WEIGHTED BERGMAN KERNELS AND VIRTUAL BERGMAN KERNELS

GUY ROOS

Abstract. We introduce the notion of virtual Bergman kernel and study some of its applications.

Contents

Introduction  1
1. Virtual Bergman kernels  2
2. Virtual Bergman kernels for bounded symmetric domains  7
3. Tables for bounded symmetric domains  9
4. Open problems  11
References  12

Introduction

Let Ω ⊂ ℂ^{n} be a domain and p : Ω →]0, +∞[ a weight function on Ω. Consider the “inflated domains”

Ω̂_{1} = {(z, ζ) ∈ Ω × ℂ | |ζ|^{2} < p(z)} ,  (0.1)
Ω̂_{m} = {(z, Z) ∈ Ω × ℂ^{m} | ∥Z∥^{2} < p(z)} ,  (0.2)

where ∥ ∥ is the standard Hermitian norm on ℂ^{m}.

In our joint work [1] with Yin Weiping, we computed explicitly the Bergman kernel of some “egg domains”; among them Ω̂_{1}, when Ω is a bounded symmetric domain and p a real power of the generic norm of Ω. We then obtained the Bergman kernel of the corresponding Ω̂_{m} by using the “inflation principle” of [2], which allows to deduce (for any weight function p) the Bergman kernel of Ω̂_{m} from the Bergman kernel of Ω̂_{1}. The “inflation principle” says that if the Bergman kernel of Ω̂_{1} is

K̂_{1}(z, ζ) = L_{1} (z, |ζ|^{2}) ,

then the Bergman kernel of Ω̂_{m} is

K̂_{m}(z, Z) = \frac{1}{m!} \frac{∂^{m−1}}{∂r^{m−1}} L_{1}(z, r)|_{r=∥Z∥^{2}} .  (0.3)

Date: August 25, 2004.

2000 Mathematics Subject Classification. 32A36, 32M15.

Key words and phrases. Bergman kernel, weighted Bergman kernel, virtual Bergman kernel.

One hour lecture for graduate students, SCV2004, Beijing.

It appears that the two previous steps can be unified in the following way. There exists a function L_{0}(z, r), defined in a neighborhood of Ω × {0} in Ω × [0, +∞[, such that for all m ≥ 1, the Bergman kernel of Ω̂_{m} is

K̂_{m}(z, Z) = \frac{1}{m!} \frac{∂^{m}}{∂r^{m}} L_{0}(z, r)|_{r=∥Z∥^{2}} .  (0.4)

We call L_{0}(z, r) the virtual Bergman kernel of (Ω, p). Its existence is closely related to the “Forelli-Rudin construction” ([3], [4], [5]).

In this talk, we investigate the properties of this virtual Bergman kernel. We then show how it can be explicitly computed on bounded symmetric domains, for a special but natural choice of the weight function p: p is taken to be a real power of the “generic norm” of the bounded symmetric domain. The explicit computation of the virtual Bergman kernel is then related to properties of the Hua integral.

1. Virtual Bergman kernels

1.1. Notations. Let V ≅ ℂ^{n} be a Hermitian vector space, with Hermitian norm ∥ ∥_{V} and volume form ω_{V} (z) = (\frac{i}{2π} ∂\overline{∂} ∥z∥^{2})^{n}. Let Ω be a domain in V and p : Ω →]0, +∞[ a continuous function on Ω. The space of holomorphic functions on Ω is denoted by Hol(Ω). We denote by H(Ω) the Bergman space

H(Ω) = H (Ω, ω_{V}) = {f ∈ Hol(Ω) | ∥f∥_{Ω}^{2} = ∫_{Ω} |f(z)|^{2} ω_{V} (z) < ∞}

and by H(Ω, p) the weighted Bergman space

H(Ω, p) = H (Ω, pω_{V}) = {f ∈ Hol(Ω) | ∥f∥_{Ω,p}^{2} = ∫_{Ω} |f(z)|^{2} p(z)ω_{V} (z) < ∞} .

