Balanced Metric and Berezin Quantization on the Siegel–Jacobi Ball

Balanced Metric and Berezin Quantization
on the Siegel–Jacobi Ball

Stefan BERCEANU S. Berceanu National Institute for Physics and Nuclear
Engineering, Department of Theoretical Physics,
PO BOX MG-6, Bucharest-Magurele, Romania<EMAIL_ADDRESS>http://www.theory.nipne.ro/index.php/mcp-home

Received March 03, 2016, in final form June 17, 2016; Published online June
27, 2016

We determine the matrix of the balanced metric of the Siegel–Jacobi ball and
its inverse. We calculate the scalar curvature, the Ricci form and the
Laplace–Beltrami operator of this manifold. We discuss several geometric
aspects related with Berezin quantization on the Siegel–Jacobi ball.

Jacobi group; Siegel–Jacobi ball; balanced metric; homogenous Kähler
manifolds; Laplace–Beltrami operator; scalar curvature; Ricci form; Berezin
quantization

32Q15; 81S10; 53D50; 57Q35; 81R30

##  1 Introduction

The Jacobi groups are semidirect products of appropriate semisimple real
algebraic groups of Hermitian type with Heisenberg groups [23, 33, 49, 74]. As
unimodular, nonreductive, algebraic groups of Harish-Chandra type [68], the
Jacobi groups are intensively studied in mathematics [15, 22, 72, 73, 74, 76,
78, 79]. The Siegel–Jacobi (partially bounded [78, 79]) domains are
homogeneous manifolds associated to the Jacobi groups by the generalized
Harish-Chandra embedding. It was underlined [10, 11, 12] that the
Siegel–Jacobi disk is a nonsymmetric, reductive [63], quantizable [66], Q-K.
Lu Kähler manifold [54], not Einstein with respect to the balanced metric [1,
31], with constant negative scalar curvature. In the present paper we
investigate similar geometric properties for the Siegel–Jacobi ball.

As was already emphasized [5, 7, 9], the Jacobi group is relevant in several
branches of physics, as: quantum mechanics, geometric quantization, quantum
optics, nuclear structure, signal processing [14, 48, 65, 70]. In particular,
the Jacobi group is responsible [8] for the squeezed states in quantum optics
[58]. Recently, new significant applications of the Jacobi group have been
underlined in various domains as: quantum tomography [59], quantum
teleportation [30], Vlasov kinetic equation [35], general symmetry methods for
analyzing solutions of differential equations using Lie’s prolongation method
[40, 69], establishing the link between the standard Gaussian distribution and
the Siegel–Jacobi space [60]. Berezin’s quantization and Berezin symbols
related to the Jacobi group have been also investigated [25, 26].

The real Jacobi group of index $n$ is defined as
$G^{J}_{n}({\use@mathgroup\M@U\symAMSb{R}})={\mathrm{H}}_{n}({\use@mathgroup\M@U\symAMSb{R}})\rtimes\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})$,
where ${\mathrm{H}}_{n}({\use@mathgroup\M@U\symAMSb{R}})$ is the real
Heisenberg group of real dimension $(2n+1)$ [9, 76]. Let $g=(M,X,k)$,
$g^{\prime}=(M^{\prime},X^{\prime},k^{\prime})\in
G^{J}_{n}({\use@mathgroup\M@U\symAMSb{R}})$, where
$X=(\lambda,\mu)\in{\use@mathgroup\M@U\symAMSb{R}}^{2n}$ and
$(X,k)\in{\mathrm{H}}_{n}({\use@mathgroup\M@U\symAMSb{R}})$. Then the
composition law in $G^{J}_{n}({\use@mathgroup\M@U\symAMSb{R}})$ is

$\displaystyle
gg^{\prime}=\big{(}MM^{\prime},XM^{\prime}+X^{\prime},k+k^{\prime}+XM^{\prime}JX^{\prime
t}\big{)},$

where

$\displaystyle J=\left(\begin{matrix}{\mathbb{0}}_{n}&{\mathbbm{1}}_{n}\\\
-{\mathbbm{1}}_{n}&{\mathbb{0}}_{n}\end{matrix}\right).$

Also it is considered the restricted real Jacobi group
$G^{J}_{n}({\use@mathgroup\M@U\symAMSb{R}})_{0}$, consisting only of elements
of the form above, but $g=(M,X)$.

