Journal of Lie Theory Volume 10 (2000) 1–0 ◯^{C}2000 Heldermann Verlag

Invariant differential operators and holomorphic function spaces

Zhimin Yan

Communicated by J. Faraut

Abstract. In this paper, we are interested in studying the algebra D(Ω) of invariant differential operators on a symmetric cone Ω. We will give some sets of generators of D(Ω) and calculate the eigenvalues of spherical functions under those generators. The explicit construction of our invariant differential operators in D(Ω) leads to introducing some differential operators on an irreducible bounded symmetric domain D in a complex vector space Z. Some interesting results are obtained about these differential operators and their applications to the study of spaces of holomorphic functions on D are given.

Introduction

It is known that the algebra of the invariant differential operators on a symmetric space of rank r is generated by a set of r algebraically independent elements. An important problem is to give such a set of generators explicitly. In the case that a symmetric space is a symmetric cone, a set of generators has been given by Nomura [14] in terms of some invariant polynomials. In part I of this paper, we shall further study the algebra D(Ω) of the invariant differential operators on a symmetric cone Ω. For each complex number λ, an invariant differential operator D_{λ} is introduced. It is shown that for any r distinct numbers λ_{1}, …, λ_{r}, D_{λ_{1}}, …, D_{λ_{r}} is a set of algebraically independent generators of D(Ω). We also introduce r ”canonical” invariant differential operators K_{1}, …, K_{r} which are constructed from some canonical invariant polynomials, then express D_{λ} in terms of K_{1}, …, K_{r} and vice versa. For any spherical function, its eigenvalues under D_{λ} and K_{j} can be computed explicitly.

In part II, we consider an irreducible bounded symmetric domain D in a complex vector space Z in the standard Harish- Chandra realization. In D there is a subdomain D_{T} which is the unit ball of the complexification of a real simple Euclidean Jordan algebra. Let G be the the identity component of Aut(D) and K the isotropy subgroup of G at 0. For each complex number λ, using results

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag obtained for tube type domains, we introduce a differential operator D_{λ}. For any distinct numbers λ_{1}, …, λ_{r}, we also prove that D_{λ_{1}}, …, D_{λ_{r}} is a set of r algebraically independent generators of the algebra of the differential operators on Z that commute with the action of K. An important feature of D_{λ} is that as a consequence of the commutativity with K, polynomials in the irreducible subspaces of Schmid’s decomposition are eigenfunctions of D_{λ}, and, moreover, the eigenvalues can be calculated explicitly. The applications of this result will be given in part III.

In [4], for every λ ∈ C, the space of holomorphic polynomials on the ambient space Z is equipped with the structure of a Harish-Chandra module, denoted by P^{(λ)}, and a composition series of P^{(λ)},

M_{0} ⊂ M_{1} ⊂ ⋯ ⊂ M_{q(λ)} = P^{(λ)}

is determined. Each quotient M_{j}/M_{j−1}, j = 0, 1, …, q(λ), (M_{−1} = 0) has a natural invariant Hermitian form. Of particular interest is the case when the quotient is unitarizable, that is, the corresponding Hermitian form is an inner product. In this case, one has a corresponding Hilbert space of analytic functions on which G acts unitarily. It is known that M_{j}/M_{j−1} is unitarizable, if and only if j = 0 or j = q(λ) with an appropriate λ. In part III, we shall express the invariant inner products in terms of integrals on D when the highest quotient is unitarizable. We shall also characterize M_{0}^{λ}, when it is unitarizable, by a corresponding canonical differential operator K_{j}. The space of harmonic polynomials in the sense of Upmeier [17], which is equal to M_{0}^{λ} for a particular value of λ, is described in terms of a single differential operator in [17]. Our result generalizes that of Upmeier. We shall describe those Hilbert spaces of holomorphic functions corresponing to the cases that the quotients M_{j}/M_{j−1} are unitarizable and obtain a generalization of the classical Dirichlet space. Finally, we characterize the dual and predual of the Bergman space L^{1}(D) ∩ H(D) which generalize the results in [19] to the case of all bounded symmetric domains.

After the first version of this paper was finished, the author noticed a paper of R. Howe and T. Umeda [8], which is relevant to §1 and §2 in this paper. In particular, our Theorem 1.11 is motivated by a remark in [8].

Preprints of this paper were distributed in 1992. Some of its results were then incorporated in the book [5]. Following the referee’s recommendation, the present paper is now shorter than the original preprint. We have omitted the proofs that appear also in [5].

1. Invariant Differential operators on Symmetric Cones

§1.1. Background and Notation of Symmetric Cones and Jordan Algebra Let V be a real simple Euclidean Jordan algebra, Ω the symmetric cone in V , i.e., the interior of the set of all squares in V . It is known that every irreducible symmetric cone can be obtained in this way. We fix a complete system of orthogonal primitive idempotents {c_{1}, …, c_{r}} where r is the rank of V , then the identity element e is equal to c_{1} + ⋯ + c_{r}. We denote by G(Ω) the identity component of the subgroup of GL(V) which preserves Ω, L the isotropy subgroup of G(Ω) at e. Then every element x in V can be written as

x = l. ∑_{i=1}^{r} t_{i}c_{i}, t_{i} ∈ R, l ∈ L,  (1)

and x ∈ Ω if and only if t_{i} > 0, i = 1, …, r.

There is a determinant polynomial Δ(x) and a trace polynomial tr(x) on V such that if x is written as in (1), then

Δ(x) = ∏_{i=1}^{r} t_{i},

and

tr(x) = ∑_{i=1}^{r} t_{i}.

For x ∈ V , one defines the multiplication operator L(x) : V → V by

L(x)y = xy, ∀y ∈ V

and the quadratic representation P (x) : V → V by

P (x) = 2L(x)^{2} − L(x^{2}).

For an idempotent c and k ∈ R, let

V (c, k) = {x ∈ V | L(c)x = k x},

V has the Peirce decomposition

V = ∑_{1≤i≤j≤r} V_{ij},

where V_{jj} = V (c_{j}, 1), and, for i ≠ j, V_{ij} = V (c_{i}, 1/2) ∩ V (c_{j}, 1/2). For all i ≠ j, all V_{ij} have the same dimension, which will be denoted by a.

Now the subspaces V^{(k)} = V (c_{1} + ⋯ + c_{k}, 1)(1 ≤ k ≤ r) are subalgebras of V . Let P_{k} be the orthogonal projection onto V^{(k)}. The principal minor Δ_{k}(x) is the polynomial defined on V by

Δ_{k}(x) = Δ^{(k)}(P_{k}x),

where Δ^{(k)} is the determinant polynomial with respect to the algebra V^{(k)}.

For an r− tuple of integers m = (m_{1}, …, m_{r}) with m_{1} ≥ ⋯ m_{r} ≥ 0, abbreviated as m ≥ 0, one defines a polynomial Δ_{m}(x) on V by

Δ_{m}(x) = Δ_{1}^{m_{1}−m_{2}}(x)Δ_{2}^{m_{2}−m_{3}}(x) ⋯ Δ_{r−1}^{m_{r−1}−m_{r}}(x)Δ_{r}^{m_{r}}(x).  (2)

One observes that if x ∈ Ω, then for any r−tuple of complex numbers s = (s_{1}, …, s_{r}), replacing m by s, (2) defines a function Δ_{s}(x) on Ω.

For a linear transformation X on V , ^{t}X will be its transpose with respect to the inner product ( , )_{V} , where ( , )_{V} is induced from the trace form on V , i.e., (x, y)_{V} = tr(xy).

Let G_{Ω} be the Lie algebra of G(Ω), and L_{Ω} the Lie algebra of L. One has the following Cartan decomposition corresponding to the involution θ : G_{Ω} → G_{Ω}, X → −^{t}X,

G_{Ω} = L_{Ω} + P_{Ω}.  (3)

Then

A_{Ω} = {∑_{i=1}^{r} t_{i}L(c_{i})| t_{i} ∈ R, i = 1, …, r}

is a maximal abelian subspace of P_{Ω}. One has

(exp A_{Ω}).e = {∑_{i=1}^{r} u_{i}c_{i}| u_{i} > 0, i = 1, …, r}.  (4)

When there is no confusion caused, we will identify (exp A_{Ω}).e with the subgroup A_{Ω} = exp A_{Ω} of G(Ω) or the subalgebra A_{Ω} of G_{Ω}. The following two coordinate systems on (exp A_{Ω}).e will be used in our later calculations.

(I) φ : (exp A_{Ω}).e → R^{r}, φ(∑_{i=1}^{r} u_{i}c_{i}) = (u_{1}, …, u_{r}),

(II) ψ : (exp A_{Ω}).e → R^{r}, ψ(∑_{i=1}^{r} u_{i}c_{i}) = (y_{1}, …, y_{r}),

where y_{i} = log u_{i}, i = 1, …, r.

According to our convention, (I) and (II) will also be used as coordinate systems of A_{Ω} and A_{Ω}.

We define linear functionals α_{ij}, 1 ≤ i, j ≤ r, i ≠ j on A_{Ω} by

α_{ij}(∑_{i=1}^{r} t_{i}L(c_{i})) = \frac{1}{2}(t_{j} − t_{i}),

then α_{ij}, 1 ≤ i, j ≤ r, i ≠ j consist of all the restricted roots of the pair (G_{Ω}, L_{Ω}).

Let N = exp N where N is the direct sum of all the root spaces corresponding to α_{ij} with 1 ≤ i < j ≤ r. Then G(Ω) has the following Iwasawa decomposition

G(Ω) = LA_{Ω}N.  (5)
Let ρ = \frac{1}{2}a ∑_{1≤i<j≤r} α_{ij}, and e^{ρ} be the function defined on A_{Ω} by
e^{ρ}(u) = e^{ρ(log u)}

for u ∈ A_{Ω}, where log is the inverse of exp : A_{Ω} → A_{Ω}.

§1.2. Differential Operators Associated with Polynomials In this section, most of our notation is from [5],[7] and [14].

For a real vector space E of dimension n with the inner product ( |), we will denote by P (E) the space of all complex-valued polynomials on E. For every polynomial p ∈ P (E) we define the unique linear differential operator p(\frac{∂}{∂x}) by

p(\frac{∂}{∂x})e^{(x|y)} = p(y)e^{(x|y)}, ∀y ∈ E.  (6)

The uniqueness of p(\frac{∂}{∂x}) follows immediately from (6); its existence can be seen by taking an orthonormal basis of E, expressing p in terms of the coordinates as p(x_{1}, …, x_{n}) and then formally replacing x_{j} by \frac{∂}{∂x_{j}} , (1 ≤ j ≤ n).

For a polynomial p(x, y) on E × E, similarly, we can define the unique differential operator p(x, \frac{∂}{∂x}) by the following equation

p(x, \frac{∂}{∂x})e^{(x|y)} = p(x, y)e^{(x|y)}.  (7)

We denote by W the Weyl group corresponding to the root system {α_{ij}}, then W is isomorphic to the full permutation group S_{r}. Let D(Ω) be the algebra of the invariant differential operators on Ω, D_{W} (A) the algebra of W −invariant differential operators on A_{Ω} with constant coefficients, and I_{W} (A_{Ω}) the algebra of W −invariant polynomials on A_{Ω}. See [7, Ch.5]. In our case, I_{W} (A_{Ω}) is the algebra of symmetric polynomials in r variables.

We write α for (α_{1}, …, α_{r}), u^{α} for u_{1}^{α_{1}} ⋯ u_{r}^{α_{r}} and (\frac{∂}{∂u})^{α} for (\frac{∂}{∂u_{1}} )^{α_{1}} ⋯ (\frac{∂}{∂u_{r}})^{α_{r}}.

In coordinate system I,

D_{W} (A_{Ω}) = \{p(u\frac{∂}{∂u})|p ∈ I_{W} (A_{Ω})\},  (8)

where p(u\frac{∂}{∂u}) = ∑_{α} b_{α}u^{α}(\frac{∂}{∂u})^{α} if p(y) = ∑_{α} b_{α}y^{α}.

In coordinate system II,

D_{W} (A_{Ω}) = \{p(\frac{∂}{∂y})|p ∈ I_{W} (A_{Ω})\}.  (9)

In the following, we study the structure of D(Ω) as a vector space. The Fischer inner product on P (V) is defined by

(p, q)_{F} = (p(\frac{∂}{∂x})q̄)(0),

for p, q ∈ P (V). There is a natural representation π of the group G(Ω) defined on P (V) by

(π(g)p)(x) = p(g^{−1}.x)  (10)

for g ∈ G(Ω), p ∈ P (V). The following is known, e.g. see [5].

Theorem I.1. P (V) is the orthogonal direct sum of the spaces P_{m}(V) (m ≥ 0) which are mutually inequivalent irreducible representation spaces of G(Ω). Moreover,

P_{m}(V) = span{π(g)Δ_{m} : g ∈ G(Ω)}.

For each m ≥ 0, there is a unique L-invariant polynomial φ_{m} in P_{m}(V) which is defined by

φ_{m}(x) = ∫_{L} Δ_{m}(l.x)dl.

Under the Fischer inner product ( , )_{F} every P_{m}(V) is a Hilbert space and has a reproducing kernel K^{m}(x, y) = K_{y}^{m}(x), that is,

p(y) = (p, K_{y}^{m})_{F} ∀p ∈ P_{m}(V).

If {ψ_{m}^{(i)}(x), i = 1, …, d_{m}} is an orthonormal basis of P_{m}(V), where d_{m} is the dimension of P_{m}(V), then

K^{m}(x, y) = ∑_{i=1}^{d_{m}} ψ_{m}^{(i)}(x)\overline{ψ_{m}^{(i)}(y)}.

It is known, e.g.see [5], that

K^{m}(g.x,^{t}g^{−1}.y) = K^{m}(x, y)  (11)

for all g ∈ G(Ω), and

K^{m}(x, y) = K^{m}(y, x).  (12)

Let P^{G(Ω)} be the subspace of P (V × V) spanned by {K^{m}(x, y), m ≥ 0} and

P (V)^{L} = {p ∈ P (V)|π(l)p = p, ∀l ∈ L}.

Then the spherical polynomials {φ_{m}}, m ≥ 0 form a basis of P (V)^{L}.

The following can be proved easily by using Proposition 14.1.1 in [5] or directly by using some ideas from [14].

Proposition 1.2. \{K^{m}(x, \frac{∂}{∂x}), m ≥ 0\} is a basis of the vector space D(Ω).

When we say that a function F (x, y) defined on Ω × V is polynomial in y, we mean that F (x, y) can be expanded as ∑_{α} a_{α}(x)y^{α} with only finitely many nonzero terms.

Remark 1. Every linear differential operator D on Ω defines a function F_{D}(x, y) on Ω × V , which is polynomial in y, by the following equation

D_{x}e^{(x|y)} = F_{D}(x, y)e^{(x|y)}.  (13)

Conversely, a function F (x, y) on Ω × V which is polynomial in y, determines a unique differential operator F (x, \frac{∂}{∂x}) by (13).

Remark 2. It follows from the proposition that every D ∈ D(Ω) can be extended to a differential operator on V with polynomial coefficients.

Among those K^{m}(x, \frac{∂}{∂x}), m ≥ 0, of particular interest to our study are K^{1_{j}}(x, \frac{∂}{∂x}), j = 1, …, r where 1_{j} is the r-tuple of integers with 1 as its first j th components and 0 the remaining components.

§1.3. Generators of D(Ω) The purpose of this section is to study D(Ω) as an algebra and give some generators of D(Ω). This section contains the main results of part I.

Recall that G(Ω) has the Iwasawa decomposition

G(Ω) = LA_{Ω}N.

Let R_{N}(D) denote the N−radial part of D ∈ D(Ω) defined as in [7,p.259]. One defines a linear mapping Γ from D(Ω) into the algebra of differential operators on A_{Ω} by

Γ(D) = e^{−ρ}R_{N}(D) ∘ e^{ρ}, ∀D ∈ D(Ω).

One has the following special case of a result of Harish-Chandra.

Theorem 1.3. Γ is an isomorphism of D(Ω) onto D_{W} (A).

Proof. See, e.g. [7, Cor. 5.19].

For λ ∈ R, we define

D_{λ} = Δ(x)^{1−λ}Δ(\frac{∂}{∂x}) ∘ Δ(x)^{λ},

then it is easy to verify that D_{λ} ∈ D(Ω).

Now we have

Theorem 1.4. The image of D_{λ} under the mapping Γ is given by

Γ(D_{λ}) = p_{λ}(\frac{∂}{∂y})

or equivalently,

Γ(D_{λ}) = p_{λ}(u\frac{∂}{∂u})

where p_{λ}(x) = ∏_{i=1}^{r}(x_{i} + λ + \frac{a}{4}(r − 1)) ∈ I_{W} (A_{Ω}).

Proof. See p.296 in [5].

Let S_{j}(x) be the j th elementary symmetric polynomial of x_{1}, …, x_{r}, then

p_{λ}(x) = ∑_{j=0}^{r}(λ + \frac{a}{4}(r − 1))^{r−j}S_{j}(x).  (14)

If λ_{1}, …, λ_{r} are all different, then it follows immediately from (14) that p_{λ_{1}}(x), …, p_{λ_{r}}(x) are algebraically independent generators of I_{W} (A_{Ω}). As a corollary of Theorem 1.3 and 1.4, now we have

Theorem 1.5. If λ_{1}, …, λ_{r} are distinct, then D_{λ_{1}}, …, D_{λ_{r}} are algebraically independent generators of D(Ω).

Now, we wish to express D_{λ} as a linear combination of K_{1}, …, K_{r}. Following Remark 1 after Proposition 1.2, we proceed to find the polynomial F_{D_{λ}} corresponding to D_{λ} as in next lemma.

Lemma 1.6. For λ ∈ R, x, y ∈ Ω,

D_{−λ}e^{(x|y)} = ∑_{j=1}^{r}(−1)^{j}( \begin{matrix}r \\ j \end{matrix} ) ∏_{l=1}^{j}(λ − \frac{l − 1}{2}a)\frac{1}{c_{1_{r−j}}} K^{1_{r−j}}(x, y)e^{(x|y)}  (15)

Proof. See p.295 in [5].

Now we have the following expansion Theorem 1.7. For any real number λ,

D_{λ} = ∑_{j=0}^{r}( \begin{matrix}r \\ j \end{matrix} ) ∏_{l=1}^{r−j}(λ + \frac{l − 1}{2}a)K_{j}  (16)

where K_{j} = \frac{1}{c_{1_{j}}} K^{1_{j}}(x, \frac{∂}{∂x}).

Proof. The theorem follows from Lemma 1.6 and Remark 1 in §1.2.

Remark 1. The above expansion has also been obtained independently by J.Arazy.

We let λ = −\frac{i−1}{2}a, i = 1, …, r in (16), we have r equations with a nonsingular coefficient matrix. Solving this system of equations, we obtain

Theorem 1.8. K_{i}, i = 1, …, r, are algebraically independent generators of D(Ω). Moreover,

K_{j} = \frac{j!}{r!}(\frac{2}{a})^{r−j} ∑_{l=1}^{r−j+1}(−1)^{l}( \begin{matrix}r − j \\ l \end{matrix} )D_{− \frac{l−1}{2} a}.

Corollary 1. For j = 1, …, r,

Γ(K_{j}) = \frac{j!}{r!}(\frac{2}{a})^{r−j} ∑_{l=1}^{r−j+1}(−1)^{l}( \begin{matrix}r − j \\ l \end{matrix} )p_{− \frac{l−1}{2} a}(\frac{∂}{∂y}).

Remark 2. Letting λ take distinct values λ_{1}, …, λ_{r} in (16), gives a system of r equations, as a consequence of Theorem 1.5 and 1.8, one obtains that the coefficient matrix of the system of r equations is nonsingular.

Finally, motivated by Theorem 1.7, for any complex number λ, we define

D_{λ} = ∑_{j=0}^{r}( \begin{matrix}r \\ j \end{matrix} ) ∏_{l=1}^{r−j}(λ + \frac{l − 1}{2}a)K_{j}.

By Proposition 7.1.6 in [5], analytic continuation, Theorem 1.1 and Schur’s lemma, we have

Theorem 1.9. For m ≥ 0 and any complex number λ,

D_{λ}p = ∏_{i=1}^{r}(m_{i} + λ + \frac{r − i}{2}a)p, ∀p ∈ P_{m}(V).  (17)

Next, we have Theorem 1.10. For m ≥ 0 and j = 1, …, r,

K_{j}p = ( \begin{matrix}r \\ j \end{matrix} )^{−1} ∑_{1≤i_{1}<···<i_{j}≤r} ∏_{l=1}^{j}(m_{i_{l}} + \frac{j − l}{2}a)p, ∀p ∈ P_{m}(V).  (18)

Proof. By Theorems 1.8 and 1.9, we have

K_{j}Δ_{m} = \frac{j!}{r!}(\frac{2}{a})^{r−j} ∑_{l=1}^{r−j+1}(−1)^{l}( \begin{matrix}r − j \\ l \end{matrix} ) ∏_{i=1}^{r}(m_{i} + \frac{r − l − i}{2}a)Δ_{m}.  (19)

Now, it is sufficient to show that

(r − j)! ∑_{1≤i_{1}<···<i_{j}≤r} ∏_{l=1}^{j}(m_{i_{l}} + \frac{j − l}{2}a) = (\frac{2}{a})^{r−j} ∑_{l=1}^{r−j+1}(−1)^{l}(\begin{matrix}r − j \\ l \end{matrix})∏_{i=1}^{r}(m_{i}+ \frac{r − l − i}{2}a).

(20)

When a = 2, Ω is the cone of positive definite Hermitian matrices. In this case, G(Ω) = GL(n, C) and L = U(n), then by (11.1.15) in [8]

K_{j}Δ_{m} = ( \begin{matrix}r \\ j \end{matrix} )^{−1} ∑_{1≤i_{1}<···<i_{j}≤r} ∏_{l=1}^{j}(m_{i_{l}} + j − l)Δ_{m}.

This and (19) implies that we have obtained a special case of (20)

(r − j)! ∑_{1≤i_{1}<···<i_{j}≤r}∏_{l=1}^{j}(m_{i_{l}} +j−l) = ∑_{l=1}^{r−j+1}(−1)^{l}( \begin{matrix}r − j \\ l \end{matrix} ) ∏_{i=1}^{r}(m_{i}+r−l−i).  (21)

However, both sides of (21) are polynomials of r variables m_{1}, …, m_{r}, and they are equal for all m ≥ 0. It follows easily that (21) holds for all m ∈ C^{r}. In particular, we have for all m ≥ 0, a ≠ 0

∑_{1≤i_{1}<i_{2}···<i_{j}≤r}∏_{l=1}^{j}(\frac{2}{a}m_{i_{l}} +j−l)= ∑_{l=1}^{r−j+1}(−1)^{l}( \begin{matrix}r − j \\ l \end{matrix} ) ∏_{i=1}^{r}(\frac{2}{a}m_{i} + r − l − i). (22)

This yields

(\frac{2}{a})^{j} ∑_{1≤i_{1}<···<i_{j}≤r} ∏_{l=1}^{j}(m_{i_{l}}+\frac{j − l}{2}a)=(\frac{2}{a})^{r} ∑_{l=1}^{r−j+1}(−1)^{l}( \begin{matrix}r − j \\ l \end{matrix} ) ∏_{i=1}^{r}(m_{i}+\frac{r − l − i}{2}a) (23)

proving (20).

It is known, for instance see [5], that every spherical function on Ω can be written as

φ_{s}(x) = ∫_{L} Δ_{s}(l.x)dl

for some s ∈ C^{r}.

One can readily see that replacing p by φ_{s} and m by s, (17) and (18) still hold. Therefore, we have Theorem 1.11. For every s ∈ C^{r}, then

(i) for any complex number λ

D_{λ}φ_{s} = ∏_{i=1}^{r}(s_{i} + λ + \frac{r − i}{2}a)φ_{s};  (24)

(ii) if j = 1, …, r,

K_{j}φ_{s} = ( \begin{matrix}r \\ j \end{matrix} )^{−1} ∑_{1≤i_{1}<···<i_{j}≤r} ∏_{l=1}^{j}(s_{i_{l}} + \frac{j − l}{2}a)φ_{s}.  (25)

2. Differential operators commuting with the action of K

In this part, we shall introduce and study some differential operators on Z which commute with the action of the isotropy group K. Their applications will be given in the last part of this paper.

§2.1. Some Background on Bounded Symmetric Domains Our notation follows that of [4]. Let G be the Lie algebra of G, and K the Lie algebra of K, then G is a simple real Lie algebra with Cartan decomposition

G = K + P.

G^{C} will be its complexification and G^{C} will be the adjoint group of G^{C}. A basis of root vectors {e_{α}} will be so chosen that τ e_{α} = −e_{−α},[e_{α}, e_{−α}] = h_{α}, [h_{α}, e_{±α}] = 2e_{±α}, where τ is the conjugation with respect to the real form K + iP. Φ^{+} will denote the set of positive non-compact roots, and setting

P^{±} = ∑_{α∈Φ^{+}} Ce_{±α},

one has

G^{C} = P^{−} + K^{C} + P^{+}.

Define a Hermitian inner product ( |) on P^{+} by (z|w) = −B(z, τw) where B is the Killing form.

It is known that in the Harish-Chandra realization, D is a bounded symmetric domain in P^{+}, and K acts on P^{+} by unitary transformations which coincide with the adjoint action. Let γ_{1}, …, γ_{r} be the strongly orthogonal roots of Harish-Chandra with the ordering γ_{1} > ⋯ > γ_{r}. We simply write

e_{j} = e_{γ_{j}}, (j = 1, …, r), e = ∑_{j=1}^{r} e_{j}.

The Cayley transform is defined by c = exp i(\frac{π}{4})(e − τe). We write ^{c}G for the Lie algebra of cGc^{−1} and G_{T} for the fixed point set of G under Ad(c^{4}). Let K_{T} , P_{1}, P_{1}^{+} and P_{1}^{−} denote the intersections of K, P, P^{+}, P^{−} with G_{T}^{C} respectively, then one has the corresponding decompositions G_{T} = K_{T} + P_{1}, G_{T}^{C} = P_{1}^{−} + K_{T}^{C} + P_{1}^{+}. Ad(c^{2}) is the Cartan involution of K_{T}, the corresponding decomposition is K_{T} = L_{T} + Q_{1}. Let K_{T}^{∗} = K_{T} + iQ_{1} be its noncompact dual.

Now N_{1}^{+} = ^{c}G ∩ P_{1}^{+} is a real form of P_{1}^{+}. In particular, N_{1}^{+} has the structure of a real simple Euclidean Jordan algebra as described in [11], e coincides with the identity element of the Jordan algebra N_{1}^{+} and e_{1}, …, e_{r} form of a complete system of orthogonal promitive idempotents. P_{1}^{+} becomes a complex Jordan algebra. The operator D(w, \overline{w}), for w ∈ P_{1}^{+}, is defined by

D(w, \overline{w}) = L(w \overline{w}) + [L(w), L(\overline{w})],

where \overline{w} is the conjugate of w with respect to the real form N_{1}^{+}. Let ∥ D(w, \overline{w}) ∥ denote the operator norm, then the unit ball

D_{T} = \{z ∈ P_{1}^{+}| ∥ D(z, \overline{z}) ∥< 1\}

is equal to D ∩ P_{1}^{+}. D_{T} is a bounded symmetric domain in P_{1}^{+} (the ”tube type subdomain” of D).

K_{T} , K_{T}^{∗} and G_{T} will denote the analytic subgroups in G^{C}corresponding to the Lie algebras K_{T} , K_{T}^{∗} and G_{T} respectively. Then D_{T} is the standard realization of G_{T} /K_{T} as a bounded symmetric domain. K_{T}^{∗}.e is the symmetric cone in N_{1}^{+}, that is, the interior of the set of all squares in N_{1}^{+}.

Let H^{−} be the real span of h_{γ_{1}}, …, h_{γ_{r}}, then iH^{−} is a Cartan subalgebra of the pair (^{c}G,^{c}K), and the iH^{−}−roots of ^{c}G are ±\frac{1}{2}(γ_{j} ± γ_{k}), ±γ_{j},±\frac{1}{2}γ_{j} (1 ≤ j, k ≤ r) with respective multiplicities a, 1 and 2b. See [13].

Let P^{+j/2} be the root space in P^{+} for \frac{1}{2}γ_{j}, and P_{2}^{+}= ∑_{j} P^{+j/2}. Then P^{+} = P_{1}^{+} + P_{2}^{+}.

§2.2. Polynomials and their corresponding differential operators Let U be a complex vector space of dimension n with a Hermitian inner product ( | ) and coordinates (z_{1}, …, z_{n}), P (U) the space of holomorphic polynomials on U, and P (U × \overline{U}) the space of polynomials on U × U which are holomorphic in the first variable and antiholomorphic in the second variable.

We call D a holomorphic differential operator if in coordinates D can be expressed as

D = ∑_{α} A_{α}(z)(\frac{∂}{∂z})^{α},

where α = (α_{1}, …, α_{r}) and (\frac{∂}{∂z})^{α} = (\frac{∂}{∂z_{1}} )^{α_{1}} ⋯ (\frac{∂}{∂z_{n}})^{α_{n}}.

Each p ∈ P (U) defines a unique holomorphic differential operator p(\frac{∂}{∂z}) by

p(\frac{∂}{∂z})e^{(z|w)} = p(w̄)e^{(z|w)} ∀z, w ∈ U.

Similarly, each p(z, w) ∈ P (U × \overline{U}) defines a unique holomorphic differential operator p(z, \frac{∂}{∂z}) by

p(z, \frac{∂}{∂z})e^{(z|w)} = p(z, w)e^{(z|w)} ∀z, w ∈ U.

Two such differential operators p(z, \frac{∂}{∂z}), q(z, \frac{∂}{∂z}) are equal if and only if

p(z, w) = q(z, w).

Let u be a unitary operator on U and let it act on functions defined on U by (u.f)(z) = f(u^{−1}.z). Then for p ∈ P (U × \overline{U}), p(z, \frac{∂}{∂z}) commutes with u if and only if

p(u.z, u.w) = p(z, w), ∀z, w ∈ U.  (1)

The Fischer inner product on P (U) is defined as follows

(p, q)_{F,U} = (p(\frac{∂}{∂z})\overline{q})(0)  (2)

where \overline{q}(z) = \overline{q(\overline{z})}.

In the following, we will apply the above discussion to the complex vector spaces P_{1}^{+} and P^{+} without further mentioning. Since N_{1}^{+} is a real form of P_{1}^{+}, a holomorphic polynomial is determined by its restriction to N_{1}^{+}, thus there is one-to-one correspondence between P (P_{1}^{+}) and P (N_{1}^{+}).

Similarly, a complex-valued polynomial p(x, y) on N_{1}^{+} × N_{1}^{+} determines a unique polynomial p(z, w) in P (P_{1}^{+} × \overline{P_{1}^{+}}).

The Fischer inner products on P (N_{1}^{+}) and P (P_{1}^{+}) are denoted respectively by ( , )_{F,N_{1}^{+}} and ( , )_{F,P_{1}^{+}} .

For simplicity, we will use the same p to denote a polynomial in P (N_{1}^{+}) or in P (N_{1}^{+} × N_{1}^{+}) and its corresponding polynomial in P (P_{1}^{+}) or P (P_{1}^{+} × \overline{P_{1}^{+}}). Under this convention, it is easy to see that

(p, q)_{F,N_{1}^{+}} = (p, q)_{F,P_{1}^{+}}  (3)

The following result is known, e.g., see [4], [16], [17].

Theorem 2.1. The space P (P^{+}) of holomorphic polynomials on P^{+} (resp. P_{1}^{+}) decomposes into irreducible subspaces under Ad(K) (resp. Ad(K_{T})) as

P (P^{+}) = ⨁_{m≥0} P_{m}(P^{+})

and

P (P_{1}^{+}) = ⨁_{m≥0} P_{m}(P_{1}^{+}).

For each m ≥ 0, Δ_{m} ∈ P_{m}(P_{1}^{+}), and its extension Δ_{m}^{E} to P^{+} is in P_{m}(P^{+}). For each m ≥ 0, restriction of polynomials maps P_{m}(P^{+}) onto P_{m}(P_{1}^{+}).

For m ≥ 0, we denote by P_{m}^{R}(N_{1}^{+}) the restriction of holomorphic polynomials in P_{m}(P_{1}^{+}) to N_{1}^{+}. Then it follows from Theorem XI.2.4 in [5] that P_{m}^{R}(N_{1}^{+}) is equal to P_{m}(N_{1}^{+}) where P_{m}(N_{1}^{+}) is the corresponding irreducible subspace occuring in the decomposition in Theorem 1.1.

Let K_{1}^{m}(x, y), for m ≥ 0, be the reproducing kernel of P_{m}(N_{1}^{+}) with respect to the Fischer inner product ( , )_{F,N_{1}^{+}} , then it follows from (3) that K_{1}^{m}(z, w) is the reproducing kernel of P_{m}(P_{1}^{+}) with respect to the Fischer inner product ( , )_{F,P_{1}^{+}} . It is this fact that relates our study in part I to the following work.

For j = 1, …, r, we define

K_{j} = \frac{1}{c_{1_{j}}} K^{1_{j}}(z, \frac{∂}{∂z})

and

K_{j}^{T} = \frac{1}{c_{1_{j}}} K_{1}^{1_{j}}(z_{1}, \frac{∂}{∂z_{1}} ).

For each m ≥ 0, let K^{m}(z, w) be the reproducing kernel of P_{m}(P^{+}). Following Theorem 1.7, we define, for any complex number λ, holomorphic differential operators D_{λ} and D_{λ}^{T} respectively by

D_{λ} = ∑_{j=0}^{r}( \begin{matrix}r \\ j \end{matrix} ) ∏_{l=1}^{r−j}(λ − \frac{r − l}{2}a)K_{j},

and

D_{λ}^{T} = ∑_{j=0}^{r}( \begin{matrix}r \\ j \end{matrix} ) ∏_{l=1}^{r−j}(λ − \frac{r − l}{2}a)K_{j}^{T} .

(We note that the parameter λ has been shifted by −\frac{r−1}{2}a.)

More generally, we define, for any positive integer k,

D_{λ}^{k} = D_{λ} ∘ D_{λ+1} ∘ ⋯ D_{λ+k−1}.

It turns out, as one may expect, that D_{λ} (resp. D_{λ}^{T} ) is diagonal on the polynomial space P (P^{+}) (resp. P (P_{1}^{+})) corresponding to the Schimid decomposition. The main purpose of this section is to calculate the eigenvalues of D_{λ} on the corresponding irreducible spaces.

The idea is as follows: roughly speaking, the action of D_{λ}^{T} on P_{m}(P_{1}^{+}) is almost the same as that of D_{λ} on P_{m}(N_{1}^{+}), then the eigenvalues of D_{λ}^{T} can be immediatedly obtained from the results in §1.2. Thus what is left is to find the relation between the eigenvalues of K^{m}(z, \frac{∂}{∂z}) and K_{1}^{m}(z_{1}, \frac{∂}{∂z_{1}} ). Fortunately, they will be seen to have the same eigenvalues on the corresponding irreducible polynomial spaces.

We write (z_{1}, z_{2}) for z ∈ P^{+} with z_{1} ∈ P_{1}^{+}, z_{2} ∈ P_{2}^{+}. For a function f on P_{1}^{+}, we define its extension f^{E} on P^{+} by

f^{E}(z_{1}, z_{2}) = f(z_{1}).

For a function F on P^{+}, we define its retriction F^{R} to P_{1}^{+} by

F^{R}(z_{1}) = F (z_{1}, 0).

We note that if p ∈ P (P_{1}^{+}), then p^{E} ∈ P (P^{+}); if p ∈ P (P^{+}), then p^{R} ∈ P (P_{1}^{+}). Similarly we define p^{R} for p ∈ P (P^{+} × \overline{P^{+}}).

It is easy to verify that for p ∈ P (P_{1}^{+}), q ∈ P (P^{+}),
(p, q^{R})_{F,P_{1}^{+}} = (p^{E}, q)_{F,P^{+}}  (4)
The relation between K^{m}(z, w) and K_{1}^{m}(z_{1}, w_{1}) is given as follows

Lemma 2.2. For each m ≥ 0,

K^{m}(z_{1}, w_{1}) = K_{1}^{m}(z_{1}, w_{1}), ∀z_{1}, w_{1} ∈ P_{1}^{+}.

Proof. It follows from (4) and the reproducing property that for all z_{1}, w_{1},

K_{1}^{m}(z_{1}, w_{1}) = ((K_{1,w_{1}}^{m})^{E}, K_{z_{1}}^{m})_{F,P^{+}} = (K_{1,w_{1}}^{m}, (K_{z_{1}}^{m})^{R})_{F,P_{1}^{+}}
= \overline{((K_{z_{1}}^{m})^{R}, K_{1,w_{1}}^{m})_{F,P_{1}^{+}}} = K^{m}(z_{1}, w_{1}).

For a differential operator D on P^{+}, following [7], we define its projection D_{P} by

(D_{P} f)(z_{1}) = (Df^{E})(z_{1})

for any function f defined on P_{1}^{+}. Writing a polynomial p(z, w) ∈ P (P^{+} × \overline{P^{+}}) in terms of coordinates (z_{1}, …, z_{n}), it is easy to see that

(p(z, \frac{∂}{∂z}))_{P} = p^{R}(z_{1}, \frac{∂}{∂z_{1}} )  (5)

As a consequence of (5) and Lemma 2.2, we have

Lemma 2.3. For all m ≥ 0,

K^{m}(z, \frac{∂}{∂z})_{P} = K_{1}^{m}(z_{1}, \frac{∂}{∂z_{1}} ).

For each m ≥ 0, we have

K^{m}(k.z, k.w) = K^{m}(z, w), ∀z, w ∈ P^{+}, ∀k ∈ K;
K_{1}^{m}(k_{1}.z_{1}, k_{1}.w_{1}) = K^{m}(z_{1}, w_{1}), ∀z_{1}, w_{1} ∈ P_{1}^{+}, ∀k_{1} ∈ K_{T} .

and then (1) implies that

K^{m}(z, \frac{∂}{∂z}) ∘ k = k ∘ K^{m}(z, \frac{∂}{∂z}), ∀k ∈ K;

K_{1}^{m}(z_{1}, \frac{∂}{∂z_{1}} ) ∘ k_{1} = k_{1} ∘ K_{1}^{m}(z_{1}, \frac{∂}{∂z_{1}} ), ∀k_{1} ∈ K_{T} .

It follows from Theorem 2.1 and Schur’s lemma that for each n ≥ 0, K^{m}(z, \frac{∂}{∂z}) (resp. K_{1}^{m}(z_{1}, \frac{∂}{∂z_{1}} )) acts on P_{n}(P^{+}) (resp.P_{n}(P_{1}^{+})) as a scalar multiple of the identity, that is,

K^{m}(z, \frac{∂}{∂z})|_{P_{n}(P^{+})} = λ_{n,m}(P^{+}) id |_{P_{n}(P^{+})}

and

K_{1}^{m}(z_{1}, \frac{∂}{∂z_{1}} )|_{P_{n}(P_{1}^{+})} = λ_{n,m}(P_{1}^{+}) id |_{P_{n}(P_{1}^{+})},

for some constants λ_{n,m}(P^{+}) and λ_{n,m}(P_{1}^{+}).

For the same reason, we have

K^{m}(x, \frac{∂}{∂x})|_{P_{n}(N_{1}^{+})} = λ_{n,m}(N_{1}^{+}) id |_{P_{n}(N_{1}^{+})},

for some constant λ_{n,m}(N_{1}^{+}).

By the relation between K^{m}(x, y) and K^{m}(z_{1}, w_{1}), it is obvious that

λ_{n,m}(P_{1}^{+}) = λ_{n,m}(N_{1}^{+})  (6)

Now we have

Theorem 2.4. For all m ≥ 0, n ≥ 0,

λ_{n,m}(P^{+}) = λ_{n,m}(P_{1}^{+}) = λ_{n,m}(N_{1}^{+}).  (7)

Proof. For each n ≥ 0, by Theorem 2.1, Δ_{n} ∈ P_{n}(P_{1}^{+}), Δ_{n}^{E} ∈ P_{n}(P^{+}) (in [4], Δ_{n}^{E} is still denoted by Δ_{n}). Since e ∈ P_{1}^{+} and Δ_{n}(e) = 1, applying Lemma 2.2, we have

λ_{n,m}(P^{+}) = λ_{n,m}(P^{+})Δ_{n}^{E}(e) = (K^{m}(z, \frac{∂}{∂z})Δ_{n}^{E})(e) = (K^{m}(z, \frac{∂}{∂z})_{P} Δ_{n})(e)
= (K_{1}^{m}(z_{1}, \frac{∂}{∂z_{1}} )Δ_{n})(e) = λ_{n,m}(P_{1}^{+})Δ_{n}(e) = λ_{n,m}(P_{1}^{+}).

This proves the theorem.

As a consequence of Theorems 1.9, 1.10 and Theorem 2.4, we have our main result in this section

Theorem 2.5. For m ≥ 0, and any p ∈ P_{m}(P^{+}),

(i)

D_{λ}^{(k)}p = μ_{m}^{(k)}(λ)p,  (8)

where μ_{m}^{(k)}(λ) = ∏_{i=1}^{r} ∏_{j=0}^{k−1}(λ + m_{i} + j − \frac{i−1}{2}a);

(ii)

K_{j}p = \frac{j!}{r!}(\frac{2}{a})^{r−j} ∑_{l=0}^{r−j}(−1)^{l}( \begin{matrix}r − j \\ l \end{matrix} ) ∏_{l=1}^{r}(m_{i} + \frac{r − i − l}{2}a)p

= ( \begin{matrix}r \\ j \end{matrix} )^{−1} ∑_{1≤i_{1}<···<i_{j}≤r} ∏_{l=1}^{j}(m_{i_{l}} + \frac{j − l}{2}a)p.  (9)

§2.3 The Algebra of the Holomorphic Differential Operators that Commute with the Action of K Let P^{K} be the subspace of polynomials in P (P^{+} × \overline{P^{+}}) defined by

P^{K} = \{p ∈ P (P^{+} × \overline{P^{+}})| p(k.z, k.w) = p(z, w), ∀k ∈ K\},

and D^{K} the space of holomorphic differential operators on P^{+} that commute with K.

Lemma 2.6. If p, q ∈ P^{K} and p(z, e) = q(z, e) for all z ∈ P^{+}, then p = q.

Proof. The fact that the orbit K.e is the Shilov boundary S of D, together with the K-invariant property of p and q, implies

p(z, ξ) = q(z, ξ), ∀z ∈ P^{+}, ξ ∈ S.

Since p, q are antiholomorphic in w, we have

p(z, w) = q(z, w), ∀z, w ∈ P^{+}

finishing the proof.

For m ≥ 0, define polynomials φ_{m}^{T} on P_{m}(P_{1}^{+}) and φ_{m} on P_{m}(P^{+}) by

φ_{m}^{T}(z) = ∫_{L_{T}} Δ_{m}(l.z) dl

and

φ_{m}(z) = ∫_{L} Δ_{m}(l.z) dl

where L_{T} (resp. L) is the isotropy subgroup of K_{T}^{∗} (resp.K) at e.

Proposition 2.7. {K^{m}(z, w), m ≥ 0} is a basis of P^{K}.

Proof. Suppose p(z, w) ∈ P^{K}, then p(z, e) is an L-invariant polynomial in P (P^{+}), by Theorem 2.1 in [4], we have

p(z, e) = ∑ a_{m}φ_{m}(z) = ∑ a_{m} \frac{1}{c_{m}} K^{m}(z, e).

Now Lemma 2.6 yields

p(z, w) = \frac{a_{m}}{c_{m}} K^{m}(z, w).

Finally, since {φ_{m}, m ≥ 0} are linearly indpendent, an argument similar to the above shows that {K^{m}(z, w), m ≥ 0} are linearly independent.

As in §1.2, we have the following lemma

Lemma 2.8. Every D in D^{K} determines a unique polynomial F_{D}(z, w) in P^{K}. Conversely, if p ∈ P^{K}, then p(z, \frac{∂}{∂z}) ∈ D^{K}. Moreover, if p(z, \frac{∂}{∂z}) = q(z, \frac{∂}{∂z}) in D^{K}, then p = q.

Now Proposition 2.7 and Lemma 2.8 imply

Proposition 2.9. \{K^{m}(z, \frac{∂}{∂z}), m ≥ 0\} is a basis of D^{K}.

The following result is an analogue of Theorem 1.8 Theorem 2.10. (i) \{K^{1_{1}}(z, \frac{∂}{∂z}), …, K^{1_{r}}(z, \frac{∂}{∂z})\} is a set of algebraically in-

dependent generators of D^{K}.

(ii) D^{K} is a commutative algebra.

We note that the K^{1_{j}}(z, \frac{∂}{∂z}) commute mutually since \{K^{1_{j}}(z, \frac{∂}{∂z}) act on P_{m}(P^{+}). Thus (ii) follows from (i) immediately.

To prove Theorem 2.10, we need the following lemma which is due to A.Korányi.

Lemma 2.11. φ_{m} = (φ_{m}^{T})^{E} for all m ≥ 0.

Proof. The proof follows immediately from tha fact in [9] that ∥ φ_{m} ∥_{F} = ∥ φ_{m}^{T} ∥_{F}.

Proof. (of Theorem 2.10) We will use the fact that {φ_{1_{1}}^{T}, …, φ_{1_{r}}^{T}} is a set of algebraically independent generators of the algebra of L_{T} −invariant polynomials.

First we show by induction that for each m ≥ 0, there is a polynomial P in r variables such that K^{m}(z, \frac{∂}{∂z}) = P (K^{1_{1}}(z, \frac{∂}{∂z}), …, K^{1_{r}}(z, \frac{∂}{∂z})).

In fact, since φ_{m} = (φ_{m}^{T})^{E} by Lemma 2.11, there is a polynomial P_{1} such that

c_{m}φ_{m}(z) = P_{1}(φ_{1_{1}}(z), …, φ_{1_{r}}(z)) = P_{1}(K_{e}^{1_{1}}(z), …, K_{e}^{1_{r}}(z)).

Then Lemma 2.6 gives that

K^{m}(z, w) = P_{1}(K^{1_{1}}(z, w)), …, K^{1_{r}}(z, w)).

One can easily see that the differential operator

K^{m}(z, \frac{∂}{∂z}) − P_{1}(K^{1_{1}}(z, \frac{∂}{∂z}), …, K^{1_{r}}(z, \frac{∂}{∂z}))

has order < m_{1} + ⋯ + m_{r}. By induction, there is a polynomial Q_{1} such that

K^{m}(z, \frac{∂}{∂z}) − P_{1}(K^{1_{1}}(z, \frac{∂}{∂z}), …, K^{1_{r}}(z, \frac{∂}{∂z})) = Q_{1}(K^{1_{1}}(z, \frac{∂}{∂z}), …, K^{1_{r}}(z, \frac{∂}{∂z}))

Now P = P_{1} + Q_{1} gives the solution.

Next we prove that K^{1_{1}}(z, \frac{∂}{∂z}), …, K^{1_{r}}(z, \frac{∂}{∂z}) are algebraically independent. For a monomial u_{1}^{α_{1}} ⋯ u_{r}^{α_{r}}, we define

weight(u_{1}^{α_{1}} ⋯ u_{r}^{α_{r}}) = α_{1} + 2α_{2} + … + rα_{r}.

A polynomial p in r variables is of weight i if p is the sum of monomials of weight i.

Suppose that there exists a polynomial Q in r variables such that

Q(K^{1_{1}}(z, \frac{∂}{∂z}), …, K^{1_{r}}(z, \frac{∂}{∂z})) = 0.

Then

Q(K^{1_{1}}(z, \frac{∂}{∂z}), …, K^{1_{r}}(z, \frac{∂}{∂z}))e^{(z|w)} = 0.  (10)

We write Q = ∑_{i=1}^{n} Q_{i} with weight(Q_{i}) = i and Q_{n} ≠ 0. It follows easily from (10) that

Q_{n}(K^{1_{1}}(z, w), …, K^{1_{r}}(z, w)) = 0.

Thus,

0 = Q_{n}(K^{1_{1}}(z, e), …, K^{1_{r}}(z, e))
= Q_{n}(c_{1_{1}}φ_{1_{1}}(z), …, c_{1_{r}}φ_{1_{r}}(z))
= Q_{n}(c_{1_{1}}φ_{1_{1}}^{T}(z), …, c_{1_{r}}φ_{1_{r}}^{T}(z))

for all z ∈ P_{1}^{+}. But φ_{1_{1}}^{T}, …, φ_{1_{r}}^{T} are algebraically independent, we have a contradiction. Therefore, we have proved the theorem.

From the definition of D_{λ}, Theorem 1.9, and Remark 2 in §1.3, we immediately obtain

Theorem 2.12. For any distinct numbers λ_{1}, …, λ_{r}, D_{λ_{1}}, …, D_{λ_{r}} is a set of algebraically independent generators of D^{K}.

3. Spaces of Holomorphic Functions

In this part, we shall apply our results in part II to the study of some spaces of holomorphic functions on a bounded symmetric domain.

§3.1. More notation For s = (s_{1}, …, s_{r}) ∈ C^{r}, one defines

(s)_{m} = ∏_{i=1}^{r}(s_{i} − \frac{i − 1}{2}a)_{m_{i}}

where (a)_{k} = a(a + 1) ⋯ (a + k − 1), (a)_{0} = 1. For any complex number s, we write s1 for (s, …, s). With some abuse of notation, we also write s + α for (s_{1} + α, …, s_{r} + α).

For s ∈ C^{r}, let Γ_{Ω}(s) be Gindikin’s Gamma function, that is,

Γ_{Ω}(s) = (2π)^{\frac{r(r−1)}{4}} ∏_{i=1}^{r} Γ(s_{i} − (i − 1)a/2).

Then

(λ)_{m} = \frac{Γ_{Ω}(λ + m)}{Γ_{Ω}(λ)}.

We denote by h(z) the K -invariant polynomial on P^{+} whose restriction to {∑_{i=1}^{r} a_{i}e_{i} | a_{i} ∈ R, i = 1, …, r} is given by

h(∑_{i=1}^{r} a_{i}c_{i}) = ∏_{i=1}^{r}(1 − a_{i}^{2}).

Let

h(z, w) = exp ∑_{j=1}^{n} z_{j} \frac{∂}{∂z_{j}} exp ∑_{j=1}^{n} \overline{w_{j}} \frac{∂}{∂\overline{z_{j}}} h(z),

then

h(z, w)^{−p} = K(z, w)

where p = (r − 1)a + b + 2 and K(z, w) is the Bergman kernel of D. See [4].

We write H_{λ} (λ > p − 1) for the Hilbert space of holomorphic functions f on D such that < f, f >_{λ} is finite where

⟨f, g⟩_{λ} = c_{λ} ∫_{D} f(z)\overline{g(z)}h(z)^{λ−p}dz,  (11)

and

c_{λ} = \frac{1}{π^{n}} \frac{Γ_{Ω}(λ)}{Γ_{Ω}(λ − n/r)},

then H_{λ} has K_{λ}(z, w) = h(z, w)^{−λ} as its reproducing kernel. For an element g ∈ G, one can define a linear transformation U_{λ}(g) on H_{λ} by

(U_{λ}(g)f)(z) = f(g^{−1}z)J_{g−1}(z)^{λ/p}, f ∈ H_{λ},  (12)

where J_{g} is the complex Jacobian determinant of g and we use the principal branch of the power functions. It is pointed out in [4] that on H_{λ}, (2) defines a unitary representation U_{λ} of G̃, the universal covering group of G. This is the scalar-valued holomorphic discrete series of representations of G̃.

For λ > p − 1, (1) is equal to

⟨f, g⟩_{λ} = ∑_{m}(λ)_{m}⟨f, g⟩_{F} .  (13)

When λ ≤ p − 1, there is no nonzero holomorphic function f satisfying ⟨f, f⟩_{λ} < ∞. However, for those λ > \frac{r−1}{2}a, (3) still defines a nonzero Hilbert space H_{λ} of holomorphic functions on D. (2) again defines a unitary representation of G̃, the analytic continuation of the holomorphic discrete series. For details about the holomorphic discrete series and its analytic continuation, see [4], [15],[18].

For λ ∈ C, we denote by P^{(λ)} the set P (P^{+}) equipped with the structure of a Harish-Chandra module obtained by analytic continuation of the holomorphic discrete series, see [4]. For m ≥ 0, let q(λ, m) be the multiplicity of λ as a zero of the polynomial λ′ → (λ′)_{m}. Set q(λ) = sup_{m≥0}q(λ, m). Clearly, q(λ) ≤ r. For j = 0, 1, …, q(λ), let

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

According to Theorem 5.3 in [4], q(λ) > 0 if and only if λ − \frac{r−1}{2}a or λ − \frac{r−2}{2}a is a nonpositive integer, and

M_{0}^{(λ)} ⊂ M_{1}^{(λ)} ⊂ ⋯ ⊂ M_{q(λ)}^{(λ)} = P^{(λ)}

is a composition series of P^{(λ)}. Morever, for every integer 0 ≤ j ≤ (λ), M_{j}^{(λ)}/M_{j−1}^{(λ)} has a U_{λ}-invariant Hermitian form given by

(f, g)_{λ,j} = lim_{λ′→λ} \frac{(λ′ − λ)^{j}}{(λ)_{m}} (f, g)_{F}  (14)

for f, g ∈ P_{m}(P).

The Hermitian form ( , )_{λ,0} on M_{0} is definite if and only if λ > \frac{r−1}{2}a or λ = j \frac{a}{2} with an integer 0 ≤ j ≤ r − 1. For j ≥ 1, ( , )_{λ,j} on M_{j}^{(λ)}/M_{j−1}^{(λ)} is definite if and only if j = q(λ) and \frac{r−1}{2}a − λ is an integer. In either case, j = 0 or q(λ), M_{j}^{(λ)}/M_{j−1}^{(λ)} is said to be unitarizable. We are mainly interested in the unitarizable cases. In particular, we shall express (4) in terms of integrals on D in the next section.

§3.2. Integral Formulas In this section we give some integral formulas for the invariant Hermitian form (4) when M_{j}^{(λ)}/M_{j−1}^{(λ)} is unitarizable.

According to (1.11) in [4], first we have

∫_{D} f(z)dv(z) = c ∫_{0}^{1} ⋯ ∫_{0}^{1} ∫_{K} f(k. ∑_{j=1}^{r} t_{j}c_{j})dk
· 2^{r} ∏_{j=1}^{r} t_{j}^{2b+1} ∏_{j<k} |t_{j}^{2} − t_{k}^{2}|^{a}dt_{1} ⋯ dt_{r}  (15)

where c is a constant whose exact value can be found in [10].

Next, we establish some lemmas

Lemma 3.1. If f_{m} ∈ P_{m}(P^{+}), g_{m′} ∈ P_{m′}(P^{+}), then

∫_{K} f_{m}(k.t)\overline{g_{m′}(k.t)}dk = \frac{δ_{m,m′}}{(n/r)_{m}} φ_{m}(t^{2})(f_{m}, g_{m′})_{F}

where t = ∑_{r=1}^{r} t_{i}e_{i}, t^{2} = ∑_{i=1}^{r} t_{i}^{2}e_{i} and φ_{m}(z) is the unique L-invariant polynomial in P_{m}(P^{+}), L is the isotropy subgroup of K at e.

Proof. We follow the proof of Lemma 3.1 in [4].

By K^{m}(k.z, k.w) = K^{m}(z, w), we have

f_{m}(k, t) = (f_{m}, K_{k.t}^{m})_{F} = (f_{m}, π(k)K_{t}^{m})_{F}
g_{m′}(k.t) = (g_{m′}, K_{k.t}^{m′})_{F} = (g_{m′}, π(k)K_{t}^{m′})_{F} .

Since the spaces P_{m}(P) and P_{m′}(P^{+}) are not equivalent, if m ≠ m′, and irreducible, applying the Schur orthogonality relations to the representation space P (P^{+}) of K, we have

∫_{K} f_{m}(k.t)\overline{g_{m′}(k.t)}dk
= ∫_{K}(f_{m}, π(k)K_{t}^{m})_{F} \overline{(g_{m′}, π(k)K_{t}^{m′})_{F}} dk
= \frac{δ_{m,m′}}{d_{m}} \overline{(K_{t}^{m}, K_{t}^{m′})_{F}} (f_{m}, g_{m′})_{F}
= \frac{δ_{m,m′}}{d_{m}} K^{m}(t, t)(f_{m}, g_{m′})_{F}

where d_{m} is the dimension of P_{m}(P^{+}). By Lemma 3.1, 3.2 and Theorem 3.4 in [4],

\frac{K^{m}(t, t)}{d_{m}} = \frac{φ_{m}(t^{2})}{(n/r)_{m}}

This proves the lemma.

Lemma 3.2. If t = ∑_{i=1}^{r} t_{i}e_{i}, then, for all α > p − 1

c_{λ}c ∫_{0}^{1} ⋯ ∫_{0}^{1}  φ_{m}(t) ∏_{i=1}^{r} t_{i}^{b} ∏_{i=1}^{r}(1 − t_{i})^{α−p} ∏_{i<j} |t_{i} − t_{j}|^{a}dt_{1} ⋯ dt_{r}
= \frac{(n/r)_{m}}{(α)_{m}}

Proof. The proof follows immediately from Theorem 3.6 and its proof in [4].

Rewriting (8) in Theorem 2.5, we have

D_{λ}^{(k)}p = (λ + m)_{k1}p.  (16)

A direct computation gives the following lemma.

Lemma 3.3. For any complex number λ and any positive integer k, we have

(λ)_{m+k} = (λ)_{m}(λ + m)_{k1}  (17)
(λ)_{m+k} = (λ)_{k1}(λ + k)_{m}  (18)
For z ∈ P^{+}, we define
|z| = ∏_{i=1}^{r} |t_{i}|  (19)

if z = k. ∑_{i=1}^{r} t_{i}e_{i}. By Corollary 1.3 in [12], (9) is well-defined. The set of the points in D for which |z| = 0 is of dimension less than n. We observe that when D is of tube type,

|z| = |Δ(z)|.  (20)

We have

Lemma 3.4. If α > p − 1, f_{m} ∈ P_{m}(P^{+}), g_{m′} ∈ P_{m′}(P^{+}) with m_{r} ≥ s, then

c_{α} ∫_{D} f_{m}(z)\overline{g_{m′}(z)}\frac{h(z, z)^{α−p}}{|z|^{2s}} dv(z)
= \frac{1}{(α)_{m−s}(n/r + m − s)_{s1}} (f_{m}, g_{m′})_{F} .  (21)

22 Yan

Proof. Lemma 3.1 and (5) imply that the left hand side of (11) is equal to

c_{α}c ∫_{0}^{1} ⋯ ∫_{0}^{1} ∫_{K} f_{m}(k.t)\overline{g_{m′}(k.t)}dk ∏_{i=1}^{r}(1 − t_{i}^{2})^{α−p}

× 2^{r} · ∏_{i=1}^{r} t_{i}^{−2s+2b+1} ∏_{j<k} |t_{j}^{2} − t_{k}^{2}|^{a}dt_{1} ⋯ dt_{r}
= c_{α}c\frac{1}{(n/r)_{m}} (f_{m}, g_{m′})_{F} ∫_{0}^{1} ⋯ ∫_{0}^{1} φ_{m−s}(t^{2}) ∏_{i=1}^{r}(1 − t_{i}^{2})^{α−p}
× 2^{r} · ∏_{i=1}^{r} t_{i}^{2b+1} ∏_{j<k} |t_{j}^{2} − t_{k}^{2}|^{a}dt_{1} ⋯ dt_{r}

Using the variable change t_{j}^{2} → t_{j} and Lemma 3.2, this is seen to be equal to

c_{α}c\frac{(f_{m}, g_{m′})_{F}}{(n/r)_{m}} ∫_{0}^{1} ⋯ ∫_{0}^{1} φ_{m−s}(t) ∏_{i=1}^{r}(1 − t_{j})^{α−p} ∏_{i=1}^{r} t_{i}^{b} ∏_{j<k} |t_{j} − t_{k}|^{a}dt_{1} ⋯ dt_{r}

= \frac{(f_{m}, g_{m′})_{F}}{(n/r)_{m}} \frac{(n/r)_{m−s}}{(α)_{m−s}} = \frac{1}{(α)_{m−s}(n/r + m − s)_{s1}} (f_{m}, g_{m′})_{F} .

The last equality follows from Lemma 3.3.

Case: λ > \frac{r−1}{2}a.

Now we note that when λ > \frac{r−1}{2}a, M_{0}^{λ} equals P (P^{+}). In this case, we give the following integral formula for the invariant inner product (4). Theorem 3.5. For λ > \frac{r−1}{2}a, if k is a positive integer such that λ+k > p−1, then

(f, g)_{λ,0} = \frac{c_{λ+k}}{(λ)_{k1}} ∫_{D}(D_{λ}^{k}f)(z)\overline{g(z)}h(z, z)^{λ+k−p}dV (z)  (22)

for all f, g ∈ P (P^{+}). Proof. First, from Lemma 3.3 we obtain

(λ + m)_{k1} = \frac{(λ)_{k1}(λ + k)_{m}}{(λ)_{m}} .  (23)

Using Lemma 3.4, we note that it suffices to show (12) for f, g ∈ P_{m}(P^{+}). Since D_{λ}^{k}f = (λ + m)_{k1}f,

c_{λ+k} _{∫D} (D_{λ}^{k}f)(z)\overline{g(z)}h(z, z)^{λ+k−p}dV (z)
= (λ + m)_{k1}c_{λ+k} ∫_{D} f(z)\overline{g(z)}h(z, z)^{λ+k−p}dV (z)
= (λ + m)_{k1} \frac{1}{(λ + k)_{m}} (f, g)_{F} .

The last equality follows from Cor. 3.7 in [4]. By (13), this is seen to be equal to

(λ)_{k1} \frac{1}{(λ)_{m}} (f, g)_{F} .

We have proved the Theorem.

Case: λ = \frac{r−1}{2}a − s.

Let n_{1} = r\frac{(r−1)}{2}a + r. Next we consider the case when M_{q(λ)}^{(λ)}/M_{q(λ)−1}^{(λ)} is unitarizable, i.e., λ = n_{1}/r − s, s is an integer. Two types of integral formulas for (4) will be given, the first one analogous to (12) and the second one leading to characterizing the completion of M_{q(λ)}^{(λ)}/M_{q(λ)−1}^{(λ)} as a Dirichlet-type space.

When s ≤ 0, λ ≥ \frac{r−1}{2}a + 1, this is the case we have discussed. In the following, we assume s ≥ 1.

Lemma 3.6. For λ = n_{1}/r − s, k ≥ s, we have

(i) if m_{r} < s, then (λ + m)_{k1} = 0

(ii) if m_{r} ≥ s, then

(λ + m)_{k1} = ∏_{j=0}^{k−1} ∏_{i=1}^{r}(m_{i} + 1 − s + j + \frac{r − i}{2}a) > 0.

Proof. Since n_{1}/r = (r − 1)a/2 + 1, we have

(λ + m)_{k1} = ∏_{j=0}^{k−1} ∏_{i=1}^{r}(n_{1}/r + m_{i} − s + j − \frac{i − 1}{2}a)

= ∏_{j=0}^{k−1} ∏_{i=1}^{r−1}(m_{i} + 1 − s + j + \frac{r − i}{2}a)

· ∏_{j=0}^{k−1}(m_{r} + 1 − s + j)  (24)

If m_{r} < s, then the last term in (14) is zero. If m_{r} ≥ s, each term in (14) is positive. This proves the lemma.

Let

\frac{1}{⟨λ⟩_{m}} = lim_{λ′→λ} \frac{(λ′ − λ)^{q(λ)}}{(λ)_{m}} .

Lemma 3.7. For λ = n_{1}/r − s, we have

(i) if m_{r} < s, then \frac{1}{⟨λ⟩_{m}} = 0;

(ii) if m_{r} ≥ s, then

\frac{(λ + m)_{k1}}{(λ + k)_{m}} = (λ)k̃1 \frac{1}{⟨λ⟩_{m}}  (25)

where (λ)k̃1 means that the zero factors are omitted.

Proof. By definition,

(λ)_{m} = ∏_{i=1}^{r−1}(n_{1}/r − s)_{m_{i}} · (1 − s)_{m_{r}}.

We note that q(λ, m) = q(λ) implies (1 − s)_{m_{r}} = 0. If m_{r} < s, it is easy to see that (1 − s)_{m} ≠ 0, thus q(λ, m) < q(λ), then \frac{1}{⟨λ⟩_{m}} = 0.

Now suppose m_{r} ≥ s. For λ = \frac{n_{1}}{r} − s, then

(λ)_{k1} = ∏_{i=1}^{r} ∏_{j=1}^{k}(\frac{r − i}{2}a − s + j)  (26)

(λ)_{m} = ∏_{i=1}^{r} ∏_{j=1}^{m_{i}}(\frac{r − i}{2}a + j).  (27)

Since the numbers of zero terms in both (16) and (17) are equal to q(λ) , we get

lim_{λ′→λ} \frac{(λ′)_{k1}}{(λ′)_{m}} = lim_{λ′→λ} \frac{(λ′)_{k1}}{(λ′ − λ)^{q(λ)}} · \frac{(λ′ − λ)^{q(λ)}}{(λ′)_{m}}
= (λ)k̃1 \frac{1}{⟨λ⟩_{m}}  (28)

For those λ′ such that (λ′ + k)_{m} ≠ 0, by Lemma 3.3, we have

\frac{(λ′ + m)_{k1}}{(λ′ + k)_{m}} = \frac{(λ′)_{k1}}{(λ′)_{m}} .  (29)

Letting λ′ → λ in (19) and using (18), we obtain (15).

Now we have

Theorem 3.8. If λ = n_{1}/r − s, s ≥ 1, then, for k ∈ Z with λ + k > p − 1, we have

(f, g)_{λ,q(λ)} = \frac{1}{(λ)k̃1} c_{λ+k} ∫_{D}(D_{λ}^{k}f)(z)\overline{g(z)}h(z, z)^{λ+k−p}dV (z)  (30)

for all f, g ∈ P (P^{+}).

Proof. For the same reason as in the proof of Theorem 3.5, it is enough to show (20) for f, g ∈ P_{m}(P^{+}).

(i) if m_{r} < s, by (6) and Lemma 3.6, the R.H.S. of (20) is equal to zero; by Lemma 3.7, the L.H.S. of (20) is also equal to zero. (ii) if m_{r} ≥ s, then

c_{λ+k} ∫_{D}(D_{λ}^{k}f)(z)\overline{g(z)}h(z, z)^{λ+k−p}dV (z)
= (λ + m)_{k1}c_{λ+k} ∫_{D}(f)(z)\overline{g(z)}h(z, z)^{λ+k−p}dV (z)
= (λ + m)_{k1} \frac{1}{(λ + k)_{m}} (f, g)_{F}

Now the theorem follows from Lemma 3.7.

The following result gives another integral formula for ( , )_{λ,q(λ)}.

Theorem 3.9. If λ = n_{1}/r−s, s ≥ 1, then, for k ∈ Z with k > n/r−1, k ≥ s, we have

(f, g)_{λ,q(λ)} = \frac{1}{(λ)k̃1} \frac{c_{n_{1}/r+k}}{(λ + k)_{s1}} ∫_{D}(D_{λ+b}^{s}f)(z)\overline{(D_{λ}^{k}g)(z)}\frac{h(z, z)^{k−n/r}}{|z|^{2s}} dV (z). (31)

for all f, g ∈ P (P^{+}). When D is of tube type, then (21) becomes

\frac{1}{(λ)s̃1} \frac{c_{n/r+k}}{(λ + k)_{s1}} ∫_{D}(Δ(\frac{∂}{∂z})^{s}f)(z)\overline{Δ(z)^{k−s}(Δ(\frac{∂}{∂z})^{k}g)(z)}h(z, z)^{k−n/r}dV (z) (32)

for all f, g ∈ P (P^{+}).

Proof. If m_{r} < s, as shown in the proof of Theorem 3.8, both sides of (21) are zero. It is enough to consider the case m_{r} ≥ s. By Lemma 3.4, we have

c_{n_{1}/r+k} ∫_{D}(D_{λ}^{s}f)(z)\overline{(D_{λ}^{k}g)(z)}\frac{h(z, z)^{k−n/r}}{|z|^{2s}} dV (z)

= (λ + b + m)_{s1}(λ + m)_{k1} ∫_{D} f(z)\overline{g(z)}\frac{h(z, z)^{k+n_{1}/r−p}}{|z|^{2s}} dV (z)
= (λ + b + m)_{s1}(λ + m)_{k1} \frac{1}{(n/r + m − s)_{s1}} · \frac{1}{(k + n_{1}/r)_{m−s}} (f, g)_{F}
= \frac{(λ + m)_{k1}}{(k + λ + s)_{m−s}} (f, g)_{F} .

The identity (λ + k)_{s1}(k + λ + s)_{m−s} = (λ + k)_{m} and Lemma 3.7 imply that this is equal to

(λ + k)_{s1} · \frac{(λ + m)_{k1}}{(λ + k)_{m}} (f, g)_{F} = (λ + k)_{s1}(λ)k̃1⟨λ⟩_{m}(f, g)_{F} .

This proves (21). Finally, (22) follows from the observation that when D is of tube type

(D_{λ}^{s}f)(z)\overline{(D_{λ}^{k}g)(z)}\frac{1}{|Δ(z)|^{2s}} = (Δ(\frac{∂}{∂z})^{s}f)(z)\overline{Δ(z)^{k−s}(Δ(\frac{∂}{∂z})^{k}g)(z)}.

Corollary 3.10. If λ = n_{1}/r − s ≤ 0 and s ≥ n/r − 1, then, we have

(f, g)_{λ,q(λ)} = \frac{1}{(λ)s̃1} \frac{c_{n_{1}/r+s}}{(λ + s)_{s1}} ∫_{D}(D_{λ+b}^{s}f)(z)\overline{(D_{λ}^{s}g)(z)}\frac{h(z, z)^{s−n/r}}{|z|^{2s}} dV (z)  (33)

for all f, g ∈ P (P^{+}). When D is of tube type, if n/r ≤ s, then (23) becomes

\frac{1}{(λ)s̃1} \frac{c_{n/r+s}}{(λ + s)_{s1}} ∫_{D}(Δ(\frac{∂}{∂z})^{s}f)(z)\overline{(Δ(\frac{∂}{∂z})^{s}g)(z)}h(z, z)^{s−n/r}dV (z)  (34)

for all f, g ∈ P (P^{+}).

Proof. Immediate.

Remark 1. Thanks to the remarks by A.Korányi, the constants before the integrals in Theorems 3.5,3.8 and 3.9 are much simpler than in the original version. §3.3. Characterization of M_{0}^{(λ)} In this section, we consider the remaining unitarizable case λ = \frac{j−1}{2}a, 1 ≤ j ≤ r, and we are only content with giving a characterization of M_{0}^{(λ)} in terms of differential operators. Integral formulas for the corresponding invariant inner product will be given in a forthcoming paper.

If λ = \frac{j−1}{2}a, 1 ≤ j ≤ r, then for m = (m_{1}, …, m_{j−1}, 0, …, 0), (λ)_{m} > 0, and for all other m, (λ)_{m} = 0. Hence we have

M_{0}^{(λ)} = ⨁_{m≥0,m_{j}=...=m_{r}=0} P_{m}(P^{+}).

It is shown in [17] that when D is of tube type, M_{0}^{( \frac{r−1}{2} a)} is the space of harmonic polynomials, in the sense of

Δ(\frac{∂}{∂z})p(z) = 0.  (35)

We note that for p ∈ P (P^{+}), (25) is equivalent to K_{r}p = 0. Now we generalize this result as follows

Theorem 3.11. For λ = \frac{j−1}{2}a, j = 1, …, r, we have

M_{0}^{( \frac{j−1}{2} a)} = \{p ∈ P (P^{+}) | K_{j}p = 0.\}  (36)

Proof. By Theorem 2.5 (ii), for p ∈ P_{m}(P^{+})

K_{j}p = ∑_{1≤i_{1}<···<i_{j}≤r} ∏_{l=1}^{j}(m_{i_{l}} + (j − l)\frac{a}{2})p  (37)

On the one hand, if p ∈ P_{m}(P^{+}) with m_{j} = … = m_{r} = 0, then the factor (m_{i_{j}} + (j − l)\frac{a}{2}) in each term of (27) becomes 0, since i_{j} ≥ j. Hence for any

p ∈ M_{0}^{( \frac{j−1}{2} a)}, K_{j}p = 0.

On the other hand, we note that each term in (27) is nonnegative and is positive if m_{j} > 0. Therefore, if p ∈ P_{m}(P^{+}) with m_{j} > 0, then K_{j}p ≠ 0.

Now the theorem follows.

For further relevant results in this area, see [3].

§3.4. Hilbert Spaces of Holomorphic Function

We have seen that for λ > p − 1, there is a natural Hilbert space of holomorphic functions on which U_{λ}(g) acts unitarily. Now we study the completion of M_{q(λ)}^{λ}/M_{q(λ)−1}^{λ} with respect to (4) when it is unitarizable.

For λ > \frac{r−1}{2}a or λ = \frac{j}{2}a, 0 ≤ j ≤ r − 1, let H_{λ} denote the completion of M_{0}^{(λ)} with respect to the inner product

(f, g)_{λ,0} = ∑_{m} \frac{(f_{m}, g_{m})_{F}}{(λ)_{m}} , f, g ∈ M_{0}^{(λ)}.

Lemma 3.12. Every f ∈ H(D) can be expanded as

f(z) = ∑_{m≥0} f_{m}

where f_{m} ∈ P_{m}, the series converges uniformly and absolutely on compact subsets of D.

For f, g ∈ H(D), let f = ∑_{m≥0} f_{m}, g = ∑_{m≥0} g_{m} be the expansion as in the lemma, it can be readily seen that

(i) for λ > \frac{r−1}{2}a

H_{λ} = \{f ∈ H(D)| ∑_{m} \frac{(f_{m}, f_{m})_{F}}{(λ)_{m}} < ∞\}

with the inner product

(f, g)_{λ} = ∑_{m} \frac{(f_{m}, g_{m})_{F}}{(λ)_{m}} ;

(ii) for λ = \frac{j}{2}a, 0 ≤ j ≤ r − 1,

H_{λ} = \{f ∈ H^{j}(D)| ∑_{m} \frac{(f_{m}, f_{m})_{F}}{(λ)_{m}} < ∞\}

with the inner product (f, g)_{λ} = ∑_{m} \frac{(f_{m},g_{m})_{F}}{(λ)_{m}} , where H^{j}(D) consists of holomorphic functions f = ∑_{m≥0} f_{m} which have terms f_{m} ≠ 0 only for those m with m_{j+1} = … = m_{r} = 0.

Remark 1. One can easily see that when λ > p − 1, the new definition of H_{λ} coincides with the previous one.

It is immediate that K_{λ}(z, w) is the reproducing kernel on H_{λ}, and the closure of the linear span {K_{λ}(·, w), w ∈ D} is H_{λ}. Now by the identities (K_{λ}(·, w), K_{λ}(·, z))_{λ}=K_{λ}(z, w) and J_{g}(z)K(g.z, g.w)\overline{J_{g}(w)}=K(z, w), z, w ∈ D, or by Theorem 5.3 in [4], we conclude that ( , )_{λ} is invariant under the action U_{λ}.

As a consequence of Theorem 3.5 and Lemma 3.12, we have

Theorem 3.13. For λ > \frac{r−1}{2}a, k ∈ Z with λ + k > p − 1,

H_{λ} = \{f ∈ H(D)| ∫_{D}(D_{λ}^{k}f)(z)\overline{f(z)}h(z, z)^{λ+k−p}dV (z) < ∞\}  (38)

with the inner product

(f, g)_{λ} = \frac{c_{λ+k}}{(λ)_{k1}} ∫_{D}(D_{λ}^{k}f)(z)\overline{g(z)}h(z, z)^{λ+k−p}dV (z).  (39)

Remark 2. When f, g are not in H(\overline{D}), the integrals in (28) and (29) are understood as lim_{r→1} ∫_{D}(D_{λ}^{k}f)(rz)\overline{g(rz)}h(z, z)^{λ+k−p}dV (z).

For λ > \frac{r−1}{2}a or λ = \frac{j}{2}a, 0 ≤ j ≤ r − 1, k ∈ Z with λ + 2k > p − 1, we define a norm ∥ ∥_{λ,k} on H_{λ} by

∥f∥_{λ,k}^{2} = c_{λ+2k} ∫_{D}(D_{n_{1}/r}^{k}f)(z)\overline{D_{n_{1}/r}^{k}f)(z)}h(z, z)^{λ+2k−p}dV (z).
Now let f = ∑_{m} f_{m}, by Lemma 3.4, we have

∥f∥_{λ,k}^{2} = ∑_{m≥0} \frac{((n_{1}/r + m)_{k1})^{2}}{(λ + 2k)_{m}} (f_{m}, f_{m})_{F} .

Applying the Stirling’s formula, we get, as m varies,

\frac{(λ)_{m}}{(λ + 2k)_{m}} ≈ \frac{Γ_{Ω}(λ + m)}{Γ_{Ω}(λ + 2k + m)} ≈ ∏_{i=1}^{r}(m_{i} + 1)^{−2k} ≈ ((n_{1}/r + m)_{k1})^{−2}

(when λ = \frac{j}{2}a, we only consider those m with (m_{1}, …, m_{j}, 0 …, 0)). That is

\frac{((n_{1}/r + m)_{k1})^{2}}{(λ + 2k)_{m}} ≈ \frac{1}{(λ)_{m}} .

Therefore, there exist two positive constants C_{1} and C_{2} such that

C_{1}∥f∥_{λ,k}^{2} ≤ (f, f)_{λ} ≤ C_{2}∥f∥_{λ,k}^{2}.  (40)

Now Theorem 3.11 and Lemma 3.12 imply the following result

Theorem 3.14. For k ∈ Z with λ + 2k > p − 1, we have

(i) if λ = \frac{j−1}{2}a, 1 ≤ j ≤ r, then

H_{λ} = {f ∈ H(D)|K_{j}f = 0, ∥f∥_{λ,k} < ∞};

(ii) if λ > \frac{r−1}{2}a, then

H_{λ} = {f ∈ H(D)|∥f∥_{λ,k} < ∞}.

For a nonnegative integer s, let H(s) be the space of holomorphic functions f such that f_{m} = 0 for those m with m_{r} < s if f is expanded as in Lemma 3.12.

For λ = n_{1}/r − s, s ≥ 1, we denote by H̃_{λ} the completion of M_{q(λ)}^{(λ)}/M_{q(λ)−1}^{(λ)} with respect to ( , )_{λ,q(λ)}.

As a consequence of Lemma 3.7 and Theorem 3.8, we have

Theorem 3.15. H̃_{λ} is identified with the space of holomorphic functions f in H(s) for which

∫_{D}(D_{λ}^{k}f)(z)\overline{f(z)}h(z, z)^{λ+k−p}dV (z) < ∞

with the inner product

(f, g)_{λ,q(λ)} = \frac{c_{λ+k}}{(λ)k̃1} ∫_{D}(D_{λ}^{k}f)(z)\overline{g(z)}h(z, z)^{λ+k−p}dV (z),

where k ∈ Z with λ + k > p − 1. When f, g are not in H(\overline{D}), the integral has the same meaning as in Remark 2.

By Corollary 3.10, we get Theorem 3.16. If λ = n_{1}/r − s ≤ 0 and s ≥ n/r − 1, then H̃_{λ} is identified with the space of holomorphic functions f in H(s) for which

∫_{D}(D_{λ+b}^{s}f)(z)\overline{(D_{λ}^{s}f)(z)}\frac{h(z, z)^{s−n/r}}{|z|^{2s}} dV (z) < ∞,

and the inner product is given by

(f, g)_{λ,q(λ)} = \frac{1}{(λ)s̃1} \frac{c_{n_{1}/r+s}}{(λ + s)_{s1}} ∫_{D}(D_{λ+b}^{s}f)(z)\overline{(D_{λ}^{s}g)(z)}\frac{h(z, z)^{s−n/r}}{|z|^{2s}} dV (z). (41)

In particular, when D is of tube type, then (31) becomes

(f, g)_{λ,q(λ)} = \frac{1}{(λ)s̃1} \frac{c_{n/r+s}}{(λ + s)_{s1}} ∫_{D}(Δ(\frac{∂}{∂z})^{s}f)(z)\overline{(Δ(\frac{∂}{∂z})^{s}g)(z)}h(z, z)^{s−n/r}dV (z).

(42)

When f, g are not in H(\overline{D}), the integral has the same meaning as in Remark 2.

When r = 1, s = 1, then D is the unit disc and λ = 0, we see that the integral in (32) is

∫_{D} f′(z)\overline{g′(z)}dV (z),

hence H̃_{λ} is just the classical Dirichlet space.

Thus we call H̃_{λ} the generalized Dirichlet space.

Remark 3. When D is of tube type and n/r − s is an integer, (32) is due to J. Arazy. §3.5. The Dual and Predual of the Bergman Space For q ≥ 1, let L_{a}^{q}(D) = L^{q}(D) ∩ H(D) be the Bergman space on the bounded symmetric domain D. In this section, we describe, as in the case of one variable, the dual and the predual of the Bergman space L_{a}^{1}(D) in terms of those differential operators given in §2.

First, as a consequence of (6) and the expansion

h(z, w)^{−λ} = ∑_{m}(λ)_{m}K^{m}(z, w),

in [4], we have the following result which is interesting in its own right.

Theorem 3.17. For any complex number λ and any positive integer k,

D_{λ,z}^{k}h(z, w)^{−λ} = c_{λ,k}h(z, w)^{−(λ+k)}

where c_{λ,k} = ∏_{j=1}^{k} ∏_{i=1}^{r}(λ + j − 1 − \frac{i−1}{2}).

Next, we introduce Bloch-type spaces of holomorphic functions on D. Writing D^{s} for D_{p}^{s}, define

B̃^{s}(D) = {f ∈ L_{a}^{2}(D)| sup_{z∈D} h(z, z)^{s}|(D^{s}f)(z)| < ∞},

and

B̃_{0}^{s}(D) = {f ∈ L^{2}(D)| lim_{z→∂D} h(z, z)^{s}|(D^{s}f)(z)| = 0},

where ∂D is the topological boundary of D. Then B̃^{s}(D) and B̃_{0}^{s}(D) become Banach spaces with the norm ∥f∥_{∗} = sup_{z∈D} h(z, z)^{s}|(D^{s}f)(z)|.

Let P be the Bergman projection, C(D̄) the space of continuous functions on D̄ and C_{0}(D) the subspace of C(D̄) consisting of functions which vanish on ∂D. As in the classical case, we have

Theorem 3.18. For s > \frac{r−1}{2}a,

P : L^{∞}(D) → B̃^{s}(D)
P : C(D̄) ⟶ B̃_{0}^{s}(D),
P : C_{0}(D) ⟶ B̃_{0}^{s}(D)

are bounded and onto. Therefore, for all s > \frac{r−1}{2}a, the B̃^{s}(D) are the same and the B̃_{0}^{s}(D) are the same.

For a Banach space X, we write X^{∗} for its dual.

Next result gives the dual and predual of the Bergman space L_{a}^{1}(D).

Theorem 3.19. For s > \frac{r−1}{2}a, L_{a}^{1}(D)^{∗} = B̃^{s}(D) and B̃_{0}^{s}(D)^{∗} = L_{a}^{1}(D).

The proofs of Theorems 3.18 and 3.19 are the same as those of corresponding results in [19].

Remark. Since the differential operators D_{λ}^{k} have the same actions on holomorphic functions as the integral operators studied in [20], one can also use D_{λ}^{k} to characterize the holomorphic Besov spaces introduced in [20]. Acknowledgement. The author would like to thank Professor Adam Korányi for his valuable advice and many insightful comments. The author is also grateful to Professor Jonathan Arazy for sending him his papers [1] and [2].

References

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[2] —, Integral formulas for the invariant inner products in spaces of analytic functions on the ball, in: Function Spaces, K. Jarosz (Editor), Lecture notes in Pure and Applied Mathematics, Vol. 136, Marcel Dekker, 1992, pp. 9–23.

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[4] Faraut, J., and A. Korányi, Function Spaces and Reproducing Kernels on Bounded Symmetric Domains, J. Functional Analysis 88 (1990), 64–89.

[5]  —, “Analysis on symmetric cones,” Oxford Science Publications, 1994.

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[9] Korányi, A., Hua-type Integrals, Hypergeometric Functions and Symmetric Polynomials, in: Internaltional Symposium in Honor of Hua Loo Keng, Vol 2: Analysis, 169–180, Springer-Verlag and Sci. Beijing 1991.

[10] —, The Volume of Symmetric Domains, the Koecher Gamma Function and an Integral of Selberg, Studia Sci. Math. Hungarica, 17 (1982), 129– 133.

[11] Korányi, A., and J. A. Wolf, Realization of Hermitian Symmetic Spaces as Generalized Halfplanes, Ann. of Math. 81 (1965), 165–288.

[12] Loos, O., “Bounded Symmetric Domains and Jordan Pairs,” Univ. of California, Irvine, 1977.

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[14] Nomura, T., Algebraically independent Generators of Invariant Differential Operators on a Symmetric Cone, J. Reine Angw. Math., 400 (1989), 122–133.

[15] Rossi, H., and M. Vergne, Analytic continuation of the holomorphic discrete series of a semisimple Lie group, Acta Math. 136 (1976), 1–59.

[16] Schmid, W., Die Randwerte holomorpher Funktionen auf hermitischen Räumen, Invent. Math. 9 (1969), 61–80.

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

[18] Wallach, N., The analytic continuation of the discrete series, I, II, Trans. Amer. Math. Soc. 251 (1979), 1–17 and 19–37.

[19] Yan, Z., Duality and differential operators on the Bergman spaces of bounded symmetric domains, J. Functional Analysis 105 (1992), 171–186.

[20] Zhu, K., Holomorphic Besov Spaces on Bounded Symmetric Domains, Quarterly J. Math., Oxford 46 (1995), 239–256.

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Received December 12, 1996 and in final form May 12, 1999