The Hilbert products on these spaces are denoted respectively ( | )_{Ω} and ( | )_{Ω,p}. The Bergman kernel of Ω (reproducing kernel of H(Ω)) is denoted by K_{Ω}(z, t); it is fully determined by K_{Ω}:

K_{Ω}(z) = K_{Ω}(z, z) (z ∈ Ω) ,

which we call also Bergman kernel of Ω. In the same way, the weighted Bergman kernel of (Ω, p) (reproducing kernel of H(Ω, p)) is denoted by K_{Ω,p}(z, t) and is determined by K_{Ω,p}(z) = K_{Ω,p}(z, z).

1.2. Let

Ω̂_{1} = {(z, ζ) ∈ Ω × ℂ | |ζ|^{2} < p(z)} .

We endow Ω̂_{1} with the volume form

ω_{V} (z) ∧ ω_{1}(ζ),

where ω_{1}(ζ) = \frac{i}{2π} ∂\overline{∂} |z|^{2}. The Bergman space H \binom{}{Ω_{1}} will be defined with respect to this volume form.

Consider a holomorphic function f ∈ Hol \binom{}{Ω_{1}}; such a function may be written

f(z, ζ) = ∑_{k=0}^{∞} f_{k}(z)ζ^{k},

with f_{k} ∈ Hol Ω. We compute ∥f∥2 ; Ω̂_{1}:

∥f∥2 ; Ω̂_{1} = ∫Ω̂_{1} |f(z, ζ)|^{2} ω_{V} (z) ∧ ω_{1}(ζ)

= ∫_{Ω} ω_{V} (z) (∫_{|ζ|^{2}<p(z)} |f(z, ζ)|^{2} ω_{1}(ζ))

= ∫_{Ω} ω_{V} (z) (∑_{k=0}^{∞} |f_{k}(z)|^{2} \frac{p^{k+1}(z)}{k + 1}) ,

which gives

∥f∥2 ; Ω̂_{1} = ∑_{k=0}^{∞} \frac{1}{k + 1} ∥f_{k}∥_{Ω,p^{k+1}}^{2} .  (1.1)

Proposition 1.1. Let K̂_{1} be the Bergman kernel of Ω̂_{1} and K_{Ω,p^{k}} the weighted Bergman kernel of Ω for the weight function p^{k}. Then

K̂_{1}(z, ζ) = ∑_{k=0}^{∞} (k + 1) K_{Ω,p^{k+1}}(z) |ζ|^{2k} .  (1.2)

Proof. For each k ∈ ℕ, let (φ_{jk})_{j∈J_{k}} be a Hilbert basis (complete orthonormal system) of H (Ω, p^{k}). Then it follows from (1.1) that

((k + 1)^{1/2} φ_{j,k+1}(z)ζ^{k})_{k∈ℕ, j∈J_{k+1}}

is a Hilbert basis of H \binom{}{Ω_{1}}. From the classical properties of Bergman kernels, we get

K_{Ω,p^{k}}(z) = ∑_{j∈J_{k}} |φ_{jk}(z)|^{2} ,
K̂_{1}(z, ζ) = ∑_{k∈ℕ, j∈J_{k}} (k + 1) |φ_{j,k+1}(z)|^{2} |ζ|^{2k}

= ∑_{k=0}^{∞} (k + 1) K_{Ω,p^{k+1}}(z) |ζ|^{2k} .

□

1.3. This leads to the following definition.

Definition 1.1. Let Ω be a domain in V and p : Ω →]0, +∞[ a continuous function on Ω. Denote by K_{Ω,p^{k}}(z, w) (K_{Ω,p^{k}}(z)) the weighted Bergman kernel of Ω w.r. to p^{k}. The virtual Bergman kernel of (Ω, p) is defined by

L_{Ω,p} (z, w; r) = L_{0} (z, w; r) = ∑_{k=0}^{∞} K_{Ω,p^{k}}(z, w)r^{k}.  (1.3)

The function L_{0}(z, r) = L_{0}(z, z; r), i.e.

L_{Ω,p} (z, r) = L_{0} (z, r) = ∑_{k=0}^{∞} K_{Ω,p^{k}}(z)r^{k}  (1.4)

will also be called virtual Bergman kernel of (Ω, p).

With these definitions, the relation (1.2) may be rewritten and the reproducing kernel of Ω̂_{1} is given by

K̂_{1} ((z, ζ) , (w, η)) = L_{1} (z, w; ζ\overline{η}) ,  (1.5)
K̂_{1}(z, ζ) = L_{1} (z, |ζ|^{2}) ,  (1.6)

with

L_{1}(z, w; r) = \frac{∂}{∂r} L_{0} (z, w; r) ,
L_{1}(z, r) = \frac{∂}{∂r} L_{0} (z, r) .

Remark. From the virtual Bergman kernel of (Ω, p), it is easy to recover the weighted Bergman kernels of (Ω, p^{k}) (k ∈ ℕ):

K_{Ω,p^{k}}(z) = \frac{1}{k!} \frac{∂^{k}}{∂r^{k}} L_{Ω,p} (z, r)|_{r=0} .  (1.7)

1.4. Let us recall some facts about harmonic analysis in the Hermitian unit ball B_{m}. Let H (B_{m}) = H (B_{m}, ω_{m}) be the Bergman space of B_{m}; let K_{B_{m}}(Z, T), K_{B_{m}}(Z) = K_{B_{m}}(Z, Z) be the Bergman kernel. It is well known (using for instance the automorphisms of B_{m}) that

K_{B_{m}}(Z) = \frac{1}{(1 − ∥Z∥^{2})^{m+1}} .  (1.8)

This may also be written

K_{B_{m}}(Z) = \frac{1}{m!} \frac{∂^{m}}{∂r^{m}} (\frac{1}{1 − r} )|_{r=∥Z∥^{2}} .  (1.9)

Let f ∈ H (B_{m}); then f may be written

f(Z) = ∑_{k=0}^{∞} f_{k}(Z),  (1.10)

where the f_{k} are k-homogeneous polynomials, which can be obtained through

f_{k}(Z) = ∫_{0}^{1} f (e^{2π i θ} Z) e^{−2π i kθ} d θ.

The expansion (1.10) converges uniformly on each compact of B_{m}.

For k ≠ ℓ, f_{k} and f_{ℓ} are orthogonal in H (B_{m}). This implies that H (B_{m}) is the Hilbert direct sum

H (B_{m}) = ˆ⨁_{k≥0}H_{k} (B_{m}) ,

where H_{k} (B_{m}) is the space of k-homogeneous polynomials, endowed with the scalar product of H (B_{m}). For each k ∈ ℕ, let (ϕ_{k,j})_{j∈J(k)} be an orthonormal basis of H_{k} (B_{m}); then

(ϕ_{k,j})_{k∈ℕ,j∈J(k)}

is a Hilbert basis of H (B_{m}) and

K_{B_{m}}(Z) = ∑_{k∈ℕ,j∈J(k)} |ϕ_{k,j}(Z)|^{2} = ∑_{k≥0} K_{B_{m},k}(Z),  (1.11)

where

K_{B_{m},k}(Z) = ∑_{j∈J(k)} |ϕ_{k,j}(Z)|^{2}

is the reproducing kernel of H_{k} (B_{m}). The expansions (1.11) also converge uniformly on compact subsets of B_{m}. Clearly, K_{B_{m},k} is a real polynomial, homogeneous of bidegree (k, k). Comparing with the expansion of (1.9)

K_{B_{m}}(Z) = ∑_{k=0}^{∞} \binom{k + m}{m} ∥Z∥^{2k} ,

we conclude that

K_{B_{m},k}(Z) = \binom{k + m}{m} ∥Z∥^{2k} .  (1.12)

More generally, consider the Hermitian ball B_{m}(ρ) of radius ρ. Its Bergman kernel w.r. to the same ω_{m} is