To the Jacobi group $G^{J}_{n}({\use@mathgroup\M@U\symAMSb{R}})$ it is
associated the homogeneous manifold – the Siegel–Jacobi upper half-plane –
${\mathcal{X}}^{J}_{n}\approx{\use@mathgroup\M@U\symAMSb{C}}^{n}\times{\mathcal{X}}_{n}$,
where the Siegel upper half-plane
${\mathcal{X}}_{n}={\mathrm{Sp}}(n,{\use@mathgroup\M@U\symAMSb{R}})/{\mathrm{U}}(n)$
is realized as

$\displaystyle{\mathcal{X}}_{n}:=\\{V\in
M(n,{\use@mathgroup\M@U\symAMSb{C}})\,|\,V=S+{\rm i}R,\,S,R\in
M(n,{\use@mathgroup\M@U\symAMSb{R}}),\,R>0,\,S^{t}=S,\,R^{t}=R\\}.$

The (complex version of the) Jacobi group of index $n$ is defined as
$G^{J}_{n}={\mathrm{H}}_{n}\rtimes\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}$,
where ${\mathrm{H}}_{n}$ denotes the $(2n+1)$-dimensional Heisenberg group [7,
9, 76]. The composition law is

$\displaystyle(g_{1},\alpha_{1},t_{1})(g_{2},\alpha_{2},t_{2})=\big{(}g_{1}g_{2},g_{2}^{-1}\times\alpha_{1}+\alpha_{2},t_{1}+t_{2}+\Im\big{(}g^{-1}_{2}\times\alpha_{1}\bar{\alpha}_{2}\big{)}\big{)},$

where $\alpha_{i}\in{\use@mathgroup\M@U\symAMSb{C}}^{n}$,
$t_{i}\in{\use@mathgroup\M@U\symAMSb{R}}$ and
$g_{i}\in\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}$,
$i=1,2$ have the form

$\displaystyle g=\left(\begin{matrix}p&q\\\
\bar{q}&\bar{p}\end{matrix}\right),\qquad p,q\in
M(n,{\use@mathgroup\M@U\symAMSb{C}}),$ (1.1)

and $g\times\alpha=p\alpha+q\bar{\alpha}$, and
$g^{-1}\times\alpha={p}^{*}\alpha-q^{t}\bar{\alpha}$.

The Siegel–Jacobi ball, denoted ${\mathcal{D}}^{J}_{n}$ [7], is the
homogeneous manifold associated with the Jacobi group $G^{J}_{n}$, whose
points are in ${\use@mathgroup\M@U\symAMSb{C}}^{n}\times{\mathcal{D}}_{n}$,
where ${\mathcal{D}}_{n}$ denotes the Siegel (open) ball. The non-compact
hermitian symmetric space
$\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}/\operatorname{U}(n)$
admits a matrix realization as a bounded homogeneous domain:

$\displaystyle{\mathcal{D}}_{n}:=\big{\\{}W\in
M(n,{\use@mathgroup\M@U\symAMSb{C}})\colon
W=W^{t},\,N>0,\,N:={\mathbbm{1}}_{n}-W\bar{W}\big{\\}}.$ (1.2)

${\mathcal{D}}_{n}$ is a hermitian symmetric space of type CI (cf. [36, Table
V, p. 518]), identified with the symmetric bounded domain of type II,
${\mathfrak{R}}_{\text{II}}$ in Hua’s notation [38].

