∈ NA, and decompose ℓ ∘ τ uniquely as ℓ ∘ τ = τ′ ∘ ℓ′ with τ′ ∈ NA and ℓ′ ∈ L. Then, using (2.2) and (2.9), we get

D_{ℓ}(λ)(f ∘ τ) = D_{ℓ}(λ)(N_{s} ∘ ℓ ∘ τ) = D_{ℓ}(λ)(N_{s} ∘ τ′ ∘ ℓ′)
= (D_{ℓ}(λ)(N_{s} ∘ τ′)) ∘ ℓ′ = N_{s}(τ′(e))(D_{ℓ}(λ)N_{s}) ∘ ℓ′

= N_{s}(τ′(e)) \frac{Γ_{Ω}(s + λ + ℓ)}{Γ_{Ω}(s + λ)} N_{s} ∘ ℓ′ = \frac{Γ_{Ω}(s + λ + ℓ)}{Γ_{Ω}(s + λ)} N_{s} ∘ τ′ ∘ ℓ′

= \frac{Γ_{Ω}(s + λ + ℓ)}{Γ_{Ω}(s + λ)} N_{s} ∘ ℓ ∘ τ = \frac{Γ_{Ω}(s + λ + ℓ)}{Γ_{Ω}(s + λ)} f ∘ τ
= (D_{ℓ}(λ)f) ∘ τ.

Corollary 2.5 The operators D_{ℓ}^{ν}(λ) are G(Ω)-invariant.

3 Integral formulas via the shifting method

In this section we develop general shifting techniques (introduced in [Y3], for the case of integer shifts). The simplest case where this technique is applied is the case of the Dirichlet space D = H_{0,1} over the unit disk D (see Section 2). For any α ∈ C and β ∈ C \ P(D) we define an operator S_{α,β} on H(D) via

S_{α,β}( ∑_{m≥0} f_{m}) := ∑_{m≥0} \frac{(α)_{m}}{(β)_{m}} f_{m}.

Theorem 5 of [A4] and the known estimate

\frac{(x)_{m}}{(y)_{m}} ≈ ∏_{j=1}^{r}(m_{j} + 1)^{x−y}, ∀x, y ∈ R

(an easy consequence of (1.9) and Stirling’s formula) ensures that S_{α,β} is continuous on H(D). For β ∈ P(D) we define operators S_{α,β,j}, 0 ≤ j ≤ q(β), on the space of analytic functions on D of the form f = ∑_{m∈S_{j}(β)} f_{m} by

S_{α,β,j}f := lim_{ξ→β}(ξ − β)^{j}S_{α,β}f = ∑_{m∈S_{j}(β)\S_{j−1}(β)} \frac{(α)_{m}}{(β)_{m,j}} f_{m},

where (β)_{m,j} are defined by (1.21). Again, S_{α,β,j} is continuous in the topology of H(D). Also, S_{α,β,0} = S_{α,β}.

Proposition 3.1 Let α, β > (r − 1)\frac{a}{2}. Then ⟨f, g⟩_{β} = ⟨S_{α,β}f, g⟩_{α} for every f, g ∈ H_{β}.

Proof: By (1.17) the operator Sα,β^{\frac{1}{2}} : H_{β} → H_{α} defined by

Sα,β^{\frac{1}{2}}( ∑_{m≥0} f_{m}) := ∑_{m≥0} (\frac{(α)_{m}}{(β)_{m}} )^{\frac{1}{2}} f_{m}

is a surjective isometry, and ∥f∥_{β}^{2} = ∥Sα,β^{\frac{1}{2}}f∥_{α}^{2} = ⟨S_{α,β}f, f⟩_{α}. Now the result follows by polarization.

In a similar way one proves the following result.

Proposition 3.2 Let α > (r − 1)\frac{a}{2} and let β ∈ P(D). Then for every 0 ≤ j ≤ q(β) and all f, g ∈ H_{β,j},

⟨f, g⟩_{β,j} = ⟨S_{α,β,j}f, g⟩_{α}.  (3.1)

The operators S_{α,β,j} allow the computation of the invariant hermitian forms ⟨f, g⟩_{β,j} by “shifting” the point β to the point α. This is the “shifting method”. One typically chooses either α = \frac{d}{r} or α > p − 1, so the forms ⟨f, g⟩_{β,j} can be computed by the integral-type inner products of H^{2}(D) or L_{a}^{2}(D, μ_{α}). In order for the shifting method to be useful, one has to identify the operators S_{α,β,j} as differential or pseudodifferential operators. Essentially, this is our aim in the rest of the paper. Yan’s paper [Y3] deals with the case where ℓ := α − β is a sufficiently large natural number. The following result is a minor generalization of a result of [Y3].

Theorem 3.1 Let λ > λ_{r} = \frac{d}{r} − 1 and let ℓ ∈ N. Then for all f, g ∈ H_{λ}

⟨f, g⟩_{λ} = α(λ, ℓ)⟨D_{ℓ}(λ)f, g⟩_{λ+ℓ},  (3.2)

where

α(λ, ℓ) = \frac{Γ_{Ω}(λ)}{Γ_{Ω}(λ + ℓ)} = \frac{1}{(λ)_{(ℓ,ℓ,...,ℓ)}} .

We include a short proof for the sake of completeness.

Proof: Let f, g ∈ H_{λ} with expansions f = ∑_{m≥0} f_{m} and g = ∑_{m≥0} g_{m} respectively. Then

⟨D_{ℓ}(λ)f, g⟩_{λ+ℓ} = ∑_{m≥0} \frac{μ_{ℓ,m}(λ)}{(λ + ℓ)_{m}} ⟨f_{m}, g_{m}⟩_{F}

= \frac{Γ_{Ω}(λ + ℓ)}{Γ_{Ω}(λ)} ∑_{m≥0} \frac{⟨f_{m}, g_{m}⟩_{F}}{(λ)_{m}} = α(λ, ℓ)^{−1} ⟨f, g⟩_{λ}.

Corollary 3.1 Let λ > λ_{r} = \frac{d}{r} − 1, and ℓ ∈ N be so that λ + ℓ > p − 1. Then H_{λ+ℓ} = L_{a}^{2}(D, μ_{λ+ℓ}), and for every f, g ∈ L_{a}^{2}(D, μ_{λ+ℓ}),

⟨f, g⟩_{λ} = α(λ, ℓ) c(λ + ℓ) ∫_{D}(D_{ℓ}(λ)f)(z) \overline{g(z)} h(z, z)^{λ+ℓ−p} dm(z).

Our main result in this section is a generalization of both Theorem 3.1 and the other results of [Y3] to the case of invariant hermitian forms associated with the pole set P(D) = ∪_{j=1}^{r}(λ_{j} − N). Since W(D) ⊂ P(D), this covers cases not studied in [A1].

Theorem 3.2 Let λ ∈ P(D), ℓ ∈ N and assume that λ + ℓ ≥ \frac{d}{r} = λ_{r} + 1. Let q = q(λ), 0 ≤ j ≤ q, and ν = q − j. Then for all f, g ∈ H_{λ,j},

⟨f, g⟩_{λ,j} = γ⟨D_{ℓ}^{ν}(λ)f, g⟩_{λ+ℓ},  (3.3)

where γ = γ(λ, ℓ) is the non-zero constant

γ := \frac{1}{q!}(\frac{∂}{∂ξ})^{q} ((ξ)_{(ℓ,ℓ,...,ℓ)})_{|ξ=λ} .  (3.4)

In particular, if λ + ℓ > p − 1, then

⟨f, g⟩_{λ,j} = γ c(λ + ℓ) ∫_{D}(D_{ℓ}^{ν}(λ)f)(z) \overline{g(z)} dm(z).  (3.5)

Moreover, if λ_{r} − λ ∈ N and ℓ is chosen so that λ + ℓ = \frac{d}{r} = λ_{r} + 1, then

⟨f, g⟩_{λ,j} = γ ∫_{S}(D_{ℓ}^{ν}(λ)f)(ξ) \overline{g(ξ)} dσ(ξ).  (3.6)

We shall also give a new proof of the following known result (see [FK1], Theorem 5.3) and of a part of Theorem 1.4 above, based on our analysis of the structure of zeros of the polynomials (·)_{m}. Recall that H_{λ,j} is said to be unitarizable if ⟨·, ·⟩_{λ,j} is either positive definite or negative definite.

Theorem 3.3 Let λ, ℓ, q, and j be as in Theorem 3.2. Then H_{λ,j} is unitarizable if and only if either

(a) j = q and λ_{r} − λ ∈ N, or

(b) j = 0 and λ ∈ W_{d}(D) = {λ_{j}}_{j=1}^{r}.

For the proof of Theorems 3.2 and 3.3 we consider separately the cases j = 0, j = q, and 1 ≤ j ≤ q − 1.

Case 1: j = 0. Since λ ∈ P(D), there is a smallest k ∈ {1, 2, …, r} and a unique s ∈ N so that λ = λ_{k} − s. We claim that S_{0}(λ) = {m ≥ 0; m_{k} ≤ s}. Indeed, if m ≥ 0, then ∏_{i=1}^{k−1} ∏_{ν=0}^{m_{i}−1}(λ + ν − λ_{i}) ≠ 0, by the minimality of k. The term ∏_{ν=0}^{m_{k}−1}(λ + ν − λ_{k}) = ∏_{ν=0}^{m_{k}−1}(ν − s) is non-zero if and only if m_{k} ≤ s. If m_{k} ≤ s and k < n ≤ r then

∏_{ν=0}^{m_{k}−1}(λ + ν − λ_{k}) = ∏_{ν=0}^{m_{k}−1}((λ_{k} − λ_{n}) + (ν − s)) ≠ 0

because m_{n} ≤ m_{k} ≤ s. This establishes the claim. Notice that since λ + ℓ ≥ λ_{r} + 1, we have (λ + ℓ)_{m} > 0 for any m ≥ 0. Also, deg_{λ}((·)_{(ℓ,ℓ,...,ℓ)}) = q by Lemma 2.3. It follows that for m ∈ S_{0}(λ), deg_{λ}(μ_{ℓ,m}) = q, and

μ_{ℓ,m}^{q}(λ) = \frac{1}{q!}(\frac{∂}{∂ξ})^{q}μ_{ℓ,m}(ξ)_{|ξ=λ} = \frac{1}{q!}(\frac{∂}{∂ξ})^{q}(\frac{(ξ + ℓ)_{m}}{(ξ)_{m}} (ξ)_{(ℓ,ℓ,...,ℓ)})_{|ξ=λ}

= \frac{(λ + ℓ)_{m}}{(λ)_{m}} \frac{1}{q!}(\frac{∂}{∂ξ})^{q}(ξ)_{(ℓ,ℓ,...,ℓ)} _{|ξ=λ} = γ \frac{(λ + ℓ)_{m}}{(λ)_{m}} .

Hence, for f, g ∈ H_{λ,0},

⟨D_{ℓ}^{q}(λ)f, g⟩_{λ+ℓ} = ∑_{m∈S_{0}(λ)} μ_{ℓ,m}^{q}(λ) \frac{⟨f_{m}, g_{m}⟩_{F}}{(λ + ℓ)_{m}}

= γ ∑_{m∈S_{0}(λ)} \frac{⟨f_{m}, g_{m}⟩_{F}}{(λ)_{m}} = γ⟨f, g⟩_{λ,0}.

This proves Theorem 3.2 in case j = 0. If λ ∈ W_{d}(D), i.e. λ = λ_{k} and s = 0, then (λ)_{m} > 0 for every m ∈ S_{0}(λ), namely 0 = m_{k} = m_{k+1} = ⋯ = m_{r}. If λ ∈ P(D) \ W_{d}(D), then λ = λ_{k} − s with 1 ≤ s. In this case (λ)_{m} assumes both positive and negative values as m ranges over S_{0}(λ). Indeed, if m and n are defined by m_{i} = n_{i} = 0 for 1 ≤ i ≤ k − 1 and k < i ≤ r, and m_{k} = 0, n_{k} = s − 1, then (λ)_{m} and (λ)_{n} have different signs. Thus ⟨·, ·⟩_{λ,0} is not definite (positive or negative), and thus H_{λ,0} is not unitarizable. This proves Theorem 3.3 in case j = 0.

Case 2: j = q. In this case ν = q − j = 0. Also, Lemma 2.3 yields deg_{λ}(μ_{ℓ,m}) = 0 if m ∈ S_{q}(λ) and deg_{λ}(μ_{ℓ,m}) ≥ 1 if m ∈ S_{q−1}(λ). It follows that for f, g ∈ H_{λ,q},

⟨D_{ℓ}(λ)f, g⟩_{λ+ℓ} = ∑_{m∈S_{q}(λ)} μ_{ℓ,m}(λ)\frac{⟨f_{m}, g_{m}⟩_{F}}{(λ + ℓ)_{m}} .

Now,

μ_{ℓ,m}(λ) = lim_{ξ→λ} \frac{(ξ + ℓ)_{m}}{(ξ)_{m}} (ξ)_{(ℓ,ℓ,...,ℓ)} = (λ + ℓ)_{m} lim_{ξ→λ} \frac{(ξ)_{(ℓ,ℓ,...,ℓ)}}{(ξ)_{m}} = γ \frac{(λ + ℓ)_{m}}{(λ)_{m,q}} ,

where γ is the non-zero constant defined in (3.4). It follows that

⟨D_{ℓ}(λ)f, g⟩_{λ+ℓ} = γ ∑_{m∈S_{q}(λ)} \frac{⟨f_{m}, g_{m}⟩_{F}}{(λ)_{m,q}} = γ⟨f, g⟩_{λ,q}.

This proves Theorem 3.2 in case j = q. To prove Theorem 3.3 in this case, assume first that λ = λ_{r} − s for some s ∈ N. We claim now that

S_{q}(λ) \ S_{q−1}(λ) = {m ≥ 0; m_{r} ≥ s + 1}.  (3.7)

Indeed, if m_{r} ≥ s + 1 then ∏_{u=0}^{m_{r}−1}(λ + u − λ_{r}) = 0. If λ ∈ λ_{i} − N, then ∏_{u=0}^{m_{i}−1}(λ + u − λ_{r}) = 0 because m_{i} ≥ m_{r} ≥ s + 1. Thus deg_{λ}((·)_{m}) = q. Conversely, if deg_{λ}((·)_{m}) = q, then in order that ∏_{u=0}^{m_{r}−1}(λ + u − λ_{r}) = 0 it is necessary that s ≤ m_{r} − 1. This establishes (3.7).

Next, let m ∈ S_{q}(λ), and let 1 ≤ i ≤ r be so that λ ∈ λ_{i} − N, say λ = λ_{i} − k_{i}. Then

lim_{ξ→λ}(ξ − λ)^{−1} ∏_{u=0}^{m_{i}−1}(ξ + u − λ_{i}) = ∏_{u=0}^{k_{i}−1}(λ + u − λ_{i}) ∏_{u=k_{i}+1}^{m_{i}−1}(λ + u − λ_{i}) = γ_{i,m} β_{i}

with β_{i} ≠ 0 and γ_{i,m} > 0. If λ ∉ λ_{i} − N we let β_{i} = ∏_{u<λ_{i}−λ}(λ + u − λ_{i}) ≠ 0 and γ_{i,m} = ∏_{u>λ_{i}−λ}(λ + u − λ_{i}) > 0. Then

(λ)_{m,q} = lim_{ξ→λ} \frac{(ξ)_{m}}{(ξ − λ)^{q}} = ∏_{i=1}^{r} γ_{i,m} β_{i}.

Hence, all the numbers {(λ)_{m,q}}_{m∈S_{q}(λ)} have constant sign (equal to sgn(∏_{i=1}^{r} β_{i})), and thus H_{λ,q} is unitarizable. Assume now that λ ∉ λ_{r} − N. Then, necessarily, the characteristic multiplicity a is odd and λ ∈ λ_{r−1} − N. Writing λ = λ_{r−1} − s, s ∈ N, it is clear by the above arguments that

S_{q}(λ) \ S_{q−1}(λ) = {m ≥ 0; m_{r−1} ≥ s + 1}.

Let m = (s+1, s+1, …, s+1, 1) and n = (s+1, s+1, …, s+1, 0). Then m, n ∈ S_{q}(λ) and (λ)_{m,q} = (λ − λ_{r})(λ)_{n,q}. Thus (λ)_{m,q} and (λ)_{n,q} have different signs, and so H_{λ,q} is not unitarizable. This proves Theorem 3.3 in case j = q.

Case 3: 1 ≤ j ≤ q − 1. Put ν = q − j. As before, ℓ ∈ N is chosen so that λ + ℓ ≥ λ_{r} + 1, and this guarantees that deg_{λ}((·)_{m+ℓ}) = q and (λ + ℓ)_{m} > 0 for all signatures m ≥ 0. Let f, g ∈ H_{λ,j}. Then

⟨D_{ℓ}^{ν}(λ)f, g⟩_{λ+ℓ} = ∑_{m∈S_{j}(λ)} μ_{ℓ,m}^{ν}(λ) \frac{⟨f_{m}, g_{m}⟩_{F}}{(λ + ℓ)_{m}} .

If m ∈ S_{j}(λ) \ S_{j−1}(λ), then

deg_{λ}(μ_{ℓ,m}) = deg_{λ} (\frac{(·)_{m+ℓ}}{(·)_{m}} ) = q − j = ν.

Thus,

μ_{ℓ,m}^{ν}(λ) = lim_{ξ→λ} \frac{μ_{ℓ,m}(ξ)}{(ξ − λ)^{ν}} = lim_{ξ→λ} \frac{(ξ + ℓ)_{m}(ξ − λ)^{−q}(ξ)_{(ℓ,ℓ,...,ℓ)}}{(ξ − λ)^{−j}(ξ)_{m}} = γ \frac{(λ + ℓ)_{m}}{(λ)_{m,j}} .

If m ∈ S_{j−1}(λ), then deg_{λ}(μ_{ℓ,m}) ≥ q − j + 1 = ν + 1, and so μ_{ℓ,m}^{ν}(λ) = 0. Thus

⟨D_{ℓ}^{ν}(λ)f, g⟩_{λ+ℓ} = γ ∑_{m∈S_{j}(λ)\S_{j−1}(λ)} \frac{(λ + ℓ)_{m}}{(λ)_{m,j}} \frac{⟨f_{m}, g_{m}⟩_{F}}{(λ + ℓ)_{m}}

= γ ∑_{m∈S_{j}(λ)\S_{j−1}(λ)} \frac{⟨f_{m}, g_{m}⟩_{F}}{(λ)_{m,j}} = γ⟨f, g⟩_{λ,j}.

Documenta Mathematica 2 (1997) 213–261

This proves Theorem 3.2 in case 1 ≤ j ≤ q − 1. To prove Theorem 3.3 in this case we need to show that as m varies in S_{j}(λ) \ S_{j−1}(λ), (λ)_{m,j} assumes both positive and negative values. Notice first that there exists a unique pair (k, s) of integers with 1 ≤ k < s ≤ r so that λ_{k} − λ and λ_{s} − λ are positive integers and

m ∈ S_{j}(λ) \ S_{j−1}(λ) ⟺ m_{k} ≥ λ_{k} − λ + 1 and m_{s} ≤ λ_{s} − λ.

In fact, s = k + 1 if the characteristic multiplicity a is even, and s = k + 2 if a is odd. Next, λ_{s} − λ = λ_{k} − λ + (s − k)\frac{a}{2} ≥ 1. Define m, n by m_{i} = n_{i} = λ_{k} − λ + 1 if 1 ≤ i ≤ k, m_{i} = n_{i} = 0 if k + 2 ≤ i ≤ r, and m_{k+1} = 0, n_{k+1} = 1. Then m, n ∈ S_{j}(λ) \ S_{j−1}(λ) and (λ)_{n,j} = (λ)_{m,j}(λ − λ_{s}). Thus (λ)_{n,j} and (λ)_{m,j} have different signs, and so H_{λ,j} is not unitarizable. This proves Theorem 3.3 in case 1 ≤ j ≤ q − 1.

A special case of Theorem 3.2 is the following essentially known result.

Corollary 3.2 Let λ ∈ P(D) be so that s = s(λ) := \frac{d}{r} − λ ∈ N. Then

(i) H_{λ,q} is unitarizable, and

⟨f, g⟩_{λ,q} = γ ∫_{S} N^{s}(ξ)(∂_{N}^{s}f)(ξ) \overline{g(ξ)} dσ(ξ), ∀f, g ∈ H_{λ,q}.

Thus, an analytic function f on D belongs to H_{λ,q} if and only if (N^{s}∂_{N}^{s})^{1/2}f ∈ H^{2}(S).

(ii) Moreover, if ℓ ∈ N is chosen so that λ + ℓ > p − 1, then

⟨f, g⟩_{λ,q} = γ′ ∫_{D}(D_{ℓ}(λ)f)(z) \overline{g(z)} h(z, z)^{λ+ℓ−p} dm(z), ∀f, g ∈ H_{λ,q}.

Consequently, an analytic function f on D belongs to H_{λ,q} if and only if (D_{ℓ}(λ))^{1/2}f ∈ L_{a}^{2}(D, μ_{λ+ℓ}).

In the last statement (D_{ℓ}(λ))^{1/2} is the positive square root of the positive operator D_{ℓ}(λ), see Corollary 2.1 Indeed, part (i) follows from Theorem 3.2 with j = q, ν = q − j = 0, ℓ = s and D_{s}(λ) = N^{s}∂_{N}^{s}. In this case H_{λ+s} = H_{\frac{d}{r}} is the Hardy space H^{2}(S) on the Shilov boundary S. Corollary 3.2 (i) for λ ∈ W_{d}(D) was proved in [A2]. The proof of part (ii) is similar.

The case where λ ∈ P(D) and s := \frac{d}{r} − λ ∈ N (i.e. the highest quotient of the composition series of U^{(λ)}-invariant spaces is unitarizable) is of particular interest.

Theorem 3.4 Let λ ∈ P(D) and assume that s := \frac{d}{r} − λ ∈ N. Then, for each φ ∈ Aut(D) and f ∈ H(D)

∂_{N}^{s}(U^{(λ)}(φ)f) = U^{(p−λ)}(φ)(∂_{N}^{s}f).  (3.8)

Namely, the operator ∂_{N}^{s} intertwines the actions U^{(λ)} and U^{(p−λ)} of Aut(D). Moreover,

⟨f, g⟩_{λ,q} = c_{1} ⟨∂_{N}^{s}f, ∂_{N}^{s}g⟩_{p−λ}, ∀f, g ∈ H_{λ,q},  (3.9)

where

c_{1}^{−1} := (\frac{d}{r})_{(s,s,...,s)} ∏_{j=1}^{r} ∏^{′}_{u=0}^{s−1}(λ + u − λ_{j}),  (3.10)

and the product ^{∏′}_{u=0}^{s−1} ranges over all non-zero terms. In particular, if λ < 1, then

⟨f, g⟩_{λ,q} = c_{1} c(p − λ) ∫_{D}(∂_{N}^{s}f)(z) \overline{(∂_{N}^{s}g)(z)} h(z, z)^{−λ} dm(z), ∀f, g ∈ H_{λ,q}. (3.11)

Proof: (3.8) is proved in [A1], Theorem 6.4. For the proof of (3.9) and (3.11) we define an inner product on the polynomials modulo M_{q−1}^{(λ)} by

[f, g] := ⟨∂_{N}^{s}f, ∂_{N}^{s}g⟩_{p−λ}, f, g ∈ H_{λ,q}.

We claim that [·, ·] is U^{(λ)}-invariant. Indeed, using (3.8) we see that for every φ ∈ Aut(D) and polynomials f and g,

[U^{(λ)}(φ)f, U^{(λ)}(φ)g] = ⟨∂_{N}^{s}(U^{(λ)}(φ)f), ∂_{N}^{s}(U^{(λ)}(φ)g)⟩_{p−λ}
= ⟨U^{(p−λ)}(φ)(∂_{N}^{s}f), U^{(p−λ)}(φ)(∂_{N}^{s}g)⟩_{p−λ}
= ⟨∂_{N}^{s}f, ∂_{N}^{s}g⟩_{p−λ} = [f, g].

Since polynomials are dense in H_{λ,q}, the fact that its inner product is the unique U^{(λ)}-invariant inner product (see [AF], [A1]) implies that

⟨f, g⟩_{λ,q} = c_{1} [f, g], ∀f, g ∈ H_{λ,q}.

The value (3.10) of c_{1} is found by taking f = g = N^{s}, and using the facts that ⟨N^{s}, N^{s}⟩_{F} = (\frac{d}{r})_{(s,s,...,s)}, [N^{s}, N^{s}] = (∂_{N}^{s}N^{s})^{2} = ⟨N^{s}, N^{s}⟩_{F}^{2}, and

⟨N^{s}, N^{s}⟩_{λ,q} = lim_{ξ→λ}(ξ − λ)^{q} \frac{⟨N^{s}, N^{s}⟩_{F}}{(ξ)_{(s,s...,s)}} = \frac{⟨N^{s}, N^{s}⟩_{F}}{∏_{j=1}^{r} ^{∏′}_{u=0}^{s−1}(λ + u − λ_{j})}.

Example: In the special case where λ = 0 and s := \frac{d}{r} ∈ N, H_{0,q} is the generalized Dirichlet space, and formula (3.11) is the generalized Dirichlet inner product

⟨f, g⟩_{0,q} = c_{1} c(p − λ) ∫_{D}(∂_{N}^{s}f)(z) \overline{(∂_{N}^{s}g)(z)} dm(z), ∀f, g ∈ H_{0,q}.

4 The expansion of the operators D_{ℓ}(λ)

Yan’s operators D_{ℓ}(λ) = N^{\frac{d}{r} −λ}∂_{N}^{ℓ}N^{λ+ℓ− \frac{d}{r}} and their derivatives play an important role in the previous section. In this section we obtain an expansion of D_{ℓ}(λ) in powers of λ. This expansion will exhibit D_{ℓ}(λ) as a polynomial in z, \frac{∂}{∂z}, and λ, showing that D_{ℓ}(λ) is a differential operator (with parameters λ and ℓ) in the ordinary sense. It also facilitates the computation of the derivatives

D_{ℓ}^{ν}(λ) = \frac{1}{ν!}(\frac{∂}{∂ξ})^{ν}D_{ℓ}(ξ)_{|ξ=λ},

needed in formulas (3.3), (3.5) and (3.6) for the forms ⟨f, g⟩_{λ,j}. Another consequence will be that for any r distinct complex numbers α_{1}, …, α_{r} the operators D_{1}(α_{1}), …, D_{1}(α_{r}) are algebraically independent generators of the ring of invariant differential operators on the cone Ω, a result obtained independently also by Korányi and Yan (see [FK2], Chapter XIV). We shall work in the framework of the cone Ω, but all the results will be valid for Z, because Ω is a set of uniqueness for analytic functions on Z.

Example 4.1. Let D ⊂ C^{d}, d ≥ 3 be a Cartan domain of rank r = 2 (called the Lie ball). The associated JB^{∗}-algebra Z = C^{d}, called the complex spin factor, is defined via

zw := (z_{1}w_{1} − z′ · w′, z_{1}w′ + w_{1}z′), z^{∗} := (\overline{z_{1}}, −\overline{z′}),

where z = (z_{1}, z′), z′ = (z_{2}, z_{3}, …, z_{d}), and z · w := ∑_{j=1}^{d} z_{j}w_{j}. The unit of Z is e := (1, 0, 0, …, 0), and the canonical frame is {e_{1}, e_{2}}, where e_{1} := \frac{1}{2}(1, i, 0, 0, …, 0), e_{2} := \frac{1}{2}(1, −i, 0, 0, …, 0). The norm polynomial and the associated differential operator are given by

N(z) := z · z = ∑_{j=1}^{d} z_{j}^{2} and ∂_{N} = N(\frac{∂}{∂z}) = \frac{1}{4} ∑_{j=1}^{d} \frac{∂^{2}}{∂z_{j}^{2}}

respectively, since (z|w) = 2z · \overline{w} is the normalized inner product. Since r = 2 and a = d − 2, the Wallach set is

W(D) = W_{d}(D) ∪ W_{c}(D), W_{d}(D) = \{0, \frac{d − 2}{2}\}, W_{c}(D) = (\frac{d − 2}{2}, ∞).

One can show that D is given by

D = \{z ∈ Z; ((∑_{j=1}^{d} |z_{j}|^{2})^{2} − |N(z)|^{2})^{\frac{1}{2}} < 1 − ∑_{j=1}^{n} |z_{j}|^{2}\}.  (4.1)

For every α ∈ C

\frac{∂^{2}}{∂z_{k}^{2}} N^{α} = \frac{∂}{∂z_{k}} (2αN^{α−1}z_{k} + N^{α} \frac{∂}{∂z_{k}})

= 2αN^{α−1} + 4αN^{α−1}z_{k} \frac{∂}{∂z_{k}} + 4α(α − 1)N^{α−2}z_{k}^{2} + N^{α} \frac{∂^{2}}{∂z_{k}^{2}} .

Since R = ∑_{j=1}^{d} z_{j} \frac{∂}{∂z_{j}}, we obtain

∂_{N} N^{α} = \frac{1}{4}(∑_{j=1}^{d} \frac{∂^{2}}{∂z_{j}^{2}} ) N^{α} = α(α − \frac{a}{2}) N^{α−1} + αN^{α−1}R + N^{α}∂_{N}.

It follows that for every α ∈ C and ℓ ∈ N,

N^{1−α}∂_{N}N^{α} = N∂_{N} + αR + α(α + \frac{d − 2}{2})I = N∂_{N} + (α)_{(1,0)}R + (α)_{(1,1)}I. (4.2)

Since

D_{ℓ}(λ) = (N^{\frac{d}{r} −λ}∂_{N}N^{1+λ− \frac{d}{r}} ) (N^{\frac{d}{r} −λ−1}∂_{N}N^{2+λ− \frac{d}{r}} ) ⋯ (N^{\frac{d}{r} +1−ℓ−λ}∂_{N}N^{ℓ+λ− \frac{d}{r}} ) ,

we finally obtain

D_{ℓ}(λ) = ∏_{j=1}^{ℓ}(N∂_{N} + (λ − \frac{d}{2} + j)R + (λ − 1 + j)(λ − \frac{d}{2} + j)I).  (4.3)

Note that the factors on the right hand sides of (4.2) and (4.3) commute, since they are G(Ω)-invariant, and the entire ring of G(Ω)-invariant operators is commutative. Also, the operators R and N∂_{N} are K-invariant. Hence the factors on the right hand sides of (4.2) and (4.3) are multipliers of the Peter-Weyl decomposition of analytic functions on D (see Corollary 2.1).

Consider a general Cartan domain of tube-type D ⊂ C^{d} with rank r. Let Ω be the associated symmetric cone in the Euclidean Jordan algebra X and fix a frame {e_{1}, …, e_{r}} of pairwise orthogonal primitive idempotents in X, whose sum is the unit element e. For 1 ≤ ν ≤ r, let ϕ_{ν} := ϕ_{1_{ν}} be the spherical polynomial associated with the signature 1_{ν} := (1, 1, …, 1, 0, 0, …, 0), where there are ν “1”’s and r − ν “0”’s. Put also ϕ_{0}(z) ≡ 1. Let {Δ_{ν}}_{ν=0}^{r} be the differential operators on Ω defined via

(Δ_{ν})f(a) := ϕ_{ν}(\frac{d}{dx})(f(P (a^{\frac{1}{2}} )x))_{|x=e},  (4.4)

where for b ∈ X, P (b) is defined via (1.1). Recall that P (b) ∈ G(Ω) for every b ∈ Ω, and that Ω = \{P (b)e; b ∈ Ω\} since P (a^{\frac{1}{2}} )e = a. Moreover, the L-invariance of the ϕ_{ν}’s and the “polar decomposition” for Ω imply that

(Δ_{ν})f(a) := ϕ_{ν}(\frac{d}{dx})(f(ψ(x)))_{|x=e}, a ∈ Ω  (4.5)

for every ψ ∈ G(Ω) for which ψ(e) = a. This implies that the operators {Δ_{ν}}_{ν=0}^{r} are G(Ω)-invariant, namely

Δ_{ν}(f ∘ ψ) = (Δ_{ν}f) ∘ ψ, ∀ψ ∈ G(Ω), ∀f ∈ C^{∞}(Ω).

We remark that (4.4) and (4.5) are equivalent to

Δ_{ν}e^{⟨x,y⟩}_{|x=a} = ϕ_{ν}(ψ^{∗}(y)) e^{⟨a,y⟩} = ϕ_{ν}(P (a^{\frac{1}{2}} )y) e^{⟨a,y⟩}, a, y ∈ Ω,  (4.6)

where ψ ∈ G(Ω) ⊂ GL(X) satisfies ψ(e) = a, ψ^{∗} is the adjoint of ψ with respect to the inner product ⟨·, ·⟩ on X, and Δ_{ν} differentiates the coordinate x. Notice also that the operators Δ_{ν} can be written as

Δ_{ν} = c_{m}K_{m}(x, \frac{∂}{∂x}),

where m = (1, 1, …, 1, 0, …, 0) (ν “ones” and r − ν zeros), and c_{m} is an appropriate constant.

For ν = 0, 1, r it is easy to compute Δ_{ν}. Clearly, Δ_{0} = I. Since N is L-invariant, ϕ_{r} = N. Using (4.6) and (1.3), we find that

Δ_{r} = N ∂_{N}.

Also, ϕ_{1}(x) = \frac{1}{r} tr(x) = \frac{1}{r} ⟨x, e⟩. Indeed, using N_{1}(x) = ⟨x, e_{1}⟩ and the fact that L is transitive on the frames, we get

ϕ_{1}(x) = ∫_{L}⟨ℓx, e_{1}⟩ dℓ = \frac{1}{r} ∑_{j=1}^{r} ∫_{L}⟨ℓx, e_{j}⟩ dℓ

= \frac{1}{r} ∫_{L}⟨ℓx, e⟩ dℓ = \frac{1}{r} ∫_{L}⟨x, ℓe⟩ dℓ = \frac{1}{r}⟨x, e⟩.

Using the fact that tr(P (a^{\frac{1}{2}} )y) = ⟨P (a^{\frac{1}{2}} )y, e⟩ = ⟨y, P (a^{\frac{1}{2}} )e⟩ = ⟨y, a⟩, ∀a, y ∈ Ω, we find that

Δ_{1} = \frac{1}{r}R,

where Rf(x) := \frac{∂}{∂t}f(tx)_{|t=1} is the radial derivative.

Our main result in this section is the expansion of D_{1}(λ) = N^{\frac{d}{r} −λ} ∂_{N} N^{1+λ− \frac{d}{r}}. This result was obtained independently by A. Korányi, see [FK2], Proposition XIV.1.5.

Theorem 4.1 For every λ ∈ C,

D_{1}(λ) = ∑_{ν=0}^{r} \binom{r}{ν} ∏_{j=ν+1}^{r}(λ − λ_{j}) Δ_{ν}.  (4.7)

Proof: For x ∈ Ω, the function α → N(x)^{α} is entire in α. Hence both sides of (4.7) are entire in λ, and it is therefore enough to prove (4.7) for λ with ℜλ < 0. Let α = λ_{r} − λ. Since ℜλ > λ_{r}, we get for every x ∈ Ω

N(x)^{−α} = \frac{1}{Γ_{Ω}(α)} ∫_{Ω} e^{−⟨x,t⟩}N(t)^{α} dμ_{Ω}(t),

where dμ_{Ω}(t) := N(t)^{− \frac{d}{r}} dt is the G(Ω)-invariant measure on Ω. Fix a, y ∈ Ω and put f_{y}(x) := e^{⟨x,y⟩}. Then

(N^{α+1}∂_{N}N^{−α}f_{y})(a)
= \frac{N(a)^{α+1}}{Γ_{Ω}(α)} N(\frac{d}{dx}) ∫_{Ω} e^{⟨x,y−t⟩}N(t)^{α}dμ_{Ω}(t) _{|x=a}

= \frac{N(a)^{α+1}}{Γ_{Ω}(α)} ∫_{Ω} e^{⟨a,y−t⟩}N(y − t)N(t)^{α} dμ_{Ω}(t)

= \frac{f_{y}(a)}{Γ_{Ω}(α)} ∫_{Ω} e^{−⟨e,P (a^{\frac{1}{2}} )t⟩}N(P (a^{\frac{1}{2}} )(y − t))N(P (a^{\frac{1}{2}} )t)^{α} dμ_{Ω}(t).

Letting b = P (a^{\frac{1}{2}} )y, the substitution t := P (a^{− \frac{1}{2}} )P (b^{\frac{1}{2}} )τ gives

(N^{α+1}∂_{N}N^{−α}f_{y})(a) = \frac{f_{y}(a)}{Γ_{Ω}(α)}N(y)^{1+α}N(a)^{1+α} ∫_{Ω} e^{−⟨b,τ⟩}N(e − τ)N(τ)^{α} dμ_{Ω}(τ).

Now, the well-known “binomial formula”

N(e + x) = ∑_{ν=0}^{r} \binom{r}{ν}ϕ_{ν}(x), x ∈ X  (4.8)

(which follows from Theorem 1.2 and the knowledge of the norms of the ϕ_{ν}’s) and the fact that for every s ∈ C^{r} and b ∈ Ω

\frac{1}{Γ_{Ω}(s)} ∫_{Ω} e^{−⟨b,τ⟩}ϕ_{s}(τ) dμ_{Ω}(τ) = ϕ_{s}(b^{−1})  (4.9)

(which follows from the analogous formula for the conical functions), imply

∫_{Ω} e^{−⟨b,τ⟩}N(e − τ)N(τ)^{α} dμ(τ) = ∑_{ν=0}^{r} \binom{r}{ν} ∫_{Ω} e^{−⟨b,τ⟩}ϕ_{1_{ν}+α}(τ) dμ_{Ω}(τ)

= ∑_{ν=0}^{r} \binom{r}{ν} Γ_{Ω}(1_{ν} + α) ϕ_{1_{ν}+α}(b^{−1}) = N(b)^{−α} ∑_{ν=0}^{r} \binom{r}{ν} Γ_{Ω}(1_{ν} + α) ϕ_{ν}(b^{−1}).

We claim that for every b ∈ Ω and 1 ≤ ν ≤ r,

ϕ_{ν}(b^{−1}) = ϕ_{r−ν}(b) N(b)^{−1}.  (4.10)

Indeed, using (4.8) we have N(e + tb^{−1}) = ∑_{ν=0}^{r} \binom{r}{ν} ϕ_{ν}(b^{−1}) t^{ν}, as well as

N(e + tb^{−1}) = N(P (b^{− \frac{1}{2}} )(b + te)) = N(b)^{−1} t^{r} N(e + t^{−1}b)
= N(b)^{−1} t^{r} ∑_{k=0}^{r} \binom{r}{k} ϕ_{k}(b) t^{−k}.

Comparing the coefficients of t^{ν} in the two expansions, we obtain (4.10). It follows that

(N^{α+1} ∂_{N}N^{−α}f_{y})(a)
= \frac{f_{y}(a) N(y)^{1+α} N(a)^{1+α}}{Γ_{Ω}(α) N(b)^{1+α}} ∑_{ν=0}^{r}(−1)^{ν} \binom{r}{ν} Γ_{Ω}(1_{ν} + α) ϕ_{r−ν}(b)

= f_{y}(a) ∑_{ν=0}^{r}(−1)^{ν} \binom{r}{ν} \frac{Γ_{Ω}(1_{ν} + α)}{Γ_{Ω}(α)} ϕ_{r−ν}(b)

= f_{y}(a) ∑_{ν=0}^{r} \binom{r}{ν} ∏_{j=1}^{ν}(λ_{j} − α) ϕ_{r−ν}(P (a^{\frac{1}{2}} )y).

Documenta Mathematica 2 (1997) 213–261

Comparing this with (4.6), we conclude that

N^{α+1} ∂_{N}N^{−α} = ∑_{ν=0}^{r} \binom{r}{ν} ∏_{j=1}^{ν}(λ_{j} − α) Δ_{r−ν} = ∑_{k=0}^{r} \binom{r}{k} ∏_{j=1}^{r−k}(λ_{j} − α) Δ_{k}.

Using the relations α = λ_{r} − λ and \frac{d}{r} = 1 + λ_{r}, we obtain (4.7).

Remark: The “binomial formula” (4.8) yields that for every ν = 1, 2, …, r and every x ∈ X,

ϕ_{ν}(x) = ∑_{1≤i_{1}<i_{2}<...<i_{ν}≤r} λ_{i_{1}} λ_{i_{2}} ⋯ λ_{i_{ν}} /\binom{r}{ν} = S_{r,ν}(λ)/\binom{r}{ν},

where λ = (λ_{1}, λ_{2}, …, λ_{r}) is the sequence of eigenvalues of x, and S_{r,ν} is the elementary symmetric polynomial of degree ν in r variables.

Combining the definition D_{ℓ}(λ) = ∏_{k=0}^{ℓ−1} D_{1}(λ + k) with Theorem 4.1, we obtain

Corollary 4.1 For every λ ∈ C and ℓ ∈ N,

D_{ℓ}(λ) = ∏_{k=0}^{ℓ−1} ∑_{ν=0}^{r} \binom{r}{ν} ∏_{j=ν+1}^{r}(λ + k − λ_{j}) Δ_{ν}.  (4.11)

For any signature m ≥ 0 let Δ_{m} be the differential operator associated with the spherical polynomial ϕ_{m} via

(Δ_{m}f)(a) := ϕ_{m}(\frac{d}{dx}) f(P (a^{\frac{1}{2}} ))_{|x=e}, a ∈ Ω.  (4.12)

Equivalently,

Δ_{m}e^{⟨x,y⟩}_{|x=a} = ϕ_{m}(P (a^{\frac{1}{2}} )y) e^{⟨a,y⟩}, a ∈ Ω.  (4.13)

Again, one can replace in (4.12) and (4.13) P (a^{\frac{1}{2}} ) by any ψ ∈ G(Ω) satisfying ψ(e) = a. Hence the operators Δ_{m} are G(Ω)-invariant, namely

Δ_{m}(f ∘ ψ) = (Δ_{m}f) ∘ ψ, ∀ψ ∈ G(Ω).

Theorem 4.2 For every λ ∈ C and ℓ ∈ N,

D_{ℓ}(λ) = ∑_{m≥0}^{(ℓ)} \frac{Γ_{Ω}(\frac{d}{r} + ℓ) Γ_{Ω}(\frac{d}{r} − λ − m^{∗})}{Γ_{Ω}(\frac{d}{r} + ℓ − m^{∗}) Γ_{Ω}(\frac{d}{r} − ℓ − λ)} \frac{d_{m}}{(\frac{d}{r})_{m}} Δ_{m}

(4.14)

= (\frac{d}{r} − λ − ℓ)_{(ℓ,...,ℓ)} ∑_{m≥0}^{(ℓ)} \frac{(−ℓ)_{m}}{(λ)_{m}} \frac{d_{m}}{(\frac{d}{r})_{m}} Δ_{m}.

Here m^{∗} := (m_{r}, m_{r−1}, …, m_{1}), d_{m} = dim(P_{m}), and the summation _{∑m≥0}^{(ℓ)} extends over all m = (m_{1}, m_{2}, …, m_{r}) ∈ N^{r} with ℓ ≥ m_{1} ≥ m_{2} ≥ … ≥ m_{r} ≥ 0.

Proof: The general binomial formula (1.15) and the relations

K_{m}(x, e) = \frac{ϕ_{m}}{∥ϕ_{m}∥_{F}^{2}} , ∥ϕ_{m}∥_{F}^{2} = \frac{(\frac{d}{r})_{m}}{d_{m}}

(see [FK2], Chapter XI) imply for ℓ ∈ N and x ∈ X

N(e + x)^{ℓ} = c ∑_{m≥0}^{(ℓ)} \frac{d_{m}}{(\frac{d}{r})_{ℓ−m^{∗}} (\frac{d}{r})_{m}} ϕ_{m}(x),  (4.15)

where c := (\frac{d}{r})_{(ℓ,ℓ,...,ℓ)}, and m^{∗} and _{∑m≥0}^{(ℓ)} are as in Theorem 4.2. Indeed, by (1.15),

N(e + x)^{ℓ} = ∑_{m≥0}(−ℓ)_{m} \frac{(−1)^{|m|} d_{m}}{(\frac{d}{r})_{m}} ϕ_{m}(x).

From this (4.15) follows by the fact that (−ℓ)_{m} = 0 if m_{1} > ℓ, whereas in case m_{1} ≤ ℓ,

(−ℓ)_{m} (−1)^{|m|} = \frac{(\frac{d}{r})_{(ℓ,ℓ,...,ℓ)}}{(\frac{d}{r})_{ℓ−m^{∗}}} .

As in the proof of Theorem 4.1, it is enough to prove that for every α ∈ C with ℜα > λ_{r} and every ℓ ∈ N,

N^{α+ℓ} ∂_{N}^{ℓ} N^{−α} = c ∑_{m≥0}^{(ℓ)} \frac{(α)_{ℓ−m^{∗}} d_{m}}{(\frac{d}{r})_{ℓ−m^{∗}} (\frac{d}{r})_{m}} Δ_{m}.  (4.16)

From this one obtains (4.14) by the substitution α = \frac{d}{r} − ℓ − λ. To prove (4.16), fix a, y ∈ Ω and let f_{y}(x) := e^{⟨x,y⟩}. Then

(N^{α+ℓ} ∂_{N}^{ℓ} N^{−α}f_{y})(a) = \frac{N(a)^{α+ℓ} f_{y}(a)}{Γ_{Ω}(α)} ∫_{Ω} e^{−⟨a,t⟩} N(y − t)^{ℓ} N(t)^{α} dμ_{Ω}(t)

= \frac{N(b)^{α+ℓ} f_{y}(a)}{Γ_{Ω}(α)} ∫_{Ω} e^{−⟨b,u⟩} N(e − u)^{ℓ} N(u)^{α} dμ_{Ω}(u),

by the substitutions b = P (a^{\frac{1}{2}} )y and u = P (b^{− \frac{1}{2}} )P (a^{\frac{1}{2}} )t. Using (4.15), (4.9), and

ϕ_{m}(x^{−1}) = ϕ_{ℓ−m^{∗}}(x) N(x)^{−ℓ}  (4.17)

(a consequence of [FK2], Proposition VII.1.5), we obtain

(N^{α+ℓ} ∂_{N}^{ℓ} N^{−α}f_{y})(a) = c \frac{f_{y}(a)}{Γ_{Ω}(α)} ∑_{m≥0}^{(ℓ)} \frac{Γ_{Ω}(m + α) d_{m}}{(\frac{d}{r})_{ℓ−m^{∗}} (\frac{d}{r})_{m}} ϕ_{ℓ−m^{∗}}(P (a^{\frac{1}{2}} )y).

With the change of variables n := ℓ − m^{∗}, the fact that d_{m} = d_{n} (use (4.17) or the general formula for d_{m} in [U1]), the definition (4.12), and

(N^{α+ℓ} ∂_{N}^{ℓ} N^{−α}f_{y})(a) = c f_{y}(a) ∑_{n≥0}^{(ℓ)} \frac{(α)_{ℓ−n^{∗}} d_{n}}{(\frac{d}{r})_{ℓ−n^{∗}} (\frac{d}{r})_{n}} ϕ_{n^{∗}}(P (a^{\frac{1}{2}} )y),

we obtain (4.16).

Corollary 4.2 The operators {Δ_{k}}_{k=1}^{r} are algebraically independent generators of the ring Diff(Ω)^{G(Ω)} of G(Ω)-invariant differential operators on Ω.

Proof: Comparing the two expansions (4.11) and (4.14) of D_{ℓ}(λ), we see that

Δ_{m} ∈ C[Δ_{1}, Δ_{2}, …, Δ_{r}]

for every signature m ≥ 0. Since {ϕ_{m}}_{m≥0} is a basis for the space of spherical polynomials, the one-to-one correspondence between spherical polynomials and the elements of Diff(Ω)^{G(Ω)} (see [FK2], Chapter XIV) implies that {Δ_{m}}_{m≥0} is a basis of Diff(Ω)^{G(Ω)}. Thus Diff(Ω)^{G(Ω)} = C[Δ_{1}, Δ_{2}, …, Δ_{r}]. Since the minimal number of algebraic generators of Diff(Ω)^{G(Ω)} is r = rank(Ω) [He], it follows that Δ_{1}, Δ_{2}, …, Δ_{r} are algebraically independent.

The divided differences of a C^{1}-function f on R are defined by

f^{[1]}(t_{0}, t_{1}) := \frac{f(t_{0}) − f(t_{1})}{t_{0} − t_{1}}

for t_{0} ≠ t_{1}, and f^{[1]}(t_{0}, t_{0}) := f′(t_{0}). The higher order divided differences of a smooth enough function f are defined inductively by

f^{[n]}(t_{0}, t_{1}, …, t_{n}) := g^{[1]}(t_{n−1}, t_{n}),

where g(x) := f^{[n−1]}(t_{0}, t_{1}, …, t_{n−2}, x). Then f^{[n]}(t_{0}, t_{1}, …, t_{n}) is symmetric in t_{0}, t_{1}, …, t_{n}, and

f^{[n]}(t, t, …, t) = \frac{1}{n!} \frac{d^{n}}{dt^{n}} f(t).

Moreover, if f is analytic in a domain D ⊂ C, then

f^{[n]}(t_{0}, t_{1}, …, t_{n}) = \frac{1}{2πi} ∫_{Γ} \frac{f(ξ)}{∏_{j=0}^{n}(ξ − t_{j})} dξ

for all t_{0}, t_{1}, …, t_{n} ∈ D and every Jordan curve Γ in D whose interior contains t_{0}, t_{1}, …, t_{n} and is contained in D. The divided differences of vector-valued maps are defined in the same way and have analogous properties. For convenience we put also f^{[0]}(t) := f(t).

Theorem 4.3 Let α_{1}, α_{2}, …, α_{r} ∈ C be distinct. Then {D_{1}(α_{j})}_{j=1}^{r} are algebraically independent generators of Diff(Ω)^{G(Ω)}. Moreover, for ℓ = 1, 2, …, r,

Δ_{ℓ} = D_{1}^{[r−ℓ]}(λ_{ℓ}, λ_{ℓ+1}, …, λ_{r})/\binom{r}{ν},  (4.18)

where D_{1}^{[r−ℓ]}(λ_{ℓ}, …, λ_{r}) are the divided differences of order r − ℓ of D_{1}(λ), evaluated at (λ_{ℓ}, λ_{ℓ+1}, …, λ_{r}).

Proof: Let h_{k}(x) := \binom{r}{ℓ} ∏_{j=k+1}^{r}(x − λ_{j}), 0 ≤ k ≤ r. Then h_{k}^{[m]}(x_{0}, x_{1}, …, x_{m}) ≡ 0

whenever m > r − k, and h_{k}^{[r−k]}(x_{0}, x_{1}, …, x_{r−k}) ≡ \binom{r}{ℓ} for all choices of x_{0}, x_{1}, …, x_{r−k}. By Theorem 4.2, D_{1}(α) = ∑_{k=0}^{r} h_{k}(α) Δ_{k}. Hence, for 1 ≤ ℓ ≤ r,

D_{1}^{[r−ℓ]}(α_{ℓ}, α_{ℓ+1}, …, α_{r}) = ∑_{k=0}^{ℓ} h_{k}^{[r−ℓ]}(α_{ℓ}, α_{ℓ+1}, …, α_{r}) Δ_{k}.

Solving this system of equations for the Δ_{k}’s, we see that Diff(Ω)^{G(Ω)} = C[Δ_{1}, Δ_{2}, …, Δ_{r}] coincides with the ring generated by the operators {D_{1}^{[r−ℓ]}(α_{ℓ}, α_{ℓ+1}, …, α_{r})}_{ℓ=1}^{r}. If the {α_{j}}_{j=1}^{r} are distinct, then

D_{1}^{[r−ℓ]}(α_{ℓ}, α_{ℓ+1}, …, α_{r}) ∈ C[D_{1}(α_{1}), D_{1}(α_{2}), …, D_{1}(α_{r})].

Hence,

Diff(Ω)^{G(Ω)} = C[Δ_{1}, Δ_{2}, …, Δ_{r}] = C[D_{1}(α_{1}), D_{1}(α_{2}), …, D_{1}(α_{r})].

The operators {D_{1}(α_{j}}_{j=1}^{r} are algebraically independent, since Diff(Ω)^{G(Ω)} cannot be algebraically generated by less than r elements. If α_{j} = λ_{j} for j = 1, 2, …, r, then h_{k}^{[r−ℓ]}(α_{ℓ}, …, α_{r}) = 0 for k < ℓ. Thus, for ℓ = 1, 2, …, r,

D_{1}^{[r−ℓ]}(α_{ℓ}, α_{ℓ+1}, …, α_{r}) = h_{ℓ}^{[r−ℓ]}(α_{ℓ}, α_{ℓ+1}, …, α_{r}) Δ_{ℓ} = \binom{r}{ℓ} Δ_{ℓ}.

Remark: The first statement in Theorem 4.3 was proved independently also by A. Korányi [FK2] and Z. Yan [Y1]. Our result is slightly stronger, giving the exact formula (4.18).

Combining Theorems 3.2 and 4.2 (or, 4.1) we obtain integral formulas for the invariant hermitian forms ⟨·, ·⟩_{λ,j}, λ ∈ P(D), 0 ≤ j ≤ q(λ).

Corollary 4.3 Let λ ∈ P(D), ℓ ∈ N and assume that λ + ℓ ≥ \frac{d}{r} = λ_{r} + 1. Let q = q(λ), 0 ≤ j ≤ q, and ν = q − j. Consider the G(Ω)-invariant differential operator

T_{λ,j} := γ ∑_{m≥0}^{(ℓ)} c_{m}(λ, ℓ) \frac{d_{m}}{(\frac{d}{r})_{m}} Δ_{m},  (4.19)

where γ is given by (3.4), and for every m ≥ 0 with m_{1} ≤ ℓ

c_{m}(λ, ℓ) := \frac{1}{ν!}(\frac{∂}{∂ξ})^{ℓ} (\frac{Γ_{Ω}(\frac{d}{r} + ℓ) Γ_{Ω}(\frac{d}{r} − ξ − m^{∗})}{Γ_{Ω}(\frac{d}{r} + ℓ − m^{∗}) Γ_{Ω}(\frac{d}{r} − ℓ − ξ)})_{|ξ=λ} .  (4.20)

Then T_{λ,j} is defined on all analytic functions on D, and for all f, g ∈ H_{λ,j}

⟨f, g⟩_{λ,j} = ⟨T_{λ,j}f, g⟩_{λ+ℓ}.  (4.21)

In particular, if λ + ℓ > p − 1 or λ + ℓ = \frac{d}{r} then we have

⟨f, g⟩_{λ,j} = ∫_{D}(T_{λ,j}f)(z) \overline{g(z)} dμ_{λ+ℓ}(z) and ⟨f, g⟩_{λ,j} = ∫_{S}(T_{λ,j}f)(ξ) \overline{g(ξ)} dσ(ξ)

(4.22)

respectively.

The case λ = λ_{r} is particularly simple, since then \frac{d}{r} − λ_{r} = 1, and we can use (4.7) rather than (4.14).

Corollary 4.4 Let D be a Cartan domain of tube type and rank r ≥ 2 in C^{d}, d ≥ 3. Then

⟨f, g⟩_{λ_{r},0} = ⟨β ∑_{ν=0}^{r−1} \binom{r}{ν} ∏_{i=2}^{r−ν} λ_{i} Δ_{ν}f, g⟩_{H^{2}(S)}, where β := ∏_{i=2}^{r} λi.  (4.23)

Proof: In this case q = q(λ_{r}) = 1, j = 0, and ν = q − j = 1. We choose ℓ = 1, so λ_{r} + ℓ = \frac{d}{r}. In order to apply Theorem 3.2 we use Theorem 4.1, and compute

D_{1}^{1}(λ_{r}) = \frac{∂}{∂ξ} D_{1}(ξ)_{|ξ=λ} = \frac{∂}{∂ξ} (∑_{ν=0}^{r} \binom{r}{ν} ∏_{i=ν+1}^{r}(ξ − λ_{i}) Δ_{ν})_{|ξ=λr}

= ∑_{ν=0}^{r−1} \binom{r}{ν} ∏_{i=ν+1}^{r−1}(λ_{r} − λ_{i}) Δ_{ν} = ∑_{ν=0}^{r−1} \binom{r}{ν} ∏_{i=2}^{r−ν} λ_{i} Δ_{ν}.

Using this, (4.23) follows from

β := \frac{∂}{∂ξ} (∏_{i=1}^{r}(ξ − λ_{i}))_{ξ=λ} = ∏_{i=1}^{r−1}(λ_{r} − λ_{i}) = ∏_{i=2}^{r} λ_{i}.

Example 4.2. Let D be the Cartan domain of rank r = 2 in C^{d} (the Lie ball), d ≥ 3. Then

⟨f, g⟩_{\frac{d−2}{2} ,0} = ⟨(\frac{2}{d − 2} R + I)f, g⟩_{H^{2}(S)}.  (4.24)

Namely, in this case λ = λ_{2} = \frac{d−2}{2}, q = q(λ) = 1, j = 0, and ν = q − j = 1. With ℓ = 1, λ + ℓ = \frac{d}{2} = λ_{2} + 1 = \frac{d}{r} we get by using Theorem 3.2 and Corollary 3.2,

⟨f, g⟩_{\frac{d−2}{2} ,0} = γ⟨D_{1}^{1}(\frac{d − 2}{2})f, g⟩_{\frac{d}{2}}

= γ ⟨(R + \frac{d − 2}{2}I)f, g⟩_{H^{2}(S)} = ⟨(\frac{2}{d − 2} R + I)f, g⟩_{H^{2}(S)}.

Since the Shilov boundary S of D is given by

S = {e^{iθ}(x_{1}, ix_{2}, ix_{3}, …, ix_{d}); θ ∈ R, ∑_{j=1}^{d} x_{j}^{2} = 1} ≡ S^{1} · S^{d−1},

the unique K-invariant probability measure on S is dσ(e^{iθ}(x_{1}, ix′)) = \frac{dθ}{2π} dν_{d−1}(x), where ν_{d−1} is the unique O(d − 1)-invariant probability measure on S^{d−1}. Thus (4.24) provides a very concrete formula for the inner product ⟨·, ·⟩_{\frac{d−2}{2} ,0}.

5 Integration over boundary orbits of Aut(D)

In this section we obtain formulas for the invariant inner products in terms of integration over an orbit of Aut(D) on the boundary ∂D. We focus on the inner products ⟨·, ·⟩_{λ_{2},0} = ⟨·, ·⟩_{\frac{a}{2}} , and conjecture that our method can be generalized for the derivation of similar formulas for the inner products ⟨·, ·⟩_{λ_{j} ,0} = ⟨·, ·⟩_{λ_{j}}, λ_{j} = (j − 1)\frac{a}{2}, j = 3, 4, …, r, in terms of integration on an appropriate boundary orbit. (Notice that the case j = 1 is trivial, since λ_{1} = 0 and H_{0,0} = H_{0} = C1).

In order to describe the facial structure of a Cartan domain of tube-type D ⊂ C^{d} [Lo], [A1], let S_{ℓ} be the compact, real analytic manifold of tripotents in Z of rank ℓ = 1, 2, …, r. The group K acts transitively and irreducibly on S_{ℓ}. Let σ_{ℓ} be the unique K-invariant probability measure on S_{ℓ} given by

∫_{S_{ℓ}} f dσ_{ℓ} = ∫_{K} f(k(v_{ℓ})) dk,  (5.1)

where v_{ℓ} is any fixed element of S_{ℓ}. For any tripotent v let Z = Z_{1}(v)+Z_{\frac{1}{2}} (v)+Z_{0}(v) be the corresponding Peirce decomposition. Then D_{v} := D∩Z_{0}(v) is a Cartan domain of tube-type, which is the open unit ball of the JB^{∗}-algebra Z_{0}(v). If v ∈ S_{ℓ} then the rank of D_{v} is r_{v} := r − ℓ, its characteristic multiplicity is a_{v} := a if ℓ ≤ r − 2 and a_{v} = 0 if ℓ = r − 1, and the genus is p_{v} = p − ℓ a. The set v + D_{v} is a face of the closure \overline{D} of D. For any function f on \overline{D} let f_{v} be the function on \overline{D_{v}} defined by

f_{v}(z) := f(v + z), z ∈ \overline{D_{v}}.  (5.2)

The fundamental polynomial “h” of Z_{0}(v) is defined by

h_{v}(z, w) := h(z, w), z, w ∈ Z_{0}(v).  (5.3)

For ℓ = 1, 2, …, r, ∂_{ℓ}D := ∪_{v∈S_{ℓ}}(v + D_{v}) is an orbit of G: ∂_{ℓ}D = G(v_{ℓ}). If v ∈ S_{r} is a maximal tripotent, then D_{v} = Z_{0}(v) = {0}. Hence ∂_{r}D = S_{r} = S is the Shilov boundary. In particular, S is a G-orbit. The only tripotent of rank 0 is 0 ∈ Z, and D = D_{0} is also a G-orbit. Thus the decomposition of \overline{D} into G-orbits is

\overline{D} = D ∪ ⋃_{ℓ=1}^{r} ∂_{ℓ}D.

For every tripotent v ∈ Z and λ > p_{v} − 1 consider the probability measure μ_{v,λ} on D_{v}, defined via

∫_{D_{v}} f dμ_{v,λ} := c_{v,λ} ∫_{D_{v}} f(z) h_{v}(z, z)^{λ−p_{v}} dm_{v}(z),  (5.4)

where m_{v} is the Lebesgue measure on D_{v} and c_{v,λ} is the normalization factor. Similarly, one defines a probability measure σ_{v} on the Shilov boundary S_{v} of D_{v}, via

∫_{S_{v}} f dσ_{v} := ∫_{K_{v}} f(k(v′)) dk,

where v′ is any tripotent orthogonal to v and K_{v} := {k ∈ K; k(Z_{ν}(v)) = Z_{ν}(v)}, ν = 0, 1/2, 1, so that K_{v}(v′) = S_{v}. The combination of μ_{v,λ} and σ_{ℓ} yields K-invariant probability measures μ_{ℓ,λ} on ∂_{ℓ}D, 1 ≤ ℓ ≤ r − 1, λ > p − ℓ a − 1, via

∫_{∂_{ℓ}D} fdμ_{ℓ,λ} := ∫_{S_{ℓ}} (∫_{D_{v}} f_{v}(z) dμ_{v,λ}(z)) dσ_{ℓ}(v).

Next, consider the “sphere bundle” B_{ℓ}, 1 ≤ ℓ ≤ r, whose base is S_{ℓ} and the fiber at each v ∈ S_{ℓ} is v + S_{v} (where S_{v} := ∂_{r−ℓ}D_{v} is the Shilov boundary of D_{v}). The group K acts on B_{ℓ} naturally, and this action is transitive. The combination of the measures σ_{v}, v ∈ S_{ℓ} and σ_{ℓ} yields K-invariant probability measures ν_{ℓ} on B_{ℓ} via

∫_{B_{ℓ}} f dν_{ℓ} := ∫_{S_{ℓ}} (∫_{S_{v}} f(v + ξ) dσ_{v}(ξ)) dσ_{ℓ}(v).

For v ∈ S_{ℓ}, consider the symmetric cone Ω_{v} in Z_{0}(v), and let Δ_{1}^{(v)}, Δ_{2}^{(v)}, …, Δ_{r−ℓ}^{(v)} be the canonical generators of the ring Diff(Ω_{v})^{G(Ω_{v})} as in section 4. We also denote

Δ_{0}^{(v)} = I, Δ^{(v)} := (Δ_{1}^{(v)}, Δ_{2}^{(v)}, …, Δ_{r−ℓ}^{(v)}), and λ_{j} = (j − 1)\frac{a}{2}, 0 ≤ j ≤ r.

Conjecture: For every 2 ≤ j ≤ r and every λ > λ_{j−1} there exists a positive function p_{j,λ} ∈ C^{∞}([0, ∞)^{j−1}), so that the inner product ⟨·, ·⟩_{λ_{j}} = ⟨·, ·⟩_{λ_{j} ,0} is given by

⟨f, g⟩_{λ_{j}} = ∫_{S_{r−j+1}}⟨p_{j,λ}(Δ^{(v)})f_{v}, g_{v}⟩_{H_{λ}(Dv)} dσ_{r−j+1}(v).  (5.5)

Moreover, if λ = λ_{j−1} + 1 = dim(D_{v})/rank(D_{v}), then p_{j} := p_{j,λ} is a polynomial with positive coefficients.

If λ is chosen appropriately then (5.5) becomes an integral formula for ⟨f, g⟩_{λ_{j}}. For instance, if λ = λ_{j−1} + 1 in (5.5), then we have H_{λ}(D_{v}) = H^{2}(S_{v}), and (5.5) becomes

⟨f, g⟩_{λ_{j}} = ∫_{S_{r−j+1}} (∫_{S_{v}}(p_{j,λ}(Δ^{(v)})f_{v})(ξ) \overline{g_{v}(ξ)}dσ_{v}(ξ)) dσ_{r−j+1}(v).  (5.6)

Also, if λ > (j − 2)a + 1 in (5.5) then H_{λ}(D_{v}) = L_{a}^{2}(D_{v}, μ_{v,λ}), and (5.5) becomes

⟨f, g⟩_{λ_{j}} = ∫_{S_{r−j+1}} (∫_{D_{v}}(p_{j}(Δ^{(v)})f_{v})(z) \overline{g_{v}(z)} dμ_{v,λ}(z)) dσ_{r−j+1}(v).  (5.7)

Note that the integral in (5.7) can be expressed as an integral on ∂_{r−j+1}D with respect to dμ_{r−j+1,λ}. Similarly, (5.6) is an integral on B_{r−j+1} with respect to ν_{r−j+1}.

Integral formulas for ⟨f, g⟩_{a/2} via integration on ∂_{r−1}D

In what follows we shall establish (5.5) for j = 2 (i.e. λ_{2} = \frac{a}{2}) in two important special cases, namely for Cartan domains of type I and IV. Our method suggests an approach for the general case. For j = 2 (5.5) becomes

⟨f, g⟩_{\frac{a}{2}} = ∫_{S_{r−1}} dσ_{r−1}(v)⟨p_{λ}(R^{(v)})f_{v}, g_{v}⟩_{H_{λ}(D)},  (5.8)

where p_{λ}(x) = p_{2,λ}(x) ∈ C^{∞}([0, ∞)) is a positive function, Δ_{1}^{(v)} = R^{(v)}, where R^{(v)} is the localized radial derivative (i.e. the radial derivative in Z_{0}(v)), and D_{v} ≡ D = {z ∈ C; |z| < 1}. We will show that in our two cases

p_{λ}(x) = \frac{Γ(x + λ)}{Γ(λ)Γ(x + 1)} q(x),

where q(x) is a polynomial with positive rational coefficients. In particular, for λ = 1, 2, …, p_{λ}(x) itself is a polynomial with positive rational coefficients. If λ is chosen appropriately, then (5.8) becomes an integral formula analogous to (5.6) or (5.7). For λ = 1, (5.8) becomes

⟨f, g⟩_{\frac{a}{2}} = ∫_{S_{r−1}} dσ_{r−1}(v)⟨p_{1}(R^{(v)})f_{v}, g_{v}⟩_{H^{2}(T)},  (5.9)

and for λ > 1, (5.8) becomes

⟨f, g⟩_{\frac{a}{2}} = ∫_{S_{r−1}} dσ_{r−1}(v)⟨p_{λ}(R^{(v)})f_{v}, g_{v}⟩_{L^{2}(D,μ_{λ})}.  (5.10)

Lemma 5.1 The right hand side of (5.5) is K-invariant. Consequently, the right hand sides of (5.6), (5.7), (5.8), (5.9), and (5.10) are K-invariant.

Proof: Let ℓ = r − j + 1, and note that for each fixed smooth function f the maps S_{ℓ} ∋ v ↦ Δ_{i}^{(v)}(f_{v}), 1 ≤ i ≤ j − 1, are K-invariant, in the sense that

Δ_{i}^{(k(v))}(f_{k(v)}) ∘ k = Δ_{i}^{(v)}((f ∘ k)_{v}), ∀k ∈ K, ∀v ∈ S_{ℓ}.

From this it follows that if v_{ℓ} ∈ S_{ℓ} is any fixed element, then

∫_{S_{ℓ}}⟨p_{j,λ}(Δ^{(v)})f_{v}, g_{v}⟩_{H_{λ}(Dv)}dσ_{ℓ}(v)
= ∫_{K}⟨p_{j,λ}(Δ^{(v_{ℓ})})(f ∘ k)_{v_{ℓ}}, (g ∘ k)_{v_{ℓ}}⟩_{H_{λ}(Dv_{ℓ})}dk.

The K-invariance of the right hand side of (5.5) follows from the invariance of the Haar measure dk.

Since M_{0}^{( \frac{a}{2} )} = ∑_{m=0}^{∞} P_{(m,0,0,...)} and

⟨f, g⟩_{\frac{a}{2}} = ∑_{m=(m,0,...,0),0≤m<∞} \frac{⟨f_{m}, g_{m}⟩_{F}}{(\frac{a}{2})_{m}} ,

in order to establish (5.8) it is enough, by the K-invariance of both sides, to find positive functions p_{λ}(x) ∈ C^{∞}([0, ∞)) so that (5.8) holds for the functions f(z) = g(z) = N_{1}^{m}(z), m ≥ 0. This is equivalent to

∫_{S_{r−1}} dσ_{r−1}(v)⟨p_{λ}(R^{(v)})(N_{1}^{m})_{v}, (N_{1}^{m})_{v}⟩_{H_{λ}(D)} = \frac{m!}{(\frac{a}{2})_{m}} .  (5.11)

Fix a frame e_{1}, e_{2}, …, e_{r} in Z. Then N_{1}(z) = (z, e_{1}), where (·, ·) is the unique Kinvariant inner product on Z for which (v, v) = 1 for every minimal tripotent v. Let e′ := e_{2} + e_{3} + … + e_{r}. Then for z = k(ξe_{1} + e′) with k ∈ K and ξ ∈ T, we have

N_{1}^{m}(z) = (ξk(e_{1}) + k(e′), e_{1})^{m} = ∑_{ℓ=0}^{m} \binom{m}{ℓ} (k(e_{1}), e_{1})^{ℓ} (k(e′), e_{1})^{m−ℓ} ξ^{ℓ}.

Thus, for v = k(e′), m ≥ 0 and any continuous function f we have

(f(R^{(v)})N_{1}^{m})(z) = ∑_{ℓ=0}^{m} \binom{m}{ℓ} (k(e_{1}), e_{1})^{ℓ} (k(e′), e_{1})^{m−ℓ} f(ℓ) ξ^{ℓ}.

Let us define

J_{m,ℓ} := ∫_{K} |(k(e_{1}), e_{1})|^{2ℓ} |(k(e′), e_{1})|^{2(m−ℓ)} dk, 0 ≤ ℓ ≤ m < ∞.  (5.12)

It follows that the function p_{λ} should satisfy

∫_{S_{r−1}} dσ_{r−1}(v) ⟨p_{λ}(R^{(v)})(N_{1}^{m})_{v}, (N_{1}^{m})_{v}⟩_{H_{λ}(D)} = ∑_{ℓ=0}^{m} J_{m,ℓ} \binom{m}{ℓ}^{2} \frac{ℓ!}{(λ)_{ℓ}} p_{λ}(ℓ).

Thus (5.11) becomes

∑_{ℓ=0}^{m} J_{m,ℓ} \binom{m}{ℓ}^{2} q_{ℓ} = \frac{m!}{(\frac{a}{2})_{m}} , m = 0, 1, 2, …,  (5.13)

where the numbers

q_{ℓ} := \frac{ℓ!}{(λ)_{ℓ}} p_{λ}(ℓ), ℓ = 0, 1, 2, …  (5.14)

do not depend on λ. The infinite system of linear equations (5.13) in the unknowns {q_{ℓ}}_{ℓ=0}^{∞} corresponds to the lower triangular matrix A = (a_{m,ℓ})_{m,ℓ=0}^{∞}, where a_{m,ℓ} =

J_{m,ℓ} \binom{m}{ℓ}^{2} for m ≥ ℓ, and a_{m,ℓ} = 0 for m < ℓ. Since a_{m,m} > 0 for m = 0, 1, 2, …, there exists a unique solution {q_{ℓ}}_{ℓ=0}^{∞} to (5.13). There are many smooth functions which interpolate the values {q_{ℓ}}_{ℓ=0}^{∞}. We will show that q_{ℓ} > 0 for every ℓ ≥ 0, and prove that {q_{ℓ}}_{ℓ=0}^{∞} can be interpolated by a polynomial of degree r − 1 with positive coefficients. For Cartan domains of type I and IV, we will solve the system (5.13) by calculating explicitly the numbers J_{m,ℓ} and applying powers of the difference operator

δ(f)(t) := f(t) − f(t − 1), t ∈ R.

If f is defined only on [0, ∞) then we define δ(f) := δ(F), where F (t) := f(t) for 0 ≤ t and F (t) = 0 for 0 > t. Similarly, δ can be defined on two-sided sequences (i.e. on functions on Z) or on sequences (i.e. functions on N). The powers of δ are defined inductively by δ^{n+1} := δ ∘ δ^{n}.

Case 1: Cartan domains of type I. Let D = D(I_{r,r}) := {z ∈ M_{r,r}(C); ∥z∥ < 1}. The rank of D is r, the dimension is d = r^{2}, the genus is p = 2r, and the characteristic multiplicity is a = 2. To every k ∈ K there correspond u, w ∈ U(r) (the unitary group) so that det(u) = det(w), and

k(z) = uzw^{∗}, z ∈ D.  (5.15)

Thus ∫_{K} f(k(z)) dk = ∫_{U(r)} ∫_{U(r)} f(uzw^{∗}) du dw, where dk is the Haar measure of K. Choose the canonical frame of matrix units e_{j} := e_{j,j}, j = 1, 2, …, r, and denote e = ∑_{j=1}^{r} e_{j} and e′ := e − e_{1} = ∑_{j=2}^{r} e_{j}.

Proposition 5.1 Let D = D(I_{r,r}). Then for every integers m, ℓ with 0 ≤ ℓ ≤ m < ∞, we have

J_{m,ℓ} = \frac{(r − 1) (ℓ!)^{2} (m − ℓ)! (m − ℓ + r − 2)!}{(r)_{m} (m + r − 1)!}.  (5.16)

Proof: Let k ∈ K be given by (5.15). Then (k(e_{1}), e_{1}) = u_{1,1}\overline{w_{1,1}} and (k(e′), e_{1}) = ∑_{j=2}^{r} u_{1,j}\overline{w_{1,j}}. Thus, for 0 ≤ ℓ ≤ m < ∞,

J_{m,ℓ} = ∫_{U(r)} ∫_{U(r)} |u_{1,1}|^{2ℓ}|w_{1,1}|^{2ℓ}| ∑_{j=2}^{r} u_{1,j}\overline{w_{1,j}}|^{2(m−ℓ)} du dw.

This integral can be written as an integral on the product of the unit spheres ∂B_{r} ⊂ C^{r} with respect to the U(r)-invariant probability measure σ:

J_{m,ℓ} = ∫_{∂B_{r}} ∫_{∂B_{r}} |ξ_{1}^{ℓ}|^{2}|η_{1}^{ℓ}|^{2}|(ξ′, η′)|^{2(m−ℓ)} dσ(ξ) dσ(η),

where ξ′ := (ξ_{2}, …, ξ_{r}) and η′ := (η_{2}, …, η_{r}). Now, by the U(r)-invariance,

∫_{∂B_{r}} |ξ_{1}^{ℓ}|^{2}|(ξ′, η′)|^{2(m−ℓ)} dσ(ξ)
= ∥η′∥^{2(m−ℓ)} ∫_{∂B_{r}} |ξ_{1}^{ℓ}|^{2}|ξ_{2}^{m−ℓ}|^{2} dσ(ξ)

= ∥η′∥^{2(m−ℓ)}∥ξ_{1}^{ℓ}ξ_{2}^{m−ℓ}∥_{H_{r}(D)}^{2} = ∥η′∥^{2(m−ℓ)} \frac{ℓ!(m − ℓ)!}{(r)_{m}}.

It follows by using [Ru], 1.4.5, that

J_{m,ℓ} = \frac{ℓ!(m − ℓ)!}{(r)_{m}} ∫_{∂B_{r}} |η_{1}^{ℓ}|^{2}(1 − |η_{1}|^{2})^{m−ℓ} dσ(η)

= \frac{ℓ!(m − ℓ)!}{(r)_{m}}(r − 1) ∫_{0}^{1} t^{ℓ}(1 − t)^{m−ℓ+r−2} dt

= \frac{ℓ!(m − ℓ)!}{(r)_{m}}(r − 1)B(ℓ + 1, m − ℓ + r − 1)

= \frac{(r − 1)(ℓ!)^{2}(m − ℓ)!(m − ℓ + r − 2)!}{(r)_{m}(m + r − 1)!}.

Corollary 5.1 For D = D(I_{r,r}) the system of equations (5.13) is equivalent to the system

∑_{ℓ=0}^{m} \frac{(m − ℓ + r − 2)!}{(m − ℓ)!} q_{ℓ} = (r − 2)! \binom{m + r − 1}{r − 1}^{2}, m = 0, 1, 2, ….  (5.17)

Proposition 5.2 For every r ≥ 2 there exists a polynomial q(x) = q_{r}(x) of degree r − 1 with positive rational coefficients, so that q(ℓ) = q_{ℓ} for ℓ = 0, 1, 2, …, where {q_{ℓ}}_{ℓ=0}^{∞} is the unique solution of (5.17).

For small values of r it is easy to solve (5.17) explicitly by applying powers of δ. Thus,

q_{2}(x) = 2x + 1, q_{3}(x) = 3x^{2} + 3x + 1, and q_{4}(x) = \frac{1}{3}(10x^{3} + 15x^{2} + 11x + 3).

The proof in the general case requires more preparation. Define

f_{n}(x) := (x + 1)_{n} = ∏_{j=1}^{n}(x + j), n ≥ 1, and g_{n}(x) := ∏_{j=0}^{n}(x + j)^{2}, n ≥ 0. (5.18)

Then g_{n}(x + 1) = f_{n+1}(x)^{2}, and

(δ^{k}f_{n})(x) = n(n − 1) ⋯ (n − k + 1) f_{n−k}(x), k ≥ 0,  (5.19)

where δ is defined by δ(f)(x) := f(x) − f(x − 1). Indeed, (5.19) is trivial for k = 0. For k = 1 and all n we have

δ(f_{n})(x) = ∏_{j=1}^{n}(x + j) − ∏_{j=1}^{n}(x + j − 1) = ∏_{j=1}^{n−1}(x + j) (x + n − x) = n f_{n−1}(x).

Assuming (5.19) for k, let n > k and compute δ^{k+1}(f_{n})(x) = n(n − 1) ⋯ (n − k + 1) δ(f_{n−k})(x) = n(n − 1) ⋯ (n − k + 1)(n − k)f_{n−k−1}(x). This establishes (5.19).

Next, define an operator σ, analogous to δ, via

(σf)(x) := f(x) + f(x − 1), x ∈ R.

Clearly, δσ = σδ, and both σ and δ commute with all the translation operators

(τ_{c}f)(x) := f(x + c).

Denote by P_{+} the set of polynomials in one variable with non-negative coefficients.

Lemma 5.2 Let f(x) be a polynomial and let n, m ∈ N. If δ^{n}f ∈ P_{+}, then δ^{n+j}τ_{m/2}f ∈ P_{+} for every integer 0 ≤ j ≤ m.

Proof: Since δ commutes with translations, we may assume that n = 0 and m = 1. It is therefore enough to check that δτ_{1/2}x^{k} ∈ P_{+} for every k ∈ N. This follows from the binomial expansion:

δτ_{1/2}x^{k} = (x + \frac{1}{2})^{k} − (x − \frac{1}{2})^{k} = ∑_{j=0}^{[ \frac{k−1}{2} ]} \binom{k}{2j + 1}2^{−2j}x^{k−2j−1}.

Lemma 5.3 Let f(x) be a polynomial and let n ∈ N. Assume that δ^{j}σ^{n−j}f ∈ P_{+} for every 0 ≤ j ≤ n. Then δ^{j}σ^{n−j} ((x + c)^{k}f(x)) ∈ P_{+} for every k ∈ N, c ≥ \frac{n}{2} and 0 ≤ j ≤ n.

Proof: Again, since δ and σ commute with translations, it is enough to assume that k = 1. We shall prove the assertion by induction on n. The case n = 0 is trivial since P_{+} is closed under sums and products. Assume that n > 0 and that the assertion holds for n − 1. A computation yields

δ ((x + \frac{n}{2})f(x)) = (x + \frac{n − 1}{2})(δf)(x) + \frac{1}{2}(σf)(x)  (5.20)

and

σ ((x + \frac{n}{2})f(x)) = (x + \frac{n − 1}{2})(σf)(x) + \frac{1}{2}(δf)(x).  (5.21)

If 0 < j ≤ n then using (5.20) we get

δ^{j}σ^{n−j} ((x + \frac{n}{2})f(x)) = δ^{j−1}σ^{(n−1)−(j−1)} ((x + \frac{n − 1}{2})(δf)(x) + \frac{1}{2}(σf)(x)) .

By assumption,

δ^{j−1}σ^{(n−1)−(j−1)}σf = δ^{j−1}σ^{n−(j−1)}f ∈ P_{+}, for 0 < j ≤ n.

Similarly,

δ^{j−1}σ^{(n−1)−(j−1)}δf = δ^{j}σ^{n−j}f ∈ P_{+} for 0 < j ≤ n.

Thus, by the induction hypothesis on n − 1,

δ^{j−1}σ^{(n−1)−(j−1)} ((x + \frac{n − 1}{2})δf(x)) ∈ P_{+}, for 0 < j ≤ n.

Next, using (5.21) we get

σ^{n} ((x + \frac{n}{2})f(x)) = σ^{n−1} ((x + \frac{n − 1}{2})σf(x) + \frac{1}{2}δf(x)) .

By assumption, σ^{n−1}δf(x) ∈ P_{+} and δ^{ℓ}σ^{n−1−ℓ}σf(x) ∈ P_{+} for 0 ≤ ℓ ≤ n − 1. Thus, by the induction hypothesis, δ^{ℓ}σ^{n−1−ℓ} ((x + \frac{n−1}{2})σf(x)) ∈ P_{+} for 0 ≤ ℓ ≤ n − 1, and in particular σ^{n−1} ((x + \frac{n−1}{2})σf(x)) ∈ P_{+}. It follows that σ^{n} ((x + \frac{n}{2})f(x)) ∈ P_{+}. This completes the induction step.

Lemma 5.4 Let g_{n}(x) be the polynomial defined by (5.18). Then δ^{i}σ^{j}g_{n} ∈ P_{+} whenever i + j ≤ n.

Proof: We proceed by induction on n. The case n = 0 is trivial, since g_{0}(x) = x^{2} ∈ P_{+}. Assume that n > 0 and that δ^{i}σ^{j}g_{n−1} ∈ P_{+} whenever i + j ≤ n − 1. A computation yields

δg_{n}(x) = 2(n + 1)(x + \frac{n − 1}{2}) g_{n−1}(x)  (5.22)

and

σg_{n}(x) = 2 ((x + \frac{n − 1}{2})^{2} + (\frac{n + 1}{2})^{2}) g_{n−1}(x).  (5.23)

Now assume i + j ≤ n. If i > 0, (5.22) yields

δ^{i}σ^{j}g_{n}(x) = δ^{i−1}σ^{j}(δg_{n}(x)) = 2(n + 1)δ^{i−1}σ^{j} ((x + \frac{n − 1}{2})g_{n−1}(x)) ,

and by induction hypothesis and Lemma 5.3

δ^{i−1}σ^{j} ((x + \frac{n − 1}{2})g_{n−1}(x)) ∈ P_{+},

so that δ^{i}σ^{j}g_{n} ∈ P_{+}. If i = 0 and 0 ≤ j ≤ n, then (5.23) implies

σ^{j}g_{n}(x) = σ^{j−1}(σg_{n}(x)) = 2σ^{j−1} (((x + \frac{n − 1}{2})^{2} + (\frac{n + 1}{2})^{2}) g_{n−1}(x)) .

The polynomial σ^{j−1}g_{n−1} belongs to P_{+} by the induction hypothesis. Also, the induction hypothesis (δ^{i}σ^{j−1}g_{n−1} ∈ P_{+} whenever i + j ≤ n) and Lemma 5.3 imply that

δ^{i}σ^{j−1} ((x + \frac{n − 1}{2})g_{n−1}(x)) ∈ P_{+} whenever i + j ≤ n.

In particular, σ^{j−1} ((x + \frac{n−1}{2})g_{n−1}(x)) ∈ P_{+}. Hence σ^{j}g_{n} ∈ P_{+} ∀ 0 ≤ j ≤ n.

Corollary 5.2 (i) δ^{j}g_{n} ∈ P_{+} for all j, n ∈ N satisfying 0 ≤ j ≤ n.

(ii) δ^{j} ((x + \frac{m}{2})g_{n}(x)) ∈ P_{+} for all j, n, m ∈ N satisfying 0 ≤ j ≤ n + m.

(iii) δ^{j}f_{n}(x)^{2} ∈ P_{+} for all j, n ∈ N satisfying 0 ≤ j ≤ n + 1.

Proof: (i) is a special case of Lemma 5.4, and (ii) follows by (i) and Lemma 5.2. Since f_{n}(x)^{2} = g_{n−1}(x + 1), (iii) follows from Lemma 5.2 with m = 2.

Remark The result in part (iii) of Corollary 5.2 is best possible in the sense that δ^{n+2}(f_{n}^{2})^{2}) need not be in P_{+}. Indeed, δ^{6}(f_{4}^{2})^{2}) is not in P_{+}.

Proof of Proposition 5.2: In terms of the polynomials (5.18), the system of equations (5.17) with unknowns q_{ℓ} has the form

∑_{ℓ=0}^{m} f_{r−2}(m − ℓ) q_{ℓ} = \frac{f_{r−1}(m)^{2}}{(r − 1) (r − 1)!}, m ≥ 0.  (5.24)

Applying powers of the operator δ with respect to the variable m and using (5.19), we get by induction on k that

δ^{k} (∑_{ℓ=0}^{m} f_{r−2}(m − ℓ) q_{ℓ}) = (r − 2)(r − 3) ⋯ (r − k − 1) ∑_{ℓ=0}^{m} f_{r−2−k}(m − ℓ) q_{ℓ}

for 0 ≤ k ≤ r − 2 (here f_{0}(x) ≡ 1). From this it follows that

δ^{r−1} (∑_{ℓ=0}^{m} f_{r−2}(m − ℓ) q_{ℓ}) = (r − 2)! q_{m}, m ≥ 0.

Applying δ^{r−1} to both sides of (5.24), Corollary 5.2 (iii) implies that there exists a polynomial q(x) of degree r − 1 with positive rational coefficients so that q_{m} = q(m), ∀m ≥ 0.

Theorem 5.1 Let D = D(I_{r,r}). Then for every f, g ∈ H\frac{a}{2} (D) and λ > 0 we have

⟨f, g⟩_{\frac{a}{2}} = ∫_{S_{r−1}} dσ_{r−1}(v)⟨p_{λ}(R^{(v)})f_{v}, g_{v}⟩_{H_{λ}(D)},

where p_{λ}(x) := Γ(x + λ) Γ(λ)^{−1} Γ(x + 1)^{−1} q(x), and q(x) is the polynomial of degree r − 1 with positive rational coefficients as in Proposition 5.2.

Case 2: Cartan domains of type IV. Let D ⊂ C^{d}, d ≥ 3, be the Cartan domain of rank r = 2 (see Examples 4.1 and 4.2), and fix a frame {e_{1}, e_{2}}. Since a = d − 2, (5.13) becomes

∑_{ℓ=0}^{m} \binom{m}{ℓ}^{2} J_{m,ℓ} q_{ℓ} = \frac{m!}{(\frac{a}{2} − 1)_{m}} , m ≥ 0,  (5.25)

where for 0 ≤ ℓ ≤ m

J_{m,ℓ} = ∫_{K} |(k(e_{1}), e_{1})|^{2ℓ}|(k(e_{2}), e_{1})|^{2(m−ℓ)} dk.

Without computing the numbers J_{m,ℓ} explicitly we show that

J_{m,ℓ} = J_{m,m−ℓ}, 0 ≤ ℓ ≤ m.  (5.26)

Indeed, let k′ ∈ K satisfy k′(e_{1}) = e_{2} and k′(e_{2}) = e_{1}. Then, by invariance of the Haar measure dk,

J_{m,ℓ} = ∫_{K} |(k(k′(e_{1})), e_{1})|^{2ℓ}|(k(k′(e_{2})), e_{1})|^{2(m−ℓ)} dk

= ∫_{K} |(k(e_{2}), e_{1})|^{2ℓ}|(k(e_{1}), e_{1})|^{2(m−ℓ)} dk = J_{m,m−ℓ}.

Theorem 5.2 The polynomial

q(x) = \frac{4}{a} x + 1 = \frac{4}{d − 2} x + 1

satisfies q(ℓ) = q_{ℓ} for every ℓ ≥ 0, where {q_{ℓ}}_{ℓ=0}^{∞} is the unique solution of (5.25). Therefore, for every λ > 0 and every f, g ∈ H\frac{a}{2} (D),

⟨f, g⟩_{\frac{a}{2}} = ∫_{S_{1}}⟨p_{λ}(R^{(v)})f_{v}, g_{v}⟩_{H_{λ}(Dv)} dσ_{1}(v),

where the functions p_{λ}, 0 < λ < ∞, are given by

p_{λ}(x) = \frac{Γ(x + λ)}{Γ(λ) Γ(x + 1)} (\frac{4}{a} x + 1).  (5.27)

In particular, for λ = 1, 2, … p_{λ} is a polynomial of degree λ with positive rational coefficients.

Proof: We claim first that

∑_{ℓ=0}^{m} \binom{m}{ℓ}^{2} J_{m,ℓ} = \frac{m!}{(\frac{d}{2})_{m}} , m ≥ 0.  (5.28)

Indeed, it is clear that

∑_{ℓ=0}^{m} \binom{m}{ℓ}^{2} J_{m,ℓ} = ∫_{K} (∫_{T} |(k(e^{it}e_{1} + e_{2}), e_{1})^{m}|^{2} \frac{dt}{2π} ) dk.

Interchanging the order of integration and using the transitivity of K on the frames, we get

∑_{ℓ=0}^{m} \binom{m}{ℓ}^{2} J_{m,ℓ} = ∫_{K} |(k(e), e_{1})^{m}|^{2} dk = ∥N_{1}^{m}∥2_{H^{2}(D)} = \frac{m!}{(\frac{d}{2})_{m}} , m ≥ 0,

by using the well-known fact that ∥(·, z)^{m}∥_{F}^{2} = m!(z, z)^{m} for every z ∈ Z and m ≥ 0. Using (5.26) and (5.28) we see that

∑_{ℓ=0}^{m} ℓ \binom{m}{ℓ}^{2} J_{m,ℓ} = ∑_{ℓ=0}^{m}(m − ℓ) \binom{m}{m − ℓ}^{2} J_{m,m−ℓ}

= ∑_{ℓ=0}^{m}(m − ℓ) \binom{m}{ℓ}^{2} J_{m,ℓ} = \frac{m · m!}{(\frac{d}{2})_{m}} − ∑_{ℓ=0}^{m} ℓ \binom{m}{ℓ}^{2} J_{m,ℓ}.

Thus

∑_{ℓ=0}^{m} ℓ \binom{m}{ℓ}^{2} J_{m,ℓ} = \frac{m · m!}{2(\frac{d}{2})_{m}}, m ≥ 0.  (5.29)

Combining (5.28) and (5.29), and using the fact that (\frac{d}{2})_{m} = (\frac{a}{2})_{m} \frac{( \frac{a}{2} +m)}{\frac{a}{2}}, we get for m ≥ 0

∑_{ℓ=0}^{m} \binom{m}{ℓ}^{2} J_{m,ℓ} (\frac{4}{a} ℓ + 1) = \frac{4}{a} \frac{m · m!}{2(\frac{d}{2})_{m}} + \frac{m!}{(\frac{d}{2})_{m}} = \frac{m!}{(\frac{a}{2})_{m}} .

In view of (5.14), this completes the proof.

—large The computation of ⟨f, g⟩_{p−1} by integration on ∂_{1}D

We conclude this section with the derivation of a formula for ⟨f, g⟩_{p−1} via integration on ∂_{1}D.

Proposition 5.3 Let F ∈ C(\overline{D}). Then

lim_{λ↓p−1} ∫_{D} F (z) dμ_{λ}(z) = ∫_{S_{1}} (∫_{D_{v}} F_{v}(w) dμ_{v,p−1}(w)) dσ_{1}(v),  (5.30)

where the measures μ_{v,p−1} are defined by (5.4).

Proof: Using (1.13) and (1.14) as well as (1.22), (1.23), and (1.9), we can write

∫_{D} F (z) dμ_{λ}(z) = c_{0} c(λ) ∫_{R_{+}^{r}} F^{#}(t)w(t)^{a} ∏_{j=1}^{r}(1 − t_{j})^{a} dt

= c_{0} c(λ) ∫_{0}^{1} ψ(t_{1}) (1 − t_{1})^{λ−p} dt_{1},

where

ψ(t_{1}) := ∫_{[0,t_{1})_{+}^{r−1}} F^{#}(t_{1}, t′) ∏_{1≤i<j≤r}(t_{i} − t_{j})^{a} ∏_{j=2}^{r}(1 − t_{j})^{λ−p} dt′,

and c(λ) = c_{D}(λ) is given by (1.22). Here t′ := (t_{2}, t_{3}, …, t_{r}), dt′ := dt_{2} dt_{3} … dt_{r}, and [0, t_{1})_{+}^{r−1} := {t′ ∈ R^{r−1}; t_{2} > t_{3} > … > t_{r} > 0}. Since ψ ∈ C([0, 1]), we have

lim_{ϵ↓0} (ϵ ∫_{0}^{1} ψ(t)(1 − t)^{ϵ−1} dt) = ψ(1). Since lim_{λ↓p−1} Γ(λ − p + 1) (λ − p + 1) = 1 and c(p − 1) = 0, we get

lim_{λ↓p−1} ∫_{D} F (z) dμ_{λ}(z) = b ψ(1)

= b ∫_{[0,1)_{+}^{r−1}} F^{#}(1, t′) ∏_{2≤i<j≤r}(t_{i} − t_{j})^{a} ∏_{j=2}^{r}(1 − t_{j})^{a−1} dt′,

where b := c_{0} c′(p − 1). Using the definitions (5.1), (5.3) and the fact that for v ∈ S_{1} the genus of D_{v} is p − a, we have (with the obvious meaning of the constants)

∫_{S_{1}} (∫_{D_{v}} F_{v}(w) dμ_{v,p−1}(w)) dσ_{1}(v)

= c_{De_{1}} (p − 1) ∫_{K} (∫_{D_{e_{1}}} F_{k(e_{1})}(k(ξ)) h(k(ξ), k(ξ))^{a−1} dm(k(ξ))) dk

= c_{De_{1}} (p − 1) c_{0}(D_{e_{1}})

× ∫_{K} (∫_{[0,1)_{+}^{r−1}} (∫_{K_{e_{1}}} F (k(e_{1} + k′(∑_{j=2}^{r} tj^{\frac{1}{2}} e_{j} dk′))) w(t′)^{a} ∏_{j=2}^{r}(1 − t_{j})^{a−1} dt′) dk,

where K_{e_{1}} := {k ∈ K; k(e_{1}) = e_{1}} and w(t′) := ∏_{2≤i<j≤r}(t_{i} − t_{j})^{a}. Interchanging the order of integration, and using the fact that k′(e_{1}) = e_{1} and the invariance of the Haar measure dk, we see that the last expression is equal to

c_{D_{e_{1}}} (p − 1) c_{0}(D_{e_{1}}) ∫_{[0,1)_{+}^{r−1}} F^{#}(1, t′) w(t′)^{a} ∏_{j=2}^{r}(1 − t_{j})^{a−1} dt′.

Comparing the computations for the left and right hand sides of (5.30), we see they are proportional. Taking F (z) ≡ 1, the proportionality constant is 1.

Corollary 5.3 The constant c_{0} = c_{0}(D) in the formula (1.12) is

c_{0}(D) = \frac{π^{d} Γ(\frac{a}{2})^{r−2}}{(∏_{ℓ=1}^{r−1} ℓ \frac{a}{2}) Γ(r \frac{a}{2}) ∏_{ℓ=2}^{r−1} Γ(ℓ \frac{a}{2})^{2}} .

Proof: Define v_{r} = 0, v_{ℓ} := e_{1} + … + e_{r−ℓ}, ℓ = 1, 2, …, r − 1, and γ_{ℓ} := c_{0}(D_{v_{ℓ}}). Then the above proof (with r replaced by ℓ) yields

\frac{γ_{ℓ}}{γ_{ℓ−1}} = \frac{c_{D_{v_{ℓ+1}}} ((ℓ − 1)a + 1)}{c′_{D_{v_{ℓ}}} ((ℓ − 1)a + 1)} = \frac{π^{(ℓ−1)a+1} Γ(\frac{a}{2})}{Γ((ℓ − 1)\frac{a}{2} + 1) Γ(\frac{ra}{2})}

for ℓ = 2, 3, …, r. Therefore, using the easily proved fact that γ_{1} = π, we get

c_{0}(D) = γ_{r} = \frac{γ_{r}}{γ_{r−1}} \frac{γ_{r−1}}{γ_{r−2}} ⋯ \frac{γ_{2}}{γ_{1}} γ_{1}

= π ∏_{ℓ=2}^{r} \frac{π^{(ℓ−1)a+1} Γ(\frac{a}{2})}{Γ((ℓ − 1)\frac{a}{2} + 1) Γ(\frac{ra}{2})} = \frac{π^{d} Γ(\frac{a}{2})^{r−2}}{(∏_{ℓ=1}^{r−1} ℓ \frac{a}{2}) Γ(r \frac{a}{2}) ∏_{ℓ=2}^{r−1} Γ(ℓ \frac{a}{2})^{2}} .

Proposition 5.3 allows the computation of the inner products ⟨f, g⟩_{p−1} by integrating over the boundary orbit ∂_{1}(D) = G(e_{1}) of G.

Theorem 5.3 Let f, g ∈ H_{p−1}. Then

⟨f, g⟩_{p−1} = ∫_{S_{1}} (∫_{D_{v}} f_{v}(w) \overline{g_{v}(w)} dμ_{v,p−1}(w)) dσ_{1}(v),  (5.31)

Proof: It is enough to establish (5.31) for polynomials f and g, and this case follows from Proposition 5.3 with F (z) = f(z) \overline{g(z)}.

6 Integral formulas in the context of the associated Siegel domain

In what follows we shall use the fact [FK2] that D is holomorphically equivalent to the tube domain

T (Ω) := X + iΩ

via the Cayley transform c : D → T (Ω), defined by c(z) := i(e + z)(e − z)^{−1}. For λ ∈ W (D) the operator V^{(λ)}f := (f ∘c^{−1})(Jc^{−1})^{λ/p} maps the space H_{λ} = H_{λ}(D) isometrically onto a Hilbert space of analytic functions on T (Ω), denoted by H_{λ}(T (Ω)). We will denote ⟨f, g⟩_{H_{λ}(T (Ω))} simply by ⟨f, g⟩_{λ}. It is known that the reproducing kernel of H_{λ}(T (Ω)) is

K_{λ}(z, w) = (N(\frac{z − w^{∗}}{i}))^{−λ} , z, w ∈ T (Ω).  (6.1)

Recall that for λ > p − 1 we have H_{λ}(D) = L_{a}^{2}(D, μ_{λ}), where μ_{λ} is the measure on D defined via (1.23). Using the facts that h(c^{−1}(w), c^{−1}(w)) = 4^{r}|N(w + ie)|^{−2}N(v) and J(c^{−1})(w) = (2i)^{d}N(w + ie)^{−p}, ∀w ∈ T (Ω), we get by a change of variables that

H_{λ}(T (Ω)) = L_{a}^{2}(T (Ω), ν_{λ}) = L^{2}(T (Ω), ν_{λ}) ∩ {analytic functions},

where

dν_{λ}(z) := c(λ)dx N(2y)^{λ−p} dy, z = x + iy, x ∈ X, y ∈ Ω,  (6.2)

and c(λ) is defined by (1.22). In this case V^{(λ)} extends to an isometry of L^{2}(D, μ_{λ}) onto L^{2}(T (Ω), ν_{λ}).

In this section we obtain integral formulas for the invariant inner products in the spaces H_{λ}(T (Ω)). Using the isometry V^{(λ)} : H_{λ}(D) → H_{λ}(T (Ω)) one obtains integral formulas for the inner products in the spaces H_{λ}(D). Our results are essentially implicitly contained in [VR], where the authors determine the Wallach set for Siegel domains of type II, using Lie and Fourier theoretical methods. The Jordantheoretical formalism allows us to formulate our results in a simpler way, avoiding the Lie-theoretical details. Since the Fourier-theoretical arguments in our proofs are contained in[VR], we omit all proofs.

For λ > (r − 1)\frac{a}{2} consider the measure σ_{λ} on Ω defined by dσ_{λ}(v) := β_{λ} N(v)^{\frac{d}{r} −λ} dv where β_{λ} := (2π)^{−2d}Γ_{Ω}(λ).

Proposition 6.1 Let λ > (r − 1)\frac{a}{2} and let f be a holomorphic function on T (Ω). Then the following conditions are equivalent:

(i) f ∈ H_{λ}(T (Ω));

(ii) The boundary values f(x) := lim_{Ω∋y→0} f(x + iy) exist almost everywhere on X, and the Fourier transform f̂ of f(x) is supported in \overline{Ω} and belongs to L^{2}(Ω, σ_{λ}).

Moreover, the map f ↦ f̂ is an isometry of H_{λ}(T (Ω)) onto L_{a}^{2}(Ω, σ_{λ}).

Proposition 6.1 yields the following result.

Theorem 6.1 Let λ > (r − 1)\frac{a}{2} and let f, g ∈ H_{λ}(T (Ω)). Then

⟨f, g⟩_{H_{λ}(T (Ω))} = ⟨f̂, ĝ⟩_{L^{2}(Ω,σ_{λ})} = \frac{Γ_{Ω}(λ)}{(2π)^{2d}} ∫_{Ω} f̂(t)\overline{ĝ(t)} N(t)^{\frac{d}{r} −λ} dt.

The group GL(Ω) := {φ ∈ GL(X); φ(Ω) = Ω} acts transitively on Ω . It acts also on the boundary ∂Ω, but this action is not transitive. The orbits of GL(Ω) on ∂Ω are exactly the r disjoint sets

∂_{k}Ω := GL(Ω)(e_{k}) = \{x ∈ \overline{Ω}; rank(x) = k\}, k = 0, 1, …, r − 1,

where {c_{1}, …, c_{r}} is a frame of pairwise orthogonal primitive idempotents, e_{0} := 0, and e_{k} := ∑_{j=1}^{k} c_{j}, k = 1, 2, …, r − 1. Consider the Peirce decomposition X_{ν} = X_{ν}(e_{k}) = \{x ∈ X; e_{k}x = νx\}, ν = 0, \frac{1}{2}, 1. Let Ω(k) be the symmetric cone of X_{1}(e_{k}), and let Γ_{Ω(k)} be the associated Gamma function. Let GL(Ω) = LN_{Ω}A be the Iwasawa decomposition. Then N_{Ω}A(e_{k}) = {x ∈ ∂_{k}Ω; N_{k}(x) > 0} is an open dense subset of ∂_{k}Ω, and every x ∈ N_{Ω}A(e_{k}) has a Peirce decomposition of the form x = x_{1} + x_{\frac{1}{2}} + 2(e − e_{k})(x_{\frac{1}{2}} (x_{\frac{1}{2}} x_{1}^{−1})) [La2]. Let us define a measure ν_{k} on ∂_{k}Ω with support N_{Ω}A(e_{k}) by

dν_{k}(x) := N_{k}(x_{1})^{k \frac{a}{2} − \frac{d}{r}} dx_{1} dx_{\frac{1}{2}} .  (6.3)

It has the following fundamental properties (see[VR] and [La2]).

Theorem 6.2 Let 1 ≤ k ≤ r − 1. Then the measure ν_{k} satisfies

∫_{N_{Ω}A(e_{k})} e^{−⟨y,x⟩} dν_{k}(x) = γ_{k} N(y)^{−k \frac{a}{2}} , ∀y ∈ Ω,  (6.4)

where γ_{k} := (2π)^{k(r−k) \frac{a}{2}} Γ_{Ω(k)}(k \frac{a}{2}), and

dν_{k}(φ(x)) = Det(φ)^{(k \frac{a}{2} )/ \frac{d}{r}} dν_{k}(x), ∀φ ∈ GL(Ω).  (6.5)

Since Ω is a set of uniqueness for analytic functions on T (Ω), (6.4) implies by analytic continuation

∫_{N_{Ω}A(e_{k})} e^{−⟨ \frac{z−w^{∗}}{i} ,x⟩} dν_{k}(x) = γ_{k} 2^{−k \frac{a}{2}} (N(\frac{z − w^{∗}}{i}))^{−k \frac{a}{2}} , ∀z, w ∈ T (Ω).

Thus (N(\frac{z−w^{∗}}{i}))^{−k \frac{a}{2}} is positive definite, and so k \frac{a}{2} is in the Wallach set W (D) = W (T (Ω)).

By complexification, GL(Ω) is realized as a subgroup of Aut(T (Ω)) which normalizes the translations τ_{x}(z) := z + x, i.e.

φ τ_{x} φ^{−1} = τ_{φ(x)}, ∀x ∈ X, ∀φ ∈ GL(Ω).

Let G ⊂ Aut(T (Ω)) be the semi-direct product of X and GL(Ω). It acts transitively on T (Ω). Let N ⊂ G be the semi-direct product of X and N_{Ω}. Then the Iwasawa decomposition of Aut(T (Ω))_{0} is KAN. For

α_{k} = \frac{d}{r} + k \frac{a}{2}, k = 0, 1, 2, …, r − 1

let H_{α_{k}} = H_{α_{k}}(T (Ω)) be the Hilbert space of analytic functions on T (Ω) whose

reproducing kernel is K_{α_{k}}(z, w) := (N(\frac{z−w^{∗}}{i}))^{−α_{k}}. Note that α_{r−1} = p − 1 and for

k = 0 we have α_{0} = \frac{d}{r} and ν_{0} = δ_{0}, the Dirac measure at 0.

Theorem 6.3 For k = 0, 1, …, r − 1 H_{α_{k}}(T (Ω)) consists of all analytic functions f on T (Ω) for which

∥f∥_{H_{α_{k}} (T (Ω))}^{2} := β_{k} sup_{t∈Ω} ∫_{N_{Ω}A(e_{k})} (∫_{X} |f(x + i(y + t))|^{2} dx) dν_{k}(y)  (6.6)

is finite, where

β_{k} = \frac{Γ_{Ω}(α_{k})2^{rk \frac{a}{2}}}{Γ_{Ω(k)}(k \frac{a}{2})} (2π)^{−(d+k(r−k) \frac{a}{2} )}.

Moreover, for every f, g ∈ H_{α_{k}}(T (Ω)),

⟨f, g⟩_{α_{k}} = β_{k} lim_{Ω∋t→0} ∫_{N_{Ω}A(e_{k})} (∫_{X} f(x + i(y + t)) \overline{g(x + i(y + t))} dx) dν_{k}(y).

References

[A1] J. Arazy, A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains, Contemp. Math. 185 (1995), 7-65.

[A2] J. Arazy, Realization of the invariant inner products on the highest quotients of the composition series, Arkiv Mat. 30 (1992), 1-24.

[A3] J. Arazy, Integral formulas for the invariant inner products in spaces of analytic functions in the unit ball, in Function Spaces, Lecture Notes in Pure and Applied Mathematics Vol.136, Marcel Dekker (1992), 9-23.

[A4] J. Arazy, Boundedness and compactness of generalized Hankel operators on bounded symmetric domains, J. Funct. Anal. 137 (1996), 97-151.

[AF] J. Arazy and S.D. Fisher, Invariant Hilbert spaces of analytic functions on bounded symmetric domains, In: Topics in Operator Theory, E.D. Hellinger Memorial Volume (Editors: L. de-Branges, I. Gohberg and J. Rovnyak), Operator Theory, Advances and Applications, 48 (1990), 67-91.

[Be] F. Berezin, Quantization in complex symmetric spaces, Math. USSR-Izv. 9 (1975), 341-379.

[FK1] J. Faraut and A. Korányi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), 64-89.

[FK2] J. Faraut and A. Korányi, Analysis on Symmetric Cones, Clarendon Press, Oxford (1994).

[Gi] S. Gindikin, Analysis on homogeneous domains, Russ. Math. Surv. 19 (1964), 1-89.

[Go] L. Garding, The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals, Ann. Math. 48 (1947), 785-826.

[He] S. Helgason, Differential Geometry, Lie Groups, and Symmetric spaces, Academic Press (1978).

[Hu] L.K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Translations of Mathematical Monographs, Amer. Math. Soc. (1963).

[La1] M. Lassalle, Noyau de Szegö, K-types et algèbres de Jordan, C. R. Acad. Sci. Paris 303 (1986), 1-4.

[La2] M. Lassalle, Algèbres de Jordan et ensemble de Wallach, Invent. Math. 89 (1087), 375-393.

[Lo] O. Loos, Bounded Symmetric Domains and Jordan Pairs, University of California at Irvine (1977).

[O] B. Orsted, Composition series for analytic continuations of holomorphic discrete series representations of SU(n, n), Trans. Amer. Math. Soc. 260 (1980), 563-573.

[Pel] M. Peloso, Möbius invariant spaces on the unit ball, Mich. Math. J. 39 (1992), 509-536.

[Ri] M. Riesz, L’intégrale de Riemann-Liouville et le problème de Cauchy, Acta Math. 81 (1949), 1-223.

[Ru] W. Rudin, Function theory in the unit ball of C^{n}, Springer-Verlag, New York (1980).

[Sc] W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 8 (1969), 61-80.

[T] A. Terras, Harmonic analysis on symmetric spaces and applications II, Springer-Verlag (1988).

[U1] H. Upmeier, Jordan algebras and harmonic analysis on symmetric spaces, Amer. J. Math. 108 (1986), 1-25.

[U2] H. Upmeier, Jordan algebras in analysis, operator theory, and quantum mechanics, CBMS Series in Math. 67, Amer. Math. Soc., (1987).

[VR] M. Vergne and H. Rossi, Analytic continuation of the holomorphic discrete series of a semi-simple Lie group, Acta Mathematica 136 (1976), 1-59.

[Y1] Z. Yan, Invariant Differential Operators on Symmetric Cones, preprint (1991).

[Y2] Z. Yan, Invariant Differential Operators and Holomorphic Function Spaces, preprint (1991).

[Y3] Z. Yan, Duality and Differential Operators on the Bergman Space of Bounded Symmetric Domains, J. Funct. Anal. 105 (1992), 171 - 186.

[Y4] Z. Yan, Differential operators and function spaces, Contemp. Math. 142 (1993), 121-142.

Jonathan Arazy Department of Mathematics University of Haifa Haifa 31905, Israel

jarazy@mathcs2.haifa.ac.il

Harald Upmeier

Fachbereich Mathematik Universität Marburg 35032 Marburg, Germany

upmeier@mathematik.uni-marburg.de

262

Documenta Mathematica 2 (1997)