K_{B_{m}(ρ)}(Z) = \frac{1}{ρ^{2m}} K_{B_{m}} (\frac{Z}{ρ} );

the component of bidegree (k, k) is

K_{B_{m}(ρ),k}(Z) = \frac{1}{ρ^{2m+2k}} \binom{k + m}{m} ∥Z∥^{2k} .  (1.13)

If f ∈ H (B_{m}(ρ)), its component of degree k is then given by

f_{k}(Z) = ∫_{B_{m}(ρ)} \frac{1}{ρ^{2m+2k}} \binom{k + m}{m} ⟨Z, W ⟩^{k} f(W)ω_{V} (W).  (1.14)

1.5. Now we show that the virtual Bergman kernel of (Ω, p) allows us to compute the Bergman kernel of any inflated domain

Ω̂_{m} = {(z, Z) ∈ Ω × ℂ^{m} | ∥Z∥^{2} < p(z)} .

Here Ω̂_{m} is endowed with the volume form

ω_{V} (z) ∧ ω_{m}(Z),

where ω_{m}(Z) = (\frac{i}{2π} ∂\overline{∂} ∥Z∥^{2})^{m}.

Theorem 1.2. The Bergman kernel K̂_{m} (K̂_{m}) of Ω̂_{m} is

K̂_{m} ((z, Z), (w, W)) = L_{m} (z, w; ⟨Z, W ⟩) ,  (1.15)
K̂_{m}(z, Z) = L_{m} (z, ∥Z∥^{2}) ,  (1.16)

where

L_{m}(z, w; r) = \frac{1}{m!} \frac{∂^{m}}{∂r^{m}} L_{0}(z, w; r),  (1.17)
L_{m}(z, r) = \frac{1}{m!} \frac{∂^{m}}{∂r^{m}} L_{0}(z, r).  (1.18)

Note that the relation between K̂_{1} and K̂_{m}, deduced from (1.16) and (1.18), is nothing else that the inflation principle (0.3).

6 GUY ROOS

Proof. We have

L_{m} (z, w; r) = ∑_{k=0}^{∞} \binom{k + m}{m}K_{Ω,p^{k+m}}(z, w)r^{k}.

So we want to prove that the Bergman kernel of Ω̂_{m} is

K̂_{m} ((z, Z), (w, W)) = ∑_{k=0}^{∞} \binom{k + m}{m}K_{Ω,p^{k+m}}(z, w) ⟨Z, W ⟩^{k} .

Let H_{k} \binom{}{Ω_{m}} be the subspace of functions f(z, Z) in H \binom{}{Ω_{m}}, which are khomogeneous polynomial w.r. to the variable Z. For k ≠ ℓ, f ∈ H_{k} \binom{}{Ω_{m}}, g ∈

H_{ℓ} \binom{}{Ω_{m}}, we have

(f | g)Ω̂_{m} = ∫Ω̂_{m} f(w, W)\overline{g(w, W)}ω_{V} (w) ∧ ω_{m}(W)
= ∫_{w∈Ω} (f(w, ) | g(w, ))_{B_{m}(p(w)^{1/2})} ω_{V} (w) = 0.

This implies that H \binom{}{Ω_{m}} is the Hilbert direct sum

H \binom{}{Ω_{m}} = ; ⨁_{k≥0}H_{k} \binom{}{Ω_{m}} .  (1.19)

Fix k ∈ ℕ. Let f ∈ H_{k} \binom{}{Ω_{m}}. For almost all w ∈ Ω, the function W ↦ f(w, W) belongs to H (B_{m} (p(w)^{1/2})); by (1.14),

p(w)^{m+k}f(w, Z) = ∫_{∥W ∥^{2}<p(w)} \binom{k + m}{m} ⟨Z, W ⟩^{k} f(w, W)ω_{m}(W).

By the reproducing property of K_{Ω,p^{k+m}}, we have

f (z, Z) = ∫_{w∈Ω} K_{Ω,p^{k+m}}(z, w)p(w)^{m+k}f(w, Z)ω_{V} (w).

These relations imply

f(z, Z) = ∫Ω̂_{m} K_{Ω,p^{k+m}}(z, w)\binom{k + m}{m} ⟨Z, W ⟩^{k} f(w, W)ω_{V} (w) ∧ ω_{m}(W),

which means that

K_{Ω,p^{k+m}}(z, w)\binom{k + m}{m} ⟨Z, W ⟩^{k}

is the reproducing kernel of H_{k} \binom{}{Ω_{m}}.

From (1.19), we deduce that

∑_{k=0}^{∞} K_{Ω,p^{k+m}}(z, w)\binom{k + m}{m} ⟨Z, W ⟩^{k}

is the reproducing kernel of H \binom{}{Ω_{m}}. □

2. Virtual Bergman kernels for bounded symmetric domains

In this section, we compute the virtual Bergman kernel L_{Ω,p} (z, r) = L_{0} (z, r) when Ω is an irreducible bounded circled symmetric domain and p is a power of the generic norm of Ω.

2.1. Let V be a complex finite-dimensional vector space and Ω ⊂ V an irreducible bounded circled symmetric domain. Then V is endowed with a canonical structure of positive Hermitian Jordan triple. The numerical invariants of V (or of Ω) are the rank r and the multiplicities a and b (b = 0 iff the domain is of tube type); the genus is defined by

g = 2 + a(r − 1) + b.

The generic minimal polynomial m (T, x, y) and the generic norm N(x, y) of Ω are written as

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

where m_{1}, …, m_{r} are polynomials on V × \overline{V}, homogeneous of bidegrees (1, 1) , …, (r, r) respectively. In particular, m_{1} is an Hermitian inner product on V; we endow V with the Kähler form

α(z) = \frac{i}{2π} ∂\overline{∂}m_{1}(z, z)

and with the volume form

ω = α^{n},

where n is the complex dimension of V.

(See [1] for a review of the above properties).

2.2. The Bergman kernel of Ω is then

K(z) = \frac{1}{vol Ω} \frac{1}{N (z, z)^{g}} ,  (2.1)

with vol Ω = ∫_{Ω} ω.

More generally, consider the weighted Bergman space of Ω with respect to a power of the generic norm:

H^{(μ)}(Ω) = {f ∈ Hol Ω | ∫_{Ω} |f(z)|^{2} N(z, z)^{μ}ω(z) < ∞} .

For μ > −1, the space H^{(μ)}(Ω) is non-zero and is a Hilbert space of holomorphic functions. Its reproducing kernel is

K^{(μ)}(z) = \frac{1}{∫_{Ω} N(z, z)^{μ}ω(z)}N (z, z)^{−g−μ} .  (2.2)

The denominator ∫_{Ω} N(z, z)^{μ}ω(z) of the previous formula is called the Hua integral. It has been computed for the four series of classical domains (with different normalizations of the volume element) by Hua L.K. [6].

Theorem 2.1. [1]Let Ω be an irreducible bounded circled symmetric domain. The value of the Hua integral is given by

∫_{Ω} N(z, z)^{s}ω(z) = \frac{χ(0)}{χ(s)} ∫_{Ω} ω,  (2.3)

where χ is the polynomial of degree n = dim_{ℂ} Ω, related to the numerical invariants of Ω by

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

Here (s)_{k} denotes the raising factorial

(s)_{k} = s(s + 1) ⋯ (s + k − 1) = \frac{Γ (s + k)}{Γ (s)}.

The proof uses the polar decomposition in positive Hermitian Jordan triples (which generalizes the polar decomposition of matrices) and the following generalization, due to Selberg [7], of the Beta integral:

∫_{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} d t_{1} ⋯ d t_{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)},  (2.5)

for Re x > 0, Re y > 0, Re z > min (\frac{1}{n}, \frac{Re x}{n−1}, \frac{Re y}{n−1}).

Corollary 2.2. The Bergman kernel of H^{(μ)}(Ω) is

K^{(μ)}(z) = \frac{χ(μ)}{χ(0)}N (z, z)^{−μ} K(z),  (2.6)

where K = K^{(0)} is the Bergman kernel of Ω .

2.3.

Theorem 2.3. Let Ω be an irreducible bounded circled symmetric domain. The virtual Bergman kernel of (Ω, N(z, z)^{μ}) is

L^{(μ)} (z, r) = K(z)F_{χ,μ} (\frac{r}{N(z, z)^{μ}} ) ,  (2.7)

where K is the Bergman kernel of Ω, χ the polynomial defined by (2.4) and F_{χ,μ} is the rational function

F_{χ,μ}(t) = \frac{1}{χ(0)}∑_{k=0}^{∞}χ(μk)t^{k}.  (2.8)

The proof of (2.7) is straightforward, using (1.4) and (2.6):

L^{(μ)} (z, r) = ∑_{k=0}^{∞} K^{(kμ)}(z)r^{k}

= ∑_{k=0}^{∞} \frac{χ(kμ)}{χ(0)}N (z, z)^{−kμ} K(z)r^{k}

= \frac{K(z)}{χ(0)} ∑_{k=0}^{∞} χ(kμ) (\frac{r}{N(z, z)^{μ}} )^{k} .

If the polynomial k ↦ χ(kμ) is decomposed as

\frac{χ(kμ)}{χ(0)} = ∑_{j=0}^{n} c_{μ,j} \frac{(k + 1)_{j}}{j!} ,  (2.9)

the function defined by (2.8) is

F_{χ,μ}(t) = ∑_{j=0}^{n} c_{μ,j} (\frac{1}{1 − t})^{j} .

3. Tables for bounded symmetric domains

Hereunder we give the results for each type of irreducible bounded symmetric domains, which allow the reader to apply Theorems 1.2 and 2.3 to special cases.

3.1. Classification of irreducible circled bounded symmetric domains. Here is the complete list of irreducible circled bounded symmetric domains, up to linear isomorphisms. There is some overlapping between the four infinite families, due to a finite number of isomorphisms in low dimemsions.

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

Ω = \{x ∈ V | I_{m} − x^{t}\overline{x} ≫ 0\} .

Type II_{n} (n ≥ 2). V = A_{n}(ℂ) (space of n × n alternating matrices).

Ω = \{x ∈ V | I_{n} + x\overline{x} ≫ 0\} .

Type III_{n} (n ≥ 1). V = S_{n}(ℂ) (space of n × n symmetric matrices).

Ω = \{x ∈ V | I_{n} − x\overline{x} ≫ 0\} .

Type IV_{n} (n ≠ 2). V = ℂ^{n}, q(x) = ∑ x_{i}^{2}, q(x, y) = 2 ∑ x_{i}y_{i}. The domain Ω is defined by

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

Type V. V = M_{2,1}(𝕆_{ℂ}) ≃ ℂ^{16}, exceptional type.

Type VI. V = H_{3}(𝕆_{ℂ}) ≃ ℂ^{27}, exceptional type.

3.2. Numerical invariants.

Type I_{m,n} (1 ≤ m ≤ n).

r = m, a = 2, b = n − m, g = m + n.

Type II_{2p} (p ≥ 1).

r = \frac{n}{2} = p, a = 4, b = 0, g = 2 (n − 1) .

Type II_{2p+1} (p ≥ 1).

r = [\frac{n}{2}] = p, a = 4, b = 2, g = 2(n − 1).

Type III_{n} (n ≥ 1).

r = n, a = 1, b = 0, g = n + 1.

10 GUY ROOS

Type IV_{n} (n ≠ 2).

r = 2, a = n − 2, b = 0, g = n.

Type V.

r = 2, a = 6, b = 4, g = 12.

Type VI.

r = 3, a = 8, b = 0, g = 18.

3.3. Generic norm. Type I_{m,n} (1 ≤ m ≤ n). V = M_{m,n}(ℂ) (space of m × n matrices with complex entries).

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

Type II_{n} (n ≥ 2). V = A_{2p}(ℂ) (space of n × n alternating matrices).

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

Type III_{n} (n ≥ 1). V = S_{n}(ℂ) (space of n × n symmetric matrices).

N(x, y) = Det(I_{n} − x\overline{y}).

Type IV_{n} (n ≠ 2). V = ℂ^{n}.

N(x, y) = 1 − q(x, \overline{y}) + q(x)q(\overline{y}).

Type V. V = M_{2,1}(𝕆_{ℂ}).

N(x, y) = 1 − (x|y) + (x^{♯}|y^{♯}).

Type VI. V = H_{3}(𝕆_{ℂ}).

N(x, y) = 1 − (x|y) + (x^{♯}|y^{♯}) − det x det \overline{y}.

3.4. The polynomial χ for the Hua integral. Recall that for s > −1,

χ(s) ∫_{Ω} N(x, x)^{s}ω = χ(0) ∫_{Ω} ω.

Type I_{m,n}.

χ(s) = ∏_{j=1}^{m}(s + j)_{n}.

Type II_{2p}.

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

Type II_{2p+1}.

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

Type III_{n}.

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

Type IV_{n}.

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

Type V.

χ(s) = (s + 1)_{8}(s + 4)_{8}.

Type VI.

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

4. Open problems

4.1. Understand the rational function

F_{χ,μ}(t) = \frac{1}{χ(0)}∑_{k=0}^{∞}χ(μk)t^{k} = ∑_{j=0}^{n} c_{μ,j} (\frac{1}{1 − t})^{j} ,

when χ is a polynomial from the list of subsection 3.4. Recall that the coefficients c_{μ,j} are given by

\frac{χ(kμ)}{χ(0)} = ∑_{j=0}^{n} c_{μ,j} \frac{(k + 1)_{j}}{j!} .

4.2. The virtual Bergman kernel

L_{Ω,p} (z, r) = ∑_{k=0}^{∞} K_{Ω,p^{k}}(z)r^{k}

is suitable for the computation of the Bergman kernel of “inflated domains by Hermitian balls”

Ω̂_{m} = {(z, Z) ∈ Ω × ℂ^{m} | ∥Z∥^{2} < p(z)} ,

which may also be written

Ω̂_{m} = {(z, Z) ∈ Ω × ℂ^{m} | Z ∈ (p(z))^{1/2} B_{m}} .

If F is any circled domain in ℂ^{m}, what can be said about the Bergman kernel of

Ω̂_{F} = {(z, Z) ∈ Ω × ℂ^{m} | Z ∈ (p(z))^{1/2} F } ,

for example when Ω is a bounded symmetric domain, p = N_{Ω}(z, z)^{μ} and F another bounded symmetric domain?

For some families {F } other than the family {B_{m}} of Hermitian balls, is it possible to define an analogous of the virtual Bergman kernel L_{Ω,p} and obtain an analogous of Theorem 1.2?

4.3. It would also be interesting to replace the weighted Bergman space H(Ω, p) by more general Hilbert spaces of holomorphic functions; for example, when Ω is a bounded symmetric domain, the spaces with reproducing kernel N(z, z)^{μ}, where μ is in the ”Berezin-Wallach set” of Ω (see [8]).

References

[1] Yin Weiping, Lu Keping, Roos Guy, New classes of domains with explicit Bergman kernel, Science in China Ser. A Mathematics, 47(3) (2004), 352–371.

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

[3] Forelli F., Rudin W., Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J., 24(1974), 593–602.

[4] Ligocka E., On the Forelli-Rudin construction and weighted Bergman projections, Studia Math., 94(1989), 257-272.

[5] Engliš M., A Forelli-Rudin construction and asymptotics of weighted Bergman kernels, J. Funct. Analysis, 177(2000), 257–281.

[6] Hua L.K., Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Amer. Math. Soc., Providence, RI, 1963.

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

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

[9] Faraut J., Korányi A., Analysis on Symmetric Cones, Clarendon Press, Oxford, 1994.

[10] Helgason S., Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.

[11] 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.

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

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

[14] 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.

[15] 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.

[16] Satake I., Algebraic Structures of Symmetric Domains, Iwanami Shoten, Tokyo, and Princeton Univ. Press, Princeton, NJ, 1980.

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E-mail address: guy.roos@normalesup.org