In [6, 7, 9] we have attached coherent states (CS) [64] to the Jacobi group
$G^{J}_{n}$ with support on the Siegel–Jacobi ball ${\mathcal{D}}^{J}_{n}$.
The particular case of coherent states attached to the Jacobi group
$G^{J}_{1}$ defined on the Siegel–Jacobi disk ${\mathcal{D}}^{J}_{1}$ has been
investigated in [5, 10]. In the present paper we don’t use effectively the
coherent states, but we use results obtained in geometry using the coherent
state approach. The homogeneous Kähler two-form on ${\mathcal{D}}^{J}_{n}$,
denoted $\omega_{{\mathcal{D}}^{J}_{n}}$, has been obtained in [7, 9] from the
scalar product of two coherent state vectors, and this will be the starting
point of the present investigation. The reproducing kernel function for the
Siegel–Jacobi ball was obtained previously, see [62, p. 532] and references
there. We recall that in [5] we have shown that when expressed in variables on
${\mathcal{X}}_{1}$ obtained (via the partial Cayley transform) from variables
on ${\mathcal{D}}^{J}_{1}$, the Kähler two-form
$\omega_{{\mathcal{X}}^{J}_{1}}$ is the one determined by Berndt [21, 23] and
Kähler [42, 43]. Applying the partial Cayley transform, which we recall later
in Proposition 2.5, from $\omega_{{\mathcal{X}}^{J}_{n}}$ obtained in [77], it
was obtained the homogeneous Kähler two-form on the Siegel–Jacobi ball
$\omega_{{\mathcal{D}}^{J}_{n}}$ – in fact, Yang considers a Jacobi group more
general than $G^{J}_{n}$. In [9] we have shown that when expressed in
appropriate variables, $\omega_{{\mathcal{D}}^{J}_{n}}$ is a particular case
of that obtained by Yang in [79]. In [12] we have underlined that the
homogeneous metric corresponding to $\omega_{{\mathcal{D}}^{J}_{1}}$ is a
balanced metric [1, 31]. In the present paper it is emphasized that the metric
corresponding to $\omega_{{\mathcal{D}}^{J}_{n}}$ [6, 7, 9] is the balanced
metric. We also point out several geometric aspects related with Berezin
quantization on the Siegel–Jacobi ball.

The paper is laid out as follows. In Section 2 we recall several notions which
are used in the paper. The notion of balanced metric on homogeneous Kähler
manifolds is briefly recalled in Section 2.1. Also other notions which appear
in connection with Berezin quantization, as quantizable manifold, Berezin
kernel, CS-type group, Q.-K. Lu manifold and diastasis are briefly recalled.
The homogenous metric on the Siegel upper half plane and Siegel ball, which we
need later, are recalled in Section 2.2. We have considered useful to collect
in Section 2.3 several formulas referring to the differential geometry of the
Siegel–Jacobi disk ${\mathcal{D}}^{J}_{1}$, a guide for the formulas deduced
in the present paper for the Siegel–Jacobi ball ${\mathcal{D}}^{J}_{n}$.
Section 3 contains the original results of the present paper. The balanced
metric is obtained from the Kähler potential, calculated previously [7] using
the coherent states. Do to the symmetry of the variables $W\in
M(n,{\use@mathgroup\M@U\symAMSb{C}})$ describing points on the Siegel ball
${\mathcal{D}}_{n}$, we write down the matrix associated with the metric
$h_{{\mathcal{D}}^{J}_{n}}(z,W)$ on the Siegel–Jacobi ball as a four-blocks
matrix expressed in
$(z,W)\in{\use@mathgroup\M@U\symAMSb{C}}^{n}\times{\mathcal{D}}_{n}$, but also
in a variable $\eta$ related with $(z,W)$ by
$\eta=({\mathbbm{1}}_{n}-W\bar{W})^{-1}(z+W\bar{z})$. The significance of the
change of coordinates ${\rm FC}\colon(\eta,W)\rightarrow(z,W)$ as a
homogeneous Kähler diffeomorphism was underlined in [9] in the context of the
fundamental conjecture [32] for homogeneous Kähler manifolds (Gindikin and
Vinberg [75]). In order to determine the inverse of the metric matrix attached
to the metric $h_{{\mathcal{D}}^{J}_{n}}$, we need the “inverse” of the metric
matrix ${{\mathfrak{h}}}^{k}_{{\mathcal{D}}_{n}}$ of the Siegel ball,
presented in Section 2.2 – the complication comparatively with that on the
noncompact Grassmann manifold [3] comes from the fact the matrices describing
the points on Siegel ball are symmetric – see Lemma 2.7.
$(h^{k}_{{\mathcal{D}}_{n}})^{-1}$ is calculated in Proposition B.2, where
more details on the calculation of the Laplace–Beltrami operator on Siegel
ball are given. Theorem 3.2 contains the matrix of the metric of the
Siegel–Jacobi ball and its “inverse” (in the sense of Lemma 2.7). Proposition
3.3 collects a description of the geometry of the Siegel–Jacobi ball from the
point of view of Berezin’s quantization. In Section 3.2 it is shown that the
scalar curvature of the Siegel–Jacobi ball is constant and negative. This is
compatible with Theorem 4.1 in [51] which asserts that a Kähler manifold that
admits a regular quantization has constant scalar curvature. It is also shown
that ${\mathcal{D}}^{J}_{n}$ is not an Einstein manifold with respect to the
balanced metric, but it is one with respect to the Bergman metric. Section 3.3
contains the Laplace–Beltrami operator on ${\mathcal{D}}^{J}_{n}$. Remark A.1
essentially asserts that if the metric on the homogeneous space in $M=G/H$ is
invariant to the action of $G$ on $M$, then the same is true for the
Laplace–Beltrami operator, a fact used in the proof of Theorem 3.2. In
Appendix A we reproduce also a lemma which calculates the invariant volume of
the Siegel–Jacobi ball used in [7]. In Appendix B already mentioned are
presented auxiliary calculations to the paper [57] of Maass and a remark
concerning the results obtained in [39] referring to the Laplace–Beltrami
operator on the Siegel ball and Siegel upper half plane.

The main new results of the present paper are contained in Remark 2.3, Lemma
2.7, Theorem 3.2, Propositions 3.3, 3.4 and 3.5. They generalize to the
Siegel–Jacobi ball the results recalled in Section 2.3 obtained for the
Siegel–Jacobi disk.

Notation. We denote by $\use@mathgroup\M@U\symAMSb{R}$,
$\use@mathgroup\M@U\symAMSb{C}$, $\use@mathgroup\M@U\symAMSb{Z}$, and
$\use@mathgroup\M@U\symAMSb{N}$ the field of real numbers, the field of
complex numbers, the ring of integers, and the set of non-negative integers,
respectively. We denote the imaginary unit $\sqrt{-1}$ by ${\rm i}$, and the
real and imaginary part of a complex number by $\Re$ and respectively $\Im$,
i.e., we have for $z\in{\use@mathgroup\M@U\symAMSb{C}}$, $z=\Re z+{\rm i}\Im
z$, and $\bar{z}=cc(z)=\Re z-{\rm i}\Im z$. $M(m\times
n,\use@mathgroup\M@U\symAMSb{F})\approxeq\use@mathgroup\M@U\symAMSb{F}^{mn}$
denotes the set of all $m\times n$ matrices with entries in the field
$\use@mathgroup\M@U\symAMSb{F}$. $M(n\times 1,\use@mathgroup\M@U\symAMSb{F})$
is identified with $\use@mathgroup\M@U\symAMSb{F}^{n}$. Set
$M(n,\use@mathgroup\M@U\symAMSb{F}):=M(n\times
n,\use@mathgroup\M@U\symAMSb{F})$. For any $A\in
M_{n}(\use@mathgroup\M@U\symAMSb{F})$, $A^{t}$ denotes the transpose matrix of
$A$. For $A\in M_{n}(\use@mathgroup\M@U\symAMSb{C})$, $\bar{A}$ denotes the
conjugate matrix of $A$ and $A^{*}=\bar{A}^{t}$. For $A\in
M_{n}(\use@mathgroup\M@U\symAMSb{C})$, the inequality $A>0$ means that $A$ is
positive definite. The identity matrix of degree $n$ is denoted by
${\mathbbm{1}}_{n}$ and ${\mathbb{0}}_{n}$ denotes the
$M(n,\use@mathgroup\M@U\symAMSb{F})$-matrix with all entries $0$. We consider
a complex separable Hilbert space ${\mathfrak{H}}$ endowed with a scalar
product which is antilinear in the first argument, $(\lambda
x,y)=\bar{\lambda}(x,y)$, $x,y\in{\mathfrak{H}}$,
$\lambda\in{\use@mathgroup\M@U\symAMSb{C}}\setminus 0$. If $A$ is a linear
operator, we denote by $A^{\dagger}$ its adjoint. If $\pi$ is representation
of a Lie group $G$ on the Hilbert space ${{\mathfrak{H}}}$ and
${\mathfrak{g}}$ is the Lie algebra of $G$, we denote
${\boldsymbol{X}}:=\operatorname{d}\pi(X)$ for $X\in{\mathfrak{g}}$. A complex
analytic manifold is Kählerian if it is endowed with a Hermitian metric whose
imaginary part $\omega$ has $\operatorname{d}\omega=0$ [36]. We denote the
action of a Lie group $G$ on the space $M$ by $G\times M\rightarrow M$. A
coset space $M=G/H$ is homogenous Kählerian if it caries a Kählerian structure
invariant under the group $G$ [24]. By a Kähler homogeneous diffeomorphism we
mean a diffeomorphism $\phi\colon M\rightarrow N$ of homogeneous Kähler
manifolds such that $\phi^{*}\omega_{N}=\omega_{M}$. We use Einstein
convention that repeated indices are implicitly summed over. If the Kähler-two
form has the local expression

$\displaystyle\omega_{M}(z)={\rm
i}\sum_{\alpha,\beta=1}^{n}h_{\alpha\bar{\beta}}(z)\operatorname{d}z_{\alpha}\wedge\operatorname{d}\bar{z}_{\beta},\qquad
h_{\alpha\bar{\beta}}=\bar{h}_{\beta\bar{\alpha}}=h_{\bar{\beta}\alpha},$
(1.3)

we denote

$\displaystyle h^{\alpha\bar{\beta}}:=(h_{\alpha\bar{\beta}})^{-1},$ (1.4)

i.e., we have

$\displaystyle
h_{\alpha\bar{\epsilon}}h^{\epsilon\bar{\beta}}=\delta_{\alpha\beta}.$ (1.5)

In this paper we use the following expression for the Laplace–Beltrami
operator on Kähler manifolds $M$ with the Kähler two-form (1.3):

$\displaystyle\Delta_{M}(z):=\sum_{\alpha,\beta=1}^{n}(h_{\alpha\bar{\beta}})^{-1}\frac{\partial^{2}}{\partial\bar{z}_{\alpha}\partial{{z}_{\beta}}},$
(1.6)

cf., e.g., Lemma 3 in the Appendix of [18] or in [41, equation (5.2.15), p.
253], where the Laplace–Beltrami operator is $-\Delta_{M}(z)$ and the author
uses the convention $h^{i\bar{j}}h_{k\bar{j}}=\delta_{ij}$ (see [41, equation
(5.2.14), p. 252]) instead of our notation (1.4), (1.5). This means that
$h^{l\bar{k}}$ in Jost’s notation corresponds to
$h^{k\bar{l}}=(h_{k\bar{l}})^{-1}$ in our notation.

##  2 Preliminaries

###  2.1 Balanced metric and Berezin quantization via coherent states

We consider a $G$-invariant Kähler two-form $\omega_{M}$ (1.3) on the
$2n$-dimensional homogeneous manifold $M=G/H$ derived from the Kähler
potential $f(z,\bar{z})$ [29]

$\displaystyle
h_{\alpha\bar{\beta}}=\frac{\partial^{2}f}{\partial{z}_{\alpha}\partial\bar{z}_{{\beta}}}.$
(2.1)

As was underlined in [12] for ${\mathcal{D}}^{J}_{1}$, the homogeneous
hermitian metric determined in [5, 7, 9] is in fact a balanced metric, because
it corresponds to the Kähler potential calculated as the product of two CS-
vectors

$\displaystyle f(z,\bar{z})=\ln K_{M}(z,\bar{z}),\qquad
K_{M}(z,\bar{z})=(e_{\bar{z}},e_{\bar{z}}).$ (2.2)

In the approach of Perelomov [64] to CS, it is supposed that there exists a
continuous, unitary, irreducible representation $\pi$ of a Lie group $G$ on
the separable complex Hilbert space ${{\mathfrak{H}}}$. The coherent vector
mapping $\varphi$ is defined locally, on a coordinate neighborhood
${\mathcal{V}}_{0}\subset M$ (cf. [4, 13]):

$\displaystyle\varphi\colon\
M\rightarrow\bar{{{\mathfrak{H}}}},\qquad\varphi(z)=e_{\bar{z}},$

where $\bar{{{\mathfrak{H}}}}$ denotes the Hilbert space conjugate to
${{\mathfrak{H}}}$. The vectors $e_{\bar{z}}\in\bar{{{\mathfrak{H}}}}$ indexed
by the points $z\in M$ are called Perelomov’s CS vectors. Using Perelomov’s CS
vectors, we consider Berezin’s approach to quantization on Kähler manifