NEW CLASSES OF DOMAINS WITH EXPLICIT BERGMAN KERNEL

GUY ROOS AND WEIPING YIN

Abstract. We introduce two classes of egg type domains, built on general bounded symmetric domains, for which we obtain the Bergman kernel in explicit formulas.

In 1921, S. Bergman introduced a kernel function, which is now known as the Bergman kernel function. It is well known that there exists a unique Bergman kernel function for each bounded domain in ℂ^{n}. For which domains can the Bergman kernel function be computed by explicit formulas? This is an important problem. Explicit formulas of the Bergman kernel function can help to solve important conjectures. We illustrate this point by two cases. Mostow and Siu have given a counterexample to the important conjecture that the universal covering of a compact Kähler manifold of negative sectional curvature should be biholomorphic to the ball. In their counterexample the explicit calculation of the Bergman kernel function and metric of the egg domain

{z ∈ ℂ^{2} : |z_{1}|^{2} + |z_{2}|^{14} < 1}

plays an essential role [19]. Another example is Lu Qikeng conjecture: in order to give a counterexample to the Lu Qikeng conjecture, an explicit formula for the Bergman kernel function is used in [3]. Therefore, computation of the Bergman kernel function by explicit formulas is an important research direction in several complex variables. Up to now, there are still many mathematicians working in that direction.

If the group of holomorphic automorphisms of a bounded homogeneous domain is known, then one can get its Bergman kernel function in explicit form. L.K. Hua [14] has obtained the Bergman kernel functions in explicit form for the four types of Cartan domains by using that method (it is called Hua’s method). It is well known that there are two exceptional Cartan domains, of dimension 16 and 27. Weiping Yin [25] has obtained the Bergman kernel functions in explicit form for these two exceptional Cartan domains by using the Hua’s method. A general expression of the Bergman kernel for all symmetric bounded homogeneous domains can also be given using the theory of Jordan triple systems; the Bergman kernel is then, up to a constant, a (negative) power of the ”generic minimal polynomial” (cf. Loos [18]).

For a non-symmetric homogeneous domain, we know that it is holomorphically equivalent to a Siegel domain (or N-Siegel domain in the sense of Yichao Xu); S.G. Gindikin [11] has computed the Bergman kernel functions in explicit form for homogeneous Siegel domains by using generalized power functions. Yichao Xu [24] has also obtained the Bergman kernel functions in explicit form for N-Siegel domains. In the middle of sixties, Jiaqing Zhong and Weiping Yin [31],[32] have constructed some new types of non-symmetric homogeneous domains and their extension spaces; Weiping Yin [26],[27],[28] has computed their Bergman kernel functions in explicit form by using Hua’s method. But their papers were published only at the beginning of the eighties, after the end of Cultural Revolution in China.

Date: March 31, 2003.

1991 Mathematics Subject Classification. 32H10.

Key words and phrases. Cartan domain, Bergman kernel, holomorphic automorphism group, bounded symmetric domain.

The second author was supported in part by NSF of China (grant No: 19631010) and by NSF of Beijing.

Besides homogeneous domains, domains for which the Bergman kernel function can be computed in explicit form are the egg domains (also called complex ellipsoids or complex ovals). In general, an egg domain has the following form:

E(p_{1}, …, p_{n}) = {(z_{1}, …, z_{n}) ∈ ℂ^{n} : |z_{1}|^{2p_{1}} + ⋯ + |z_{n}|^{2p_{n}} < 1},

where p_{1}, …, p_{n} are positive real numbers. Here the z_{j} (j = 1, …, n) are complex numbers. A more general case is obtained when the z_{j} are complex vectors, that is z_{j} = (z_{j1}, …, z_{jm_{j}}) and |z_{j}|^{2p_{j}} = [∑_{k=1}^{m_{j}} |z_{jk}|^{2}]^{p_{j}}; the corresponding domain is then denoted by E(p_{1}, …, p_{n}; m_{1}, …, m_{n}) or simply by E_{p;m}. S. Bergman ([2], p.82) has computed the Bergman kernel function in explicit form on E(p_{1}, 1) by summing a series; although he states that 1/p_{1} is a positive integer, his computation is valid for arbitrary p_{1}. The explicit form of the Bergman kernel functions on E(1, …, 1, p_{n}) and on E(1, …, 1, p_{n}; 1, …, 1, m) were obtained by D’Angelo in [5] and [6] respectively. In the case 1/p_{1}, …, 1/p_{n} are positive integers, Zinov’ev [33] obtained the explicit form of the Bergman kernel function on E(p_{1}, …, p_{n}). If 1/p_{1}, …, 1/p_{n−1} are positive integers and 1/p_{n} = p is a positive real number, then the Bergman kernel function on E(p_{1}, …, p_{n}) can also be computed explicitly. If p_{1}, …, p_{n} are positive integers, then the Bergman kernel function on E(p_{1}, …, p_{n}) has an explicit expression in terms of multivariate hypergeometric functions ([9, Theorem 1]).

We are not able to compute an explicit formula for the Bergman kernel function on any egg domain. If there are two (or more) numbers among the positive real numbers 1/p_{1}, …, 1/p_{n} which are not integers (nor inverse of integers), then one cannot get an explicit formula for the Bergman kernel function on E(p_{1}, …, p_{n}) or on E_{p;m}. So, one needs to estimate the Bergman kernel function on egg domains. This work has been done by Sheng Gong and Xuean Zheng [12],[13]. In his thesis of doctoral degree (Purdue Univ 1997-1998) titled “The Bergman kernel on Reinhardt domains”, Chieh-hsien Tiao has obtained an estimation for the Bergman kernel function on general Reinhardt domains [23].

During the period of the second author’s stay at the Institut des Hautes Études Scientifiques (IHES) in February 1998, he and G. Roos introduced the following four types of domains, called super-Cartan domains or Cartan-Hartogs domains (N, m, n are positive integers and K > 0 is real):

Y_{I}(N, m, n; K) := {W ∈ ℂ^{N}, Z ∈ R_{I}(m, n) : |W |^{2K} < det(I − ZZ^{T})},
Y_{II}(N, p; K) := {W ∈ ℂ^{N}, Z ∈ R_{II}(p) : |W |^{2K} < det(I − ZZ^{T})},
Y_{III}(N, q; K) := {W ∈ ℂ^{N}, Z ∈ R_{III}(q) : |W |^{2K} < det(I − ZZ^{T})},
Y_{IV} (N, n; K) := {W ∈ ℂ^{N}, Z ∈ R_{IV} (n) :
|W |^{2K} < 1 − ZZ^{T} − [(ZZ^{T})^{2} − |ZZ′|^{2}]^{1/2}}.

where R_{I}(m, n), R_{II}(p), R_{III}(q) and R_{IV} (n) denote respectively the Cartan domains of first, second, third and fourth type in the sense of L.K. Hua. Here Z^{T} denotes the conjugate and transpose of Z, det denotes the determinant of a square matrix.

Weiping Yin has got the Bergman kernel function in explicit form for these four Cartan-Hartogs domains [29],[30]. We would like to point out that if we can get the Bergman kernel in explicit form for a domain, then this domain is a good domain for research.

Quite recently, Weiping Yin has introduced the following four types of domains, which may be called Cartan egg domains:

CE_{I}(M, N, m, n; K) := {W_{1} ∈ ℂ^{M}, W_{2} ∈ ℂ^{N}, Z ∈ R_{I}(m, n) :
|W_{1}|^{2} + |W_{2}|^{2K} < det(I − ZZ^{T})},
CE_{II}(M, N, p; K) := {W_{1} ∈ ℂ^{M}, W_{2} ∈ ℂ^{N}, Z ∈ R_{II}(p) :
|W_{1}|^{2} + |W_{2}|^{2K} < det(I − ZZ^{T})},
CE_{III}(M, N, q; K) := {W_{1} ∈ ℂ^{M}, W_{2} ∈ ℂ^{N}, Z ∈ R_{III}(q) :
|W_{1}|^{2} + |W_{2}|^{2K} < det(I − ZZ^{T})},
CE_{IV} (M, N, n; K) := {W_{1} ∈ ℂ^{M}, W_{2} ∈ ℂ^{N}, Z ∈ R_{IV} (n) :

|W_{1}|^{2} + |W_{2}|^{2K} < 1 − ZZ^{T} − [(ZZ^{T})^{2} − |ZZ′|^{2}]^{1/2}}.

More generally, we will consider in this paper the two following classes of domains, built on an arbitrary irreducible bounded circled homogeneous domain Ω:

Y (q, Ω; k) := {(W, Z) ∈ ℂ^{q} × Ω; |W |^{2k} < N(Z, Z)} ,
E(p, q, Ω; k) := {(W_{1}, W_{2}, Z) ∈ ℂ^{p} × ℂ^{q} × Ω; |W_{1}|^{2} + |W |^{2k} < N(Z, Z)} ,

where N(Z, Z) is the generic norm of Ω, which is the proper generalization of det(I − ZZ^{T}).

If we are able to compute the Bergman kernel when p = q = 1, then, by using twice the Principle of inflation (which will be stated below), one can get the Bergman kernel functions of Cartan egg domains for general p, q. Therefore, we consider first Y (1, Ω, k) and E(1, 1, Ω; k). The domain E(1, 1, Ω; k) is not homogeneous, so we cannot use Hua’s method to get the Bergman kernel function in explicit form. Also the domain E(1, 1, Ω; k) is not Reinhardt, so we cannot get its Bergman kernel function by summing an infinite series. Our new method is a combination of these two methods.

In Section 1, we review the properties of Jordan triple systems, which are necessary for the general definition of Y (q, Ω; k) and E(p, q, Ω; k). Section 2 is devoted to auxiliary results: the computation of ∫_{Ω} N(x, x)^{s}α^{n}, which appears to be a simple consequence of an integral computed by Selberg, the notion of “semi-Reinhardt domains” and description of complete orthonormal systems for these domains, and the “Principle of inflation”, which allows to derive the Bergman kernel for Y (q, Ω; k) and E(p, q, Ω; k) from the special case p = q = 1. In Section 3, we study automorphisms of the domains Y (q, Ω; k) and E(p, q, Ω; k) and compute their Bergman kernels. The last section gives tables stating specific results for all types (classical and exceptional) of bounded symmetric domains.

1. Bounded symmetric domains and Jordan triple systems

Hereunder we give a review of properties of the Jordan triple structure associated to a complex bounded symmetric domain (see [18], [21]).

1.1. Jordan triple system associated to a bounded symmetric domain. Let Ω be an irreducible bounded circled homogeneous domain in a complex vector space V. Let K be the identity component of the (compact) Lie group of (linear) automorphisms of Ω leaving 0 fixed. Let ω be a volume form on V, invariant by K and by translations. Let K be the Bergman kernel of Ω with respect to ω, that is, the reproducing kernel of the Hilbert space H^{2}(Ω, ω) = Hol(Ω) ∩ L^{2}(Ω, ω). The Bergman metric at z ∈ Ω is defined by

h_{z}(u, v) = ∂_{u}\overline{∂}_{v} log K(z).

The Jordan triple product on V is defined by

h_{0}(\{uvw\}, t) = ∂_{u}\overline{∂}_{v}∂_{w}\overline{∂}_{t} log K(z) |_{z=0} .

The triple product (x, y, z) ↦ {xyz} is complex bilinear and symmetric with respect to (x, z), complex antilinear with respect to y. It satisfies the Jordan identity

{xy{uvw}} − {uv{xyw}} = {{xyu}vw} − {u{vxy}w}.

The space V endowed with the triple product {xyz} is called a (Hermitian) Jordan triple system. For x, y, z ∈ V, denote by D(x, y) and Q(x, z) the operators defined by

{xyz} = D(x, y)z = Q(x, z)y.

The Bergman metric at 0 is related to D by

h_{0}(u, v) = tr D(u, v).

A Jordan triple system is called Hermitian positive if (u|v) = tr D(u, v) is positive definite. As the Bergman metric of a bounded domain is always definite positive, the Jordan triple system associated to a bounded symmetric domain is Hermitian positive.

The quadratic representation

Q : V ⟶ End_{ℝ}(V)

is defined by Q(x)y = \frac{1}{2}\{xyx\}. The following fundamental identity for the quadratic representation is a consequence of the Jordan identity:

Q(Q(x)y) = Q(x)Q(y)Q(x).

The Bergman operator B is defined by

B(x, y) = I − D(x, y) + Q(x)Q(y),

where I denotes the identity operator in V. It is also a consequence of the Jordan identity that the following fundamental identity holds for the Bergman operator:

Q(B(x, y)z) = B(x, y)Q(z)B(y, x).

The Bergman operator gets its name from the following property:

h_{z} (B(z, z)u, v) = h_{0}(u, v) (z ∈ Ω; u, v ∈ V).

If Φ ∈ (Aut Ω)_{0}, the identity component of the automorphism group of Ω, the relation

(1.1)  B(Φx, Φy) = dΦ(x) ∘ B(x, y) ∘ dΦ(y)^{∗}

holds for x, y ∈ Ω, where ^{∗} denotes the adjoint with respect to the hermitian metric h_{0}. As a consequence, the Bergman kernel of Ω is given by

(1.2)  K(z) = \frac{1}{vol Ω} \frac{1}{det B(z, z)}.

1.2. Spectral theory. An Hermitian positive Jordan triple system is always semisimple, that is the direct sum of a finite family of simple subsystems with componentwise triple product.

As the domain Ω is assumed to be irreducible, the associated Jordan triple system V is simple, that is V is not the direct sum of two non trivial subsystems.

An automorphism f : V → V of the Jordan triple system V is a complex linear isomorphism preserving the triple product : f{u, v, w} = {fu, fv, fw}. The automorphisms of V form a group, denoted Aut V, which is a compact Lie group; we will denote by K its identity component.

An element c ∈ V is called tripotent if {ccc} = 2c.If c is a tripotent, the operator D(c, c) annihilates the polynomial T (T − 1)(T − 2).

Let c be a tripotent. The decomposition V = V_{0}(c)⊕V_{1}(c)⊕V_{2}(c), where V_{j}(c) is the eigenspace V_{j}(c) = {x ∈ V ; D(c, c)x = jx}, is called the Peirce decomposition of V (with respect to the tripotent c).

Two tripotents c_{1} and c_{2} are called orthogonal if D(c_{1}, c_{2}) = 0. If c_{1} and c_{2} are orthogonal tripotents, then D(c_{1}, c_{1)} and D(c_{2}, c_{2}) commute and c_{1} + c_{2} is also a tripotent.

A non zero tripotent c is called primitive if it is not the sum of non zero orthogonal tripotents. A tripotent c is maximal if there is no non zero tripotent orthogonal to c. The set of maximal tripotents is equal to the Shilov boundary of the domain Ω.

A frame of V is a maximal sequence (c_{1}, …, c_{r}) of pairwise orthogonal primitive tripotents. The frames of V form a manifold F, which is called the Satake-Furstenberg boundary of Ω.

Let c = (c_{1}, …, c_{r}) be a frame. For 0 ≤ i ≤ j ≤ r, let

V_{ij}(c) = {x ∈ V | D(c_{k}, c_{k})x = (δ_{i}^{k} + δ_{j}^{k})x, 1 ≤ k ≤ r} :

the decomposition V = ⨁_{0≤i≤j≤r} V_{ij}(c) is called the simultaneous Peirce decomposition with respect to the frame c.

Let V be a simple Hermitian positive Jordan triple system. Then there exist frames for V. All frames have the same number of elements, which is the rank r of V. The subspaces V_{ij} = V_{ij}(c) of the simultaneous Peirce decomposition have the following properties: V_{00} = 0 ; V_{ii} = ℂe_{i} (0 < i); all V_{ij}’s (0 < i < j) have the same dimension a; all V_{0i}’s (0 < i) have the same dimension b.

The numerical invariants of V (or of Ω) are the rank r and the two integers

a = dim V_{ij} (0 < i < j),
b = dim V_{0i} (0 < i).

The genus of V is the number g defined by

g = 2 + a(r − 1) + b.

The HPJTS V and the domain Ω are said to be of tube type if b = 0.

Let V be a simple Hermitian positive Jordan triple system. Then any x ∈ V can be written in a unique way

x = λ_{1}c_{1} + λ_{2}c_{2} + ⋯ + λ_{p}c_{p},

where λ_{1} > λ_{2} > ⋯ > λ_{p} > 0 and c_{1}, c_{2} …, c_{p} are pairwise orthogonal tripotents. The element x is regular iff p = r; then (c_{1}, c_{2}, …, c_{r}) is a frame of V. The decomposition x = λ_{1}c_{1} + λ_{2}c_{2} + ⋯ + λ_{p}c_{p} is called the spectral decomposition of x.

1.3. The generic minimal polynomial. Let V be a Jordan triple system of rank r. There exist polynomials m_{1}, …, m_{r} on V × \overline{V}, homogeneous of respective bidegrees (1, 1), …, (r, r), such that for each regular x ∈ V, the polynomial

m(T, x, y) = T^{r} − m_{1}(x, y)T^{r−1} + ⋯ + (−1)^{r}m_{r}(x, y)

satisfies

m(T, x, x) = ∏_{i=1}^{r}(T − λ_{i}^{2}),

where x = λ_{1}c_{1} + λ_{2}c_{2} + ⋯ + λ_{r}c_{r} is the spectral decomposition of x. Here \overline{V} denotes the space V with the conjugate complex structure. The polynomial

m(T, x, y) = T^{r} − m_{1}(x, y)T^{r−1} + ⋯ + (−1)^{r}m_{r}(x, y)

is called the generic minimal polynomial of V (at (x, y)). The (inhomogeneous) polynomial N : V × \overline{V} → ℂ defined by

N(x, y) = m(1, x, y)

is called the generic norm. The following identities hold:

(1.3)  det B(x, y) = N(x, y)^{g},
tr D(x, y) = g m_{1}(x, y).

1.4. The spectral norm. Let V be an HPJTS. The map x ↦ λ_{1}, where x = λ_{1}c_{1} +λ_{2}c_{2} +⋯+λ_{p}c_{p} is the spectral decomposition of x (λ_{1} > λ_{2} > ⋯ > λ_{p} > 0) is a norm on V, called the spectral norm. The bounded symmetric domain Ω is the unit ball of V for the spectral norm. It is also characterized by the set of polynomial inequalities

\frac{∂^{j}}{∂T^{j}} m(T, x, x)|_{T =1} > 0, 0 ≤ j ≤ r − 1.

Proposition 1.1. Let

Φ : F × {λ_{1} > λ_{2} > ⋯ > λ_{r} > 0} ⟶ V_{reg}

be defined by

Φ((c_{1}, …, c_{r}), (λ_{1}, …, λ_{r})) = ∑_{j=1}^{r} λ_{j}c_{j} ;

here V_{reg} is the open dense subset of regular elements of V. Then Φ is a diffeomorphism; its restriction

Φ_{0} : F × {1 > λ_{1} > λ_{2} > ⋯ > λ_{r} > 0} ⟶ Ω_{reg} = Ω ∩ V_{reg}

is a diffeomorphism onto the open dense subset Ω_{reg} of regular elements of Ω.

Let α be the Kähler form on V associated to the Hermitian inner product m_{1}:

(1.4)  α = \frac{i}{2π} ∂\overline{∂}m_{1}.

In the following, we endow V with the volume form ω = α^{n} (n = dim_{ℂ} V), so that the volume of the unit ball associated to m_{1} is equal to 1. The following property is well-known (see e.g., [16, (5.1.1)]):

Proposition 1.2. Let V be an irreducible HPJTS of dimension n, rank r and invariants a, b. The pull-back of the volume element α^{n} by Φ is

Φ^{∗}α^{n} = Θ ∧ ∏_{j=1}^{r} λ_{j}^{2b+1} ∏_{1≤j<k≤r} (λ_{j}^{2} − λ_{k}^{2})^{a} dλ_{1} ∧ … ∧ dλ_{r},

where a and b are the numerical invariants of V and Θ is a K-invariant volume form on F.

2. Auxiliary results

2.1. The Selberg integral. Using an integral of Selberg, we compute in this subsection the integral ∫_{Ω} N(x, x)^{s}α^{n}, where N(x, x) is the generic norm of the irreducible bounded symmetric domain Ω. The integral of Selberg was used by Korányi [15] for computing the volume of a general bounded symmetric domain. The integral ∫_{Ω} N(x, x)^{s}α^{n} has been calculated by Hua [14] for the four series of classical domains.

Proposition 2.1. Let vol Ω and vol F be the volumes of Ω and F w.r. to α^{n} and Θ:

vol Ω = ∫_{Ω} α^{n}, vol F = ∫_{F} Θ.

Then, for s ∈ ℂ, Re s > −1

(2.1)  ∫_{Ω} N(x, x)^{s}α^{n} = F (s) vol F = \frac{F (s)}{F (0)} vol Ω,

with

F (s) = \frac{1}{2^{r}r!} ∏_{j=1}^{r} \frac{Γ(b + 1 + (j − 1)\frac{a}{2})Γ(s + 1 + (j − 1)\frac{a}{2})Γ(j \frac{a}{2} + 1)}{Γ(s + b + 2 + (r + j − 2)\frac{a}{2})Γ(\frac{a}{2} + 1)}, (2.2)

\frac{F (s)}{F (0)} = ∏_{j=1}^{r} \frac{Γ(s + 1 + (j − 1)\frac{a}{2})Γ(b + 2 + (r + j − 2)\frac{a}{2})}{Γ(1 + (j − 1)\frac{a}{2})Γ(s + b + 2 + (r + j − 2)\frac{a}{2})}.

Proof. According to Propositions 1.1 and 1.2, we have

F (s) = ∫ ⋯ ∫_{1>λ_{1}>λ_{2}>···>λ_{r}>0} ∏_{j=1}^{r}(1 − λ_{j}^{2})^{s} ∏_{j=1}^{r} λ_{j}^{2b+1} ∏_{1≤j<k≤r} (λ_{j}^{2} − λ_{k}^{2})^{a} dλ_{1} ∧ … ∧ dλ_{r} ;

by the change of variables t_{j} = λ_{j}^{2}, we have

F (s) = \frac{1}{2^{r}} ∫ ⋯ ∫_{1>λ_{1}>λ_{2}>···>λ_{r}>0} ∏_{j=1}^{r}(1 − t_{j})^{s} ∏_{j=1}^{r} t_{j}^{b} ∏_{1≤j<k≤r} (t_{j} − t_{k})^{a} dt_{1} ∧ … ∧ dt_{r}.

Extending to integration over the cube [0, 1]^{r}, we also have

F (s) = \frac{1}{2^{r}r!} ∫_{0}^{1} ⋯ ∫_{0}^{1} ∏_{j=1}^{r}(1 − t_{j})^{s} ∏_{j=1}^{r} t_{j}^{b} ∏_{1≤j<k≤r} |t_{j} − t_{k}|^{a} dt_{1} ∧ … ∧ dt_{r}.

The above integral has been evaluated by Selberg [22]:

Theorem 2.2 (Selberg). For Re x > 0, Re y > 0, Re z > − min (\frac{1}{n}, \frac{Re x}{n−1}, \frac{Re y}{n−1}), one has

∫_{0}^{1} ⋯ ∫_{0}^{1} ∏_{j=1}^{n} t_{j}^{x−1}(1 − t_{j})^{y−1} ∏_{1≤j<k≤n} |t_{j} − t_{k}|^{2z} dt_{1} … dt_{n}

= ∏_{j=1}^{n} \frac{Γ(x + (j − 1)z)Γ(y + (j − 1)z)Γ(jz + 1)}{Γ(x + y + (n + j − 2)z)Γ(z + 1)}.

See [1] for a simple proof.

Applying this result for n ← r, x ← b + 1, y ← s + 1, z ← \frac{a}{2}, we obtain the expression of F (s):

F (s) = \frac{1}{2^{r}r!} ∏_{j=1}^{r} \frac{Γ(b + 1 + (j − 1)\frac{a}{2})Γ(s + 1 + (j − 1)\frac{a}{2})Γ(j \frac{a}{2} + 1)}{Γ(s + b + 2 + (r + j − 2)\frac{a}{2})Γ(\frac{a}{2} + 1)},

valid for Re s > −1. It follows that

\frac{F (s)}{F (0)} = ∏_{j=1}^{r} \frac{Γ(s + 1 + (j − 1)\frac{a}{2})Γ(b + 2 + (r + j − 2)\frac{a}{2})}{Γ(1 + (j − 1)\frac{a}{2})Γ(s + b + 2 + (r + j − 2)\frac{a}{2})}.

This proves Proposition 2.1.

More precisely, ∫_{Ω} N(x, x)^{s}α^{n} is a very simple rational function of s:

Theorem 2.3. Let V be an irreducible HPJTS of dimension n. There exists a polynomial χ of degree n, the zeroes of which are all negative integers or half integers, such that

χ(s) ∫_{Ω} N(x, x)^{s}α^{n}

is independent of s. The polynomial χ is equal to

(2.3)  χ(s) = ∏_{j=1}^{r} (s + 1 + (j − 1)\frac{a}{2})_{1+b+(r−j)a} .

Here we use the classical notation (s)_{k} for the polynomial of degree k (“Pochhammer polynomial”):

(s)_{k} = ∏_{j=0}^{k−1} (s + j) = s(s + 1) ⋯ (s + k − 1).

Proof. It follows from (2.1) and (2.2) that ∫_{Ω} N(x, x)^{s}α^{n} is, up to a constant, equal to

(2.4)  \frac{1}{χ(s)} = ∏_{j=1}^{r} \frac{Γ(s + 1 + (j − 1)\frac{a}{2})}{Γ(s + b + 2 + (r + j − 2)\frac{a}{2})}.

This may also be written (changing j into r + 1 − j in the denominator)

χ(s) = \frac{∏_{j=1}^{r} Γ(s + b + 2 + (2r − j − 1)\frac{a}{2})}{∏_{j=1}^{r} Γ(s + 1 + (j − 1)\frac{a}{2})}

= ∏_{j=1}^{r} (s + 1 + (j − 1)\frac{a}{2})_{1+b+(r−j)a} .

□

As a consequence of Theorem 2.3, we have for s ∈ ℂ, Re s >-1

(2.5)  ∫_{Ω} N(x, x)^{s}α^{n} = \frac{χ(0)}{χ(s)} vol Ω.

Note also that the degree of χ:

∑_{j=1}^{r} (1 + b + (r − j)a) = r + rb + \frac{r(r − 1)}{2}a

is equal to the dimension of V.

2.2. Semi-Reinhardt domains.

Definition 2.1. A bounded domain D in ℂ^{m+n} is called a semi-Reinhardt domain if 0 ∈ D and if

(e^{iθ_{1}}w_{1}, …, e^{iθ_{m}}w_{m}, e^{iθ}z_{1}, ⋯, e^{iθ}z_{n}) ∈ D

for all (w, z) ∈ D and θ_{1}, …, θ_{m}, θ real.

That means, D is “Reinhardt” w.r. to w and is circular w.r. to z. Obviously, a semi-Reinhardt domain is a circular domain, but the converse is not true.

It is well-known that: if D_{1} is a Reinhardt domain containing the origin in ℂ^{m}, then {w_{1}^{j_{1}} … w_{m}^{j_{m}}} is a complete orthogonal system for D_{1}; if D_{2} is a circular domain containing the origin in ℂ^{n}, then a complete orthonormal system for D_{2} is given by

\{P_{ki}; k ∈ ℕ, 1 ≤ i ≤ m_{k} = \binom{n + k − 1}{k}\} ,

where, for any fixed k, {P_{k1}, P_{k2}, …, P_{km_{k}}} is an orthonormal basis of the space of homogeneous polynomials of degree k in z_{1}, …, z_{n}. We give hereunder a generalization of these facts for semi-Reinhardt domains.

Let D be a semi-Reinhardt domain in ℂ^{m+n}. For each multi-index j ∈ ℕ^{m}, consider the weighted scalar product on polynomials on ℂ^{n} defined by

(f | g)^{(j)} = ∫_{D} |w^{j}|^{2} f(z)\overline{g(z)}.

As D is circular w.r. to z, polynomials of different degrees are orthogonal for this scalar product. For each k ∈ ℕ, choose a basis of the space P_{k} of homogeneous polynomials of degree k in z_{1}, …, z_{n}, which is orthonormal w.r. to ( | )^{(j)}:

{P_{ki}^{(j)}}_{1≤i≤m_{k}} ,

where m_{k} = dim P_{k} = \binom{n+k−1}{k}. Then \{P_{ki}^{(j)}; k ∈ ℕ, 1 ≤ i ≤ m_{k}\} is a complete orthonormal system for the space of holomorphic functions such that

∫_{D} |w^{j}|^{2} |f(z)|^{2} < ∞.

Theorem 2.4. Let D be a semi-Reinhardt domain in ℂ^{m+n}. Then a complete orthonormal system for D is given by

{w_{1}^{j_{1}} … w_{m}^{j_{m}}P_{ki}^{(j)}(z); j = (j_{1}, …, j_{m}) ∈ ℕ^{m}, k ∈ ℕ, 1 ≤ i ≤ m_{k}} .

Proof. Let Φ_{jki}(w, z) = w^{j}P_{ki}^{(j)}(z). As D is semi-Reinhardt, it is clear that the system {Φ_{jki}} is orthonormal.

Let f(z, w) be a holomorphic, square-integrable function on D. Let b_{jki} = ∫_{D} f(w, z)\overline{Φ_{jki}(w, z)}. It suffices to show that

(2.6)  ∫_{D} |f(w, z)|^{2} = ∑_{j,k,!≤i≤m_{k}} |b_{jki}|^{2} .

Let D_{z} = {w; (z, w) ∈ D}. Then D_{z} is “Reinhardt”, f(w, z) = ∑ b_{j}(z)w^{j},

b_{j}(z) = \frac{∫_{w∈D_{z}} f(w, z)\overline{w}^{j}}{∫_{w∈D_{z}} |w^{j}|^{2}} ,

and ∫_{w∈D_{z}} |f(w, z)|^{2} = ∑_{j} |b_{j}(z)|^{2} ∫_{w∈D_{z}} |w^{j}|^{2}. The functions b_{j}(z) are holomorphic, for they may also be obtained locally as Cauchy integrals of f(z, w) on the Shilov boundary of a polydisc in ℂ_{w}^{m}. Let Ω be the projection of D on the second factor ℂ^{n}; then Ω is circular, b_{j} is square-integrable on Ω, b_{j} = ∑_{k,i} c_{jki}P_{ki}^{(j)},with

c_{jki} = ∫_{Ω} |w^{j}|^{2} b_{j}(z)\overline{P_{ki}^{(j)}}(z) = ∫_{Ω} f(w, z)\overline{w}^{j}\overline{P_{ki}^{(j)}}b_{jki} = b_{jki}

and ∫_{Ω} |w^{j}|^{2} |b_{j}(z)|^{2} = ∑ _{k∈ℕ, 1≤i≤m_{k}} |b_{jki}|^{2}. The equality (2.6) then easily follows. □

2.3. The principle of inflation. Let Ω be a bounded complete Hartogs domain in ℂ^{n+1}:

Ω = {(z, ζ); z ∈ D, ζ ∈ ℂ, |ζ|^{2} < ϕ(z)} ,

where ϕ is a bounded, positive, continuous function on some bounded domain D in ℂ^{n}. Due to the circular symmetry w.r. to the one-dimensional variable, the Bergman kernel of Ω can be written

K_{Ω}(z, ζ) = L(z, |ζ|^{2}).

The “Principle of inflation”, given in [4], allows to compute the Bergman kernel of the “inflated” domain G, obtained from Ω when replacing ζ by an m-dimensional variable Z and |ζ|^{2} by ∥Z∥^{2} = |Z_{1}|^{2} + ⋯ + |Z_{m}|^{2}.

Theorem 2.5. ([4]) Let G be defined by

G = {(z, Z); z ∈ D, Z ∈ ℂ^{m}, ∥Z∥^{2} < ϕ(z)} .

The spaces ℂ^{n+1} and ℂ^{n+m} being respectively endowed with the translation invariant volume forms β(z) ∧ \frac{i}{2π} ∂\overline{∂}|ζ|^{2} and β(z) ∧ (\frac{i}{2π} ∂\overline{∂} ∥Z∥^{2})^{m}, the Bergman kernel function K_{G} of G is

K_{G}(z, Z) = \frac{1}{m!} \frac{∂^{m−1}}{∂r^{m−1}} L(z, r)|_{r=∥Z∥^{2}} .

Trivial, but basic example: n = 0, ϕ = 1; Ω is the unit disc, G the unit hermitian ball,

K_{Ω}(ζ) = \frac{1}{(1 − |ζ|^{2})^{2}} , K_{G}(Z) = \frac{1}{(1 − ∥Z∥^{2})^{m+1}} .

Proof. See [4, Subsection 2.2].

3. Egg domains built on bounded symmetric domains

3.1. The domains Y (q, Ω; k) and E(p, q, Ω; k). Let Ω be any bounded irreducible circled homogeneous domain. For q ∈ ℕ and k ∈ ℝ, k > 0, define the domain Y (q, Ω; k) by

Y (q, Ω; k) := {(W, Z) ∈ ℂ^{q} × Ω; ∥W ∥^{2k} < N(Z, Z)};

for p, q ∈ ℕ and k ∈ ℝ, k > 0, define the domain E(p, q, Ω; k) by

E(p, q, Ω; k) := {(W_{1}, W_{2}, Z) ∈ ℂ^{p} × ℂ^{q} × Ω; ∥W_{1}∥^{2} + ∥W_{2}∥^{2k} < N(Z, Z)} .

Note that, as it is the case for Ω, the boundary of Y (q, Ω; k) and E(p, q, Ω; k) is not smooth when the rank of the symmetric domain Ω is greater than 1.

For Ω belonging to one of the four classical series, an explicit form of the Bergman kernel for Y (q, Ω; k) has been obtained by Weiping Yin ([29], [30]).

The spaces ℂ^{p} and ℂ^{q} are equipped with their canonical Hermitian structure and with the volume forms

ω_{1}(W_{1}) = (\frac{i}{2π} ∂\overline{∂} ∥W_{1}∥^{2})^{p} ,
ω_{2}(W_{2}) = (\frac{i}{2π} ∂\overline{∂} ∥W_{2}∥^{2})^{q} ,

which give volume 1 to the Hermitian unit balls. The Bergman kernel of Y (q, Ω; k) (resp. E(p, q, Ω; k)) is considered with respect to the volume form ω_{2}(W) ∧ ω(Z) (resp. ω_{1}(W_{1}) ∧ ω_{2}(W_{2}) ∧ ω(Z)), where ω = α^{n} is defined by (1.4).

3.2. Automorphisms. We consider here the case where p = q = 1 and we write for short

Y (Ω; k) = Y (1, Ω; k),
E(Ω; k) = E(1, 1, Ω; k).

Let Φ ∈ (Aut Ω)_{0}, the identity component of the automorphism group of Ω, and let JΦ(Z) = det DΦ(Z) be the complex Jacobian of Φ at Z ∈ Ω. Let Z_{0} be the inverse image of 0 under Φ: Φ(Z_{0}) = 0. From the relations B(Φ(Z), Φ(T)) = d Φ(Z) ∘ B(Z, T) ∘ d Φ(T)^{∗}, B(Z, 0) = I and det B(Z, T) = N(Z, T)^{g}, we obtain

(3.1)  1 = JΦ(Z)N(Z, Z_{0})^{g}\overline{JΦ(Z_{0})},

which shows that N(Z, Z_{0}) never vanishes when Z ∈ Ω. In particular, we have

(3.2)  JΦ(Z) = \frac{1}{\overline{JΦ(Z_{0})}N(Z, Z_{0})^{g}}

and

(3.3)  |JΦ(Z_{0})|^{2} = \frac{1}{N(Z_{0}, Z_{0})^{g}}

Consider the holomorphic function A( , Z_{0}) defined by

(3.4)  A(Z, Z_{0}) = \frac{N(Z_{0}, Z_{0})^{\frac{1}{2}}}{N(Z, Z_{0})} ;

up to multiplication by a complex number of modulus 1, JΦ(Z) is then the g-th power of A(Z, Z_{0}). As a bounded circled symmetric domain Ω is always convex and Z ↦ A(Z, Z_{0}) is a non-vanishing holomorphic function on Ω, the holomorphic function Z ↦ ln A(Z, Z_{0}) is well-defined on Ω (say, with the initial condition that ln A(Z_{0}, Z_{0}) is real); for each λ ∈ ℂ, we then define the holomorphic function Z ↦ A(Z, Z_{0})^{λ} by A(Z, Z_{0})^{λ} = exp λ ln A(Z, Z_{0}).

Lemma 3.1. Let Ω be a bounded irreducible circled homogeneous domain of genus g. Then for each Φ ∈ (Aut Ω)_{0} and for complex numbers α_{1}, α_{2} of modulus 1, the map Ψ : E(Ω; k) → E(Ω; k) defined by

(3.5)  Ψ(W_{1}, W_{2}, Z) = (α_{1}A(Z, Z_{0})W_{1}, α_{2}A(Z, Z_{0})^{\frac{1}{k}} W_{2}, Φ(Z)),

where Z_{0} = Φ^{−1}(0), is a holomorphic automorphism of E(Ω; k). Moreover, we have

(3.6)  |JΨ(W_{1}, W_{2}, Z)| = |A(Z, Z_{0})|^{1+ \frac{1}{k} +g} .

In particular,

(3.7)  |JΨ(W_{1}, W_{2}, Z_{0})|^{2} = N(Z_{0}, Z_{0})^{−1− \frac{1}{k} −g}.

Proof. From (3.2), (3.3), (3.4) we deduce that

(3.8)  |JΦ(Z)| = |A(Z, Z_{0})|^{g} .

From this, it is easily deduced that Ψ is a holomorphic automorphism of E(Ω; k). The differential of Ψ has the form

dΨ(W_{1}, W_{2}, Z) = ( \begin{matrix}α_{1}A(Z, Z_{0}) & 0 & 0 \\ 0 & α_{2}(A(Z, Z_{0}))^{\frac{1}{k}} & 0 \\ ∗ & ∗ & dΦ(Z) \end{matrix} ) ,

which implies

|JΨ(W_{1}, W_{2}, Z)| = |A(Z, Z_{0})|^{1+ \frac{1}{k} +g} ,

that is (3.6); (3.7) then follows from A(Z_{0}, Z_{0}) = N(Z_{0}, Z_{0})^{− \frac{1}{2}} . □

As a corollary (or directly along the same lines), we have

Corollary 3.2. Let Ω be a bounded irreducible circled homogeneous domain of genus g. Then for each Φ ∈ (Aut Ω)_{0} and for a complex number α of modulus 1, the map Ψ_{0} : Y (Ω; k) → Y (Ω; k) defined by

Ψ_{0}(W, Z) = (αA(Z, Z_{0})^{\frac{1}{k}} W, Φ(Z)),

where Φ(Z_{0}) = 0, is a holomorphic automorphism of Y (Ω; k). We have

|JΨ_{0}(W, Z)| = |A(Z, Z_{0})|^{\frac{1}{k} +g}

and

(3.9)  |JΨ_{0}(W, Z_{0})|^{2} = N(Z_{0}, Z_{0})^{− \frac{1}{k} −g}.

3.3. The Bergman kernel of Y (q, Ω; k). In view of the Principle of inflation, we consider first Y (Ω; k) = Y (1, Ω; k). Let K_{Y} denote the Bergman kernel of Y (Ω; k). Assume Φ ∈ (Aut Ω)_{0} is an automorphism of Ω which maps Z ∈ D to 0. Let W^{∗} = A(Z, Z)^{\frac{1}{k}} W; then by the transformation law of the Bergman kernel and (3.9), we have

K_{Y} (W, Z) = K_{Y} (W^{∗}, 0) N(Z, Z)^{− \frac{1}{k} −g}.

So the determination of K_{Y} reduces to computing K_{Y} (W^{∗}, 0).

The domain Y (Ω; k) is clearly semi-Reinhardt in ℂ^{1+n}; applying Theorem 2.4, it has a complete orthonormal system of the form

\{W^{j}P_{ℓi}^{(j)}(Z); j ∈ ℕ, k ∈ ℕ, 1 ≤ i ≤ m_{ℓ} = \binom{n + ℓ − 1}{ℓ}\} ,

where the P_{ℓi}^{(j)}’s are polynomials of degree ℓ on V; in particular, P_{01}^{(j)} is the constant polynomial a_{j} defined by

|a_{j}|^{2} ∫_{Y (Ω;k)} |W^{j}|^{2} ω_{2}(W) ∧ ω(Z) = 1.

The Bergman kernel of Y (Ω; k) at (W, 0) is then

K_{Y} (W, 0) = ∑ |W^{j}P_{ℓi}^{(j)}(0)|^{2} = ∑ |a_{j}|^{2} |W^{j}|^{2} .

Lemma 3.3.

(3.10)  |a_{j}|^{2} = \frac{1}{χ(0) vol Ω} (j + 1) χ (\frac{j + 1}{k}) .

Proof. We have

|a_{j}|^{−2} = ∫_{Y (Ω;k)} |W^{j}|^{2} ω_{2}(W) ∧ ω(Z)

= ∫_{Z∈Ω} (∫_{|W |^{2k}<N(Z,Z)} |W |^{2j} ω_{2}(W)) ω(Z);

as

∫_{|W |^{2k}<R^{2}} |W |^{2j} ω_{2}(W) = 2 ∫_{0}^{R^{1/k}} r^{2j+1}dr = \frac{R^{2(j+1)/k}}{j + 1} ,

then, by Theorem 2.3,

|a_{j}|^{−2} = \frac{1}{j + 1} ∫_{Ω} N(Z, Z)^{(j+1)/k}ω(Z) = \frac{1}{j + 1} \frac{χ(0)}{χ((j + 1)/k)} vol Ω.

This proves (3.10). □

So we have

K_{Y} (W, 0) = ∑ |a_{j}|^{2} |W^{j}|^{2} = \frac{1}{χ(0) vol Ω} ∑_{j=0}^{∞} (j + 1) χ (\frac{j + 1}{k}) |W |^{2j} .

Let F be defined by

F (X) = ∑_{j=0}^{∞} (\frac{j + 1}{k}) χ (\frac{j + 1}{k}) X^{j};

then

K_{Y} (W, 0) = \frac{k}{χ(0) vol Ω}F (|W |^{2})

and

K_{Y} (W, Z) = K_{Y} (W^{∗}, 0) N(Z, Z)^{− \frac{1}{k} −g}

= \frac{k}{χ(0) vol Ω}F (\frac{|W |^{2}}{N(Z, Z)^{1/k}} ) N(Z, Z)^{− \frac{1}{k} −g}.

So we have proved

Theorem 3.4. The Bergman kernel of Y (Ω; k) = Y (1, Ω; k) is

(3.11)  K_{Y} (W, Z) = \frac{k}{χ(0) vol Ω}F (\frac{|W |^{2}}{N(Z, Z)^{1/k}} ) N(Z, Z)^{− \frac{1}{k} −g},

where

(3.12)  F (X) = ∑_{j=0}^{∞} (\frac{j + 1}{k}) χ (\frac{j + 1}{k}) X^{j}.

As (\frac{j+1}{k}) χ (\frac{j+1}{k}) is a polynomial function of j, F is a rational function of X (a finite linear combination of derivatives of (1 − X)^{−1}). So K_{Y} (W, Z) is the product of the Bergman kernel of Ω:

K_{Ω}(Z) = \frac{1}{vol Ω} \frac{1}{N(Z, Z)^{g}}

by

\frac{k}{χ(0)}F (\frac{|W |^{2}}{N(Z, Z)^{1/k}} ) N(Z, Z)^{− \frac{1}{k}} = G (|W |^{2} , N(Z, Z)^{1/k}) ,

where G(X_{1}, X_{2}) is a rational function of X_{1}, X_{2}.

The Bergman kernel of Y (q, Ω; k) is then obtained from K_{Y} by using the Principle of inflation (Theorem 2.5):

Corollary 3.5. The Bergman kernel of Y (q, Ω; k) is

(3.13) K_{Y (q,Ω;k)}(W, Z) = \frac{1}{q!} \frac{k}{χ(0) vol Ω}F^{(q−1)} (\frac{∥W ∥^{2}}{N(Z, Z)^{1/k}} ) N(Z, Z)^{− \frac{q}{k} −g},

where F is the rational function defined by (3.12).

3.4. The Bergman kernel of E(p, q, Ω; k). Let K_{E} denote the Bergman kernel of E(Ω; k) = E(1, 1, Ω; k). Assume Φ ∈ (Aut Ω)_{0} is an automorphism of Ω which maps Z ∈ D to 0. Let W_{1}^{∗} = A(Z, Z)W_{1}, W_{2}^{∗} = A(Z, Z)^{\frac{1}{k}} W_{2}; then by the transformation law of the Bergman kernel and (3.7), we have

K_{E}(W_{1}, W_{2}, Z) = K_{E}(W_{1}^{∗}, W_{2}^{∗}, 0) N(Z, Z)^{−1− \frac{1}{k} −g}.

So the determination of K_{E} reduces to computing K_{E}(W_{1}^{∗}, W_{2}^{∗}, 0).

The domain E(Ω; k) is semi-Reinhardt in ℂ^{2+n}; applying Theorem 2.4, it has a complete orthonormal system of the form

\{W_{1}^{j_{1}}W_{2}^{j_{2}}P_{ℓi}^{(j_{1},j_{2})}(Z); j_{1}, j_{2}, ℓ ∈ ℕ, 1 ≤ i ≤ m_{ℓ} = \binom{n + ℓ − 1}{ℓ}\} ,

where the P_{ℓi}^{(j_{1},j_{2})}’s are polynomials of degree ℓ; in particular, P_{01}^{(j_{1},j_{2})} is the constant polynomial a_{j_{1}j_{2}} defined by

|a_{j_{1}j_{2}}|^{2} ∫_{E(Ω;k)} |W_{1}^{j_{1}}W_{2}^{j_{2}} |^{2} ω_{1}(W_{1}) ∧ ω_{2}(W_{2}) ∧ ω(Z) = 1.

The Bergman kernel of E(Ω; k) at (W_{1}, W_{2}, 0) is then

(3.14)  K_{E}(W_{1}, W_{2}, 0) = ∑ |W_{1}^{j_{1}}W_{2}^{j_{2}}P_{ℓi}^{(j_{1},j_{2})}(0)|^{2}
= ∑ |a_{j_{1}j_{2}}|^{2} |W_{1}^{j_{1}} |^{2} |W_{2}^{j_{2}} |^{2} .

Lemma 3.6. Let h = j_{1} + 1 + \frac{j_{2}+1}{k}; then

(3.15)  |a_{j_{1}j_{2}}|^{2} = \frac{k}{χ(0) vol Ω} \frac{Γ (h + 1) χ(h)}{Γ(j_{1} + 1)Γ (\frac{j_{2}+1}{k})}.

Proof. We have

|a_{j_{1}j_{2}}|^{−2} = ∫_{E(Ω;k)} |W_{1}^{j_{1}}W_{2}^{j_{2}} |^{2} ω_{1}(W_{1}) ∧ ω_{2}(W_{2}) ∧ ω(Z)

= ∫_{Z∈Ω} (∫_{|W_{1}|^{2}+|W_{2}|^{2k}<N(Z,Z)} |W_{1}^{j_{1}}W_{2}^{j_{2}} |^{2} ω_{1}(W_{1}) ∧ ω_{2}(W_{2})) ω(Z)

and, by standard computations,

∫_{|W_{1}|^{2}+|W_{2}|^{2k}<R^{2}} |W_{1}^{j_{1}}W_{2}^{j_{2}} |^{2} ω_{1}(W_{1}) ∧ ω_{2}(W_{2})

= 4 ∫_{r_{1}^{2}+r_{2}^{2k}<R^{2}, r_{j}≥0} r_{1}^{2j_{1}+1}r_{2}^{2j_{2}+1}dr_{1}dr_{2}

= \frac{4}{k} ∫_{s_{1}^{2}+s_{2}^{2}<R^{2}, s_{j}≥0} s_{1}^{2j_{1}+1}s_{2}^{\frac{2(j_{2}+1)}{k} −1}ds_{1}ds_{2}

= \frac{4}{k} ∫_{0}^{R} ρ^{2j_{1}+ \frac{2(j_{2}+1)}{k} +1}dρ ∫_{0}^{π/2} cos^{2j_{1}+1} θ sin^{\frac{2(j_{2}+1)}{k} −1} θ dθ

= \frac{4}{k} \frac{R^{2(j_{1}+1+ \frac{j_{2}+1}{k})}}{2 (j_{1} + 1 + \frac{j_{2}+1}{k})} \frac{Γ(j_{1} + 1)Γ (\frac{j_{2}+1}{k})}{2Γ (j_{1} + 1 + \frac{j_{2}+1}{k})}

= \frac{1}{k} \frac{Γ(j_{1} + 1)Γ (\frac{j_{2}+1}{k})}{Γ (j_{1} + 1 + \frac{j_{2}+1}{k} + 1)}R^{2(j_{1}+1+ \frac{j_{2}+1}{k})}.

16 GUY ROOS AND WEIPING YIN

Let h = j_{1} + 1 + \frac{j_{2}+1}{k}; then, by Theorem 2.3,

|a_{j_{1}j_{2}}|^{−2} = \frac{1}{k} \frac{Γ(j_{1} + 1)Γ (\frac{j_{2}+1}{k})}{Γ (h + 1)} ∫_{Ω} N(Z, Z)^{h}ω(Z)

= \frac{1}{k} \frac{Γ(j_{1} + 1)Γ (\frac{j_{2}+1}{k})}{Γ (h + 1)} \frac{χ(0)}{χ(h)} vol Ω,

which proves (3.15). □

From (3.15) and (3.14), we obtain

K_{E}(W_{1}, W_{2}, 0) = \frac{1}{vol Ω} \frac{k}{χ(0)} ∑_{j_{1},j_{2}=0}^{∞} \frac{Γ (h + 1) χ(h)}{Γ(j_{1} + 1)Γ (\frac{j_{2}+1}{k})} |W_{1}^{j_{1}} |^{2} |W_{2}^{j_{2}} |^{2} .

Let b_{j} (1 ≤ j ≤ n + 2) be the constants defined by

h(h − 1)χ(h) = ∑_{j=1}^{n+2} b_{j} (h + 1)_{j} = ∑_{j=1}^{n+2} b_{j} \frac{Γ(h + j + 1)}{Γ(h + 1)};

so

|a_{j_{1}j_{2}}|^{2} = \frac{1}{vol Ω} \frac{k}{χ(0)} \frac{Γ (h + 1) χ(h)}{Γ(j_{1} + 1)Γ (\frac{j_{2}+1}{k})}

= \frac{k}{χ(0) vol Ω} ∑_{j=1}^{n+2} b_{j} \frac{Γ(h + j + 1)Γ(h − 1)}{Γ(h + 1)Γ(j_{1} + 1)Γ (\frac{j_{2}+1}{k})}

and

K_{E}(W_{1}, W_{2}, 0) = \frac{1}{vol Ω} \frac{k}{χ(0)}

∑_{j_{1},j_{2}=0}^{∞} ∑_{j=1}^{n+2} b_{j} \frac{Γ(h + j + 1)Γ(h − 1)}{Γ(h + 1)Γ(j_{1} + 1)Γ (\frac{j_{2}+1}{k})} |W_{1}^{j_{1}} |^{2} |W_{2}^{j_{2}} |^{2} .

Let s = (j_{2} + 1)/k, t_{1} = |W_{1}|^{2}, t_{2} = |W_{2}|^{2}, then we have

K_{E}(W_{1}, W_{2}, 0) = Λ(t_{1}, t_{2})

where

(3.16) Λ(t_{1}, t_{2}) = \frac{1}{vol Ω} \frac{k}{χ(0)} ∑_{j=1}^{n+2} b_{j} ∑_{j_{2}=0}^{∞} ( ∑_{j_{1}=0}^{∞} \frac{Γ(h + j + 1)Γ(j_{1} + s)}{Γ(h + 1)Γ(j_{1} + 1)Γ (s)}t_{1}^{j_{1}} ) t_{2}^{j_{2}}.

But

∑_{j_{1}=0}^{∞} \frac{Γ(h + j + 1)Γ(j_{1} + s)}{Γ(h + 1)Γ(j_{1} + 1)Γ (s)}t_{1}^{j_{1}} = ∑_{j_{1}=0}^{∞} \frac{Γ(j_{1} + j + 2 + s)Γ(j_{1} + s)}{Γ(j_{1} + 2 + s)Γ(j_{1} + 1)Γ (s)}t_{1}^{j_{1}}

= \frac{Γ(j + 2 + s)}{Γ(s + 2)}_{2}F_{1}(s + 2 + j, s; s + 2; t_{1})

where _{2}F_{1}(a, b; c; z) denotes the Gauss hypergeometric function

2F_{1}(a, b; c; z) = ∑_{m=0}^{∞} \frac{(a)_{m} (b)_{m}}{(c)_{m}} \frac{z^{m}}{Γ(m + 1)}.

By Euler’s transformation formula (see [10, (1.2.10)], or [7]):

2F_{1}(a, b; c; z) = (1 − z)^{c−a−b}_{2}F_{1}(c − a, c − b; c; z),

we get

2F_{1}(s + 2 + j, s; s + 2; t_{1}) = (1 − t_{1})^{−s−j}_{2}F_{1}(−j, 2; s + 2; t_{1}),

where _{2}F_{1}(−j, 2; s + 2; t_{1}) is in fact a polynomial of degree j:

2F_{1}(−j, 2; s + 2; t_{1}) = ∑_{m=0}^{j} \frac{(−j)_{m} (2)_{m}}{(s + 2)_{m}} \frac{t_{1}^{m}}{Γ(m + 1)}.

(It can be expressed with help of a Jacobi polynomial, but we will not use this fact here.) This proves

∑_{j_{1}=0}^{∞} \frac{Γ(h + j + 1)Γ(j_{1} + s)}{Γ(h + 1)Γ(j_{1} + 1)Γ (s)}t_{1}^{j_{1}} (3.17)

= Γ(j + 2 + s)(1 − t_{1})^{−s−j} ∑_{m=0}^{j} \frac{(−j)_{m} (2)_{m}}{Γ(s + 2 + m)} \frac{t_{1}^{m}}{Γ(m + 1)}.

From (3.16) and (3.17), we deduce

Λ(t_{1}, t_{2}) = \frac{1}{vol Ω} \frac{k}{χ(0)}(1 − t_{1})^{−1/k} ∑_{j=1}^{n+2} b_{j}(1 − t_{1})^{−j}

∑_{m=0}^{j}(−j)_{m} (2)_{m} \frac{t_{1}^{m}}{Γ(m + 1)} ∑_{j_{2}=0}^{∞} \frac{Γ(j + 2 + \frac{j_{2}+1}{k})}{Γ(\frac{j_{2}+1}{k} + 2 + m)} (\frac{t_{2}}{(1 − t_{1})^{1/k}} )^{j_{2}} .

Consider now, for 0 ≤ m ≤ j, the function H_{jm} defined by

H_{jm}(λ) = ∑_{p=0}^{∞} \frac{Γ(j + 2 + \frac{p+1}{k})}{Γ(\frac{p+1}{k} + 2 + m)}λ^{p} = ∑_{p=0}^{∞} (\frac{p + 1}{k} + 2 + m)_{j−m} λ^{p};

as (\frac{p+1}{k} + 2 + m)_{j−m} is a polynomial of p, H_{jm} is a rational function of λ (more precisely, a finite linear combination of derivatives of the function (1 − λ)^{−1}). So we finally obtain:

Theorem 3.7. The Bergman kernel K_{E} of E(Ω; k) is

(3.18)  K_{E}(W_{1}, W_{2}, Z) = Λ (\frac{|W_{1}|^{2}}{N(Z, Z)}, \frac{|W_{2}|^{2}}{N(Z, Z)^{1/k}} ) N(Z, Z)^{−1− \frac{1}{k} −g},

where Λ is defined by

Λ(t_{1}, t_{2}) = \frac{1}{vol Ω} \frac{k}{χ(0)}(1 − t_{1})^{−1/k}H (t_{1}, \frac{t_{2}}{(1 − t_{1})^{1/k}} ) , (3.19)

H(t_{1}, λ) = ∑_{j=1}^{n+2} b_{j}(1 − t_{1})^{−j} ∑_{m=0}^{j}(−j)_{m} (2)_{m} \frac{t_{1}^{m}}{Γ(m + 1)}H_{jm} (λ) ,

H_{jm}(λ) = ∑_{p=0}^{∞} (\frac{p + 1}{k} + 2 + m)_{j−m} λ^{p},

h(h − 1)χ(h) = ∑_{j=1}^{n+2} b_{j} (h + 1)_{j} .

We see that K_{E}(W_{1}, W_{2}, Z) is the product of the Bergman kernel of Ω:

K_{Ω}(Z) = \frac{1}{vol Ω} \frac{1}{N(Z, Z)^{g}}

by the function

\frac{k}{χ(0)}H (\frac{|W_{1}|^{2}}{N(Z, Z)}, \begin{matrix}|W_{2}| & 2 \\ (N(Z, Z) − |W_{1}|^{2}) & 1/k \end{matrix} ) N(Z, Z)^{−1− \frac{1}{k}} ,

where H(X_{1}, X_{2}) is a rational function of X_{1}, X_{2}.

The Bergman kernel K_{E(p,q,Ω;k)} of E(p, q, Ω; k) is then obtained from K_{E} using twice the Principle of inflation (Theorem 2.5):

Corollary 3.8. The Bergman kernel of E(p, q, Ω; k) is

K_{E(p,q,Ω;k)}(W_{1}, W_{2}, Z)

= \frac{1}{p!q!}Λ^{(p−1,q−1)} (\frac{∥W_{1}∥^{2}}{N(Z, Z)}, \frac{∥W_{2}∥^{2}}{N(Z, Z)^{1/k}} ) N(Z, Z)^{−p− \frac{q}{k} −g},

where

Λ^{(p−1,q−1)} (t_{1}, t_{2}) = \frac{∂^{p+q−2}Λ}{∂t_{1}^{p−1}∂t_{2}^{q−1}} (t_{1}, t_{2})

is the partial derivative of the function Λ defined by (3.19).

4. Tables

The following examples exhaust the list of simple Hermitian positive Jordan triple systems (see [17]). The HPJTS occurring in the four infinite series I_{p,q}, II_{n}, III_{n}, IV_{n} are called classical; the two HPJTS of type V and V I are called exceptional. There is some overlapping between the classical series, due to a finite number of isomorphisms in low dimension. We give hereunder for each type:

• the definition of the space V, its Jordan triple product, and the corresponding bounded circled homogeneous domain;

• the generic norm;

• the numerical invariants r, a, b, g = 2 + a(r − 1) + b;

• the polynomial

χ(s) = ∏_{j=1}^{r} (s + 1 + (j − 1)\frac{a}{2})_{1+b+(r−j)a} .

from which one can deduce the behavior of the function

s ↦ ∫_{Ω} N(x, x)^{s}α^{n}

and the Bergman kernels of Y (q, Ω; k) and E(p, q, Ω; k).

4.1. Type I_{m,n} (1 ≤ m ≤ n). V = M_{m,n}(ℂ) (space of m×n matrices with complex entries), endowed with the triple product

\{xyz\} = x^{t}\overline{y}z + z^{t}\overline{y}x.

The domain Ω is the set of m × n matrices x such that I_{m} − x^{t}\overline{x} is definite positive. The generic minimal polynomial is

m(T, x, y) = Det(T I_{m} − x^{t}\overline{y}),

where Det is the usual determinant of square matrices. The numerical invariants are r = m, a = 2, b = n − m, g = m + n. These HPJTS are of tube type only for m = n.

Applying (2.3), we obtain the polynomial χ:

χ(s) = ∏_{j=1}^{m} (s + j)_{m+n+1−2j} ,

which may also be written

χ(s) = ∏_{j=1}^{m}(s + j)_{n}.

4.2. Type II_{n} (n ≥ 2). V = A_{n}(ℂ) (space of n × n alternating matrices) with the same triple product as for Type I. The domain Ω is the set of n × n alternating matrices x such that I_{n} + x\overline{x} is definite positive.

4.2.1. Type II_{2p} (n = 2p even). The generic minimal polynomial is here given by

m(T, x, y)^{2} = Det(T I_{n} + x\overline{y}).

The numerical invariants are r = \frac{n}{2} = p, a = 4, b = 0, g = 2 (n − 1); these HPJTS are of tube type.

The polynomial χ is

χ(s) = ∏_{j=1}^{p} (s + 2j − 1)_{1+4(p−j)} = ∏_{j=1}^{p}(s + 2j − 1)_{2p−1}.

4.2.2. Type II_{2p+1} (n = 2p + 1 odd). The generic minimal polynomial is given by

T m(T, x, y)^{2} = Det(T I_{n} + x\overline{y}).

The numerical invariants are r = [\frac{n}{2}] = p, a = 4, b = 2, g = 2(n − 1); these HPJTS are not of tube type.

The polynomial χ is

χ(s) = ∏_{j=1}^{p} (s + 2j − 1)_{3+4(p−j)} = ∏_{j=1}^{p}(s + 2j − 1)_{2p+1}.

4.3. Type III_{n} (n ≥ 1). V = S_{n}(ℂ) (space of n × n symmetric matrices) with the same triple product as for Type I. The domain Ω is the set of n × n symmetric matrices x such that I_{n} − x\overline{x} is definite positive. The generic minimal polynomial is

m(T, x, y) = Det(T I_{n} − x\overline{y}).

The numerical invariants are r = n, a = 1, b = 0, g = n + 1. These HPJTS are of tube type.

The polynomial χ is

χ(s) = ∏_{j=1}^{n} (s + \frac{j + 1}{2})_{1+n−j} .

4.4. Type IV_{n} (n ≠ 2). V = ℂ^{n} with the quadratic operator defined by

Q(x)y = q(x, \overline{y})x − q(x)\overline{y},

where q(x) = ∑ x_{i}^{2}, q(x, y) = 2 ∑ x_{i}y_{i}. The domain Ω is the set of points x ∈ ℂ^{n} such that

1 − q(x, \overline{x}) + |q(x)|^{2} > 0, 2 − q(x, \overline{x}) > 0.

The generic minimal polynomial is

m(T, x, y) = T^{2} − q(x, \overline{y}) + q(x)q(\overline{y}).

The numerical invariants are r = 2, a = n − 2, b = 0, g = n. These HPJTS are of tube type.

The polynomial χ is

χ(s) = (s + 1)_{n−1} (s + \frac{n}{2}) .

4.5. Type V. V = M_{2,1}(𝕆_{ℂ}), the subspace of H_{3}(𝕆_{ℂ}) consisting in matrices of the form

( \begin{matrix}0 & a_{3} & ã_{2} \\ ã_{3} & 0 & 0 \\ a_{2} & 0 & 0 \end{matrix} )

with the same quadratic operator as for type VI (see below). Here ã denotes the Cayley conjugate of a ∈ 𝕆_{ℂ}. The generic minimal polynomial is

m(T, x, y) = T^{2} − (x|y)T + (x^{♯}|y^{♯}).

The domain Ω is the “exceptional domain of dimension 16” defined by

1 − (x|x) + (x | x^{♯}) > 0, 2 − (x|x) > 0.

The numerical invariants are r = 2, a = 6, b = 4, g = 12. This HPJTS is not of tube type.

The polynomial χ is

χ(s) = ∏_{j=1}^{r} (s + 1 + (j − 1)\frac{a}{2})_{1+b+(r−j)a} .

χ(s) = (s + 1)_{11} (s + 4)_{5} ;

it can also be written

χ(s) = (s + 1)_{8}(s + 4)_{8}.

4.6. Type VI. V = H_{3}(𝕆_{ℂ}), the space of 3 × 3 matrices with entries in the space 𝕆_{ℂ} of octonions over ℂ, which are Hermitian with respect to the Cayley conjugation; the quadratic operator is defined by

Q(x)y = (x|y)x − x^{♯} × \overline{y},

where × denotes the Freudenthal product, x^{♯} the adjoint matrix in H_{3}(𝕆_{ℂ}) and (x|y) the standard Hermitian product in H_{3}(𝕆_{ℂ}) (for details, see [20]). The domain Ω is the “exceptional domain of dimension 27” defined by

1 − (x|x) + (x^{♯}|x^{♯}) − |det x|^{2} > 0,
3 − 2(x|x) + (x^{♯}|x^{♯}) > 0,
3 − (x|x) > 0.

The generic minimal polynomial is

m(T, x, y) = T^{3} − (x|y)T^{2} + (x^{♯}|y^{♯})T − det x det \overline{y},

where det denotes the determinant in H_{3}(𝕆_{ℂ}). The numerical invariants are r = 3, a = 8, b = 0, g = 18. This HPJTS is of tube type.

The polynomial χ is

χ(s) = (s + 1)_{17}(s + 5)_{9}(s + 9)
= (s + 1)_{9}(s + 5)_{9}(s + 9)_{9}.

References

[1] Aomoto K., Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal. 18(1987), 545-549.

[2] Bergman S., Zur Theorie von pseudokonformen Abbildungen, Mat. Sb. (N.S.), 1(1) (1936), 79-96.

[3] Boas H., Counterexample to the Lu Qi-keng conjecture, Proc. Amer. Math. Soc., 97(1986), 374-375.

[4] Boas H., Fu Siqi, Straube E., The Bergman kernel function: explicit formulas and zeroes, Proc. Amer. Math. Soc., 127(3) (1999), 805-811.

[5] D’Angelo J.P., A note on the Bergman kernel, Duke Math. J., 45(2) (1978), 259-265.

[6] D’Angelo J.P., An explicit computation of the Bergman kernel function, J. Geom. Analysis, 4(1) (1994), 23-34.

[7] Euler L., Introductio in Analysis Infinitorum, Bousquet, Lausanne, 1748.

[8] Faraut Jacques, Korányi Adam, Analysis on Symmetric Cones, Clarendon Press, Oxford, 1994.

[9] Francsics G., Hanges N., The Bergman kernel and hypergeometric functions, J. Funct. Anal., 142(2) (1996), 494-510.

[10] Gasper G., Rahman M., Basic Hypergeometric Series, Encyclopedia of Mathematics and its Applications, vol.35, Cambridge University Press, Cambridge, 1990.

[11] Gindikin S.G., Analysis in homogeneous domains, Russian Math. Surveys, 19(1984), 1-89.

[12] Gong Sheng, Zheng Xuean, The Bergman kernel function of the Reinhardt domains (I, II) (in Chinese), Science in China (A series; Chinese version), 25(1995), 580-590 and 683-692.

[13] Gong Sheng, Zheng Xuean, The Bergman kernel function of some Reinhardt domains, Trans. Amer. Math. Soc., 348(5) (1996), 1771-1803.

[14] Hua L.K., Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Amer. Math. Soc., Providence, RI, 1963. (Translated from Хуа Ло-Кен, Гармонический анализ функций многих переменных в классических областях, Изд. Иностранной Литературы, Москва, 1959 ; original Chinese text published by Science Press, Peking, 1958).

[15] Korányi A., The volume of symmetric domains, the Koecher gamma function and an integral of Selberg, Studia Sci. Math. Hungar., 17, 129-133, 1982.

[16] Korányi A., Function spaces on bounded symmetric domains, pp. 183-282, in J.Faraut, S.Kaneyuki, A.Korányi, Q.-k.Lu, G.Roos, Analysis and Geometry on Complex Homogeneous Domains, Progress in Mathematics, Birkhäuser, Boston, 1999.

[17] Loos, Ottmar, Jordan Pairs, Lecture Notes in Mathematics, 460, Springer-Verlag, Berlin-Heidelberg-New York, 1975.

[18] Loos, Ottmar, Bounded symmetric domains and Jordan pairs, Math. Lectures, Univ. of California, Irvine, 1977.

[19] Mostow G D, Siu Y.-T., A compact Kähler surface of negative curvature not covered by the ball, Ann. of Math., 112(1980), 321-360.

[20] Roos, Guy, Algèbres de composition, Systèmes triples de Jordan exceptionnels, pp. 1-84, in G.Roos, J.P.Vigué, Systèmes triples de Jordan et domaines symétriques, Travaux en cours, 43, Hermann, Paris, 1992.

[21] Roos, Guy, Jordan triple systems, pp. 425-534, in J.Faraut, S.Kaneyuki, A.Korányi, Q.-k.Lu, G.Roos, Analysis and Geometry on Complex Homogeneous Domains, Progress in Mathematics, Birkhäuser, Boston, 1999.

[22] Selberg A., Bemerkninger om et multiplet integral, Norske Mat. Tidsskr. 26(1944), 71-78.

[23] Tiao, Chieh-hsien, The Bergman kernel on monomial polyhedra, Michigan Math. J., 46(1), 53-69.

[24] Xu Yichao, On the Bergman functions of homogeneous bounded domains, Sci. Sinica, Special Issue II (1979), 80-90.

[25] Yin Weiping, Two problems on Cartan domains, J. of China Univ. of Sci. and Techn., 16(2) (1986), 130-146.

[26] Yin Weiping, The unitary geometry of non-symmetric homogeneous Siegel domains of the first type(I), Acta Math. Sinica, 24(5)(1981), 753-764.

[27] Yin Weiping, The unitary geometry of non-symmetric homogeneous Siegel domains of the first type(II), Acta Math. Sinica, 24(5)(1981), 765-779.

[28] Yin Weiping, The unitary geometry of non-symmetric homogeneous Siegel domains of the first type(III), Acta Math. Sinica, 24(6)(1981), 870-891.

[29] Yin Weiping, The Bergman kernel on Super-Cartan domain of the first type, Science in China (A Series), 29(7)(1999), 607-615.

[30] Yin Weiping , The Bergman kernel on four types of super-Cartan domains, Chinese Science Bulletin (A Series), 44(13)(1999), 1391-1395.

[31] Zhong Jiaqing, Yin Weiping, Some types of non-symmetric homogeneous domains (in Chinese), Acta Math. Sinica, 24(4)(1981), 587-613.

[32] Zhong Jiaqing, Yin Weiping, Extension spaces of non-symmetric classical domains (in Chinese), Acta Math. Sinica, 24(4)(1981), 614-640.

[33] Zinov’ev B.S., On reproducing kernels for multicircular domains of holomorphy, Siberian Math.J., 15(1974), 24-33.

G.R.: 58 avenue de Versailles 75016 Paris France

E-mail address: guy.roos@laposte.net, roos@mx.amss.ac.cn

W.Y.: Dept. of Math., Capital Normal Univ., Beijing 100037, China

E-mail address: wyin@mail.cnu.edu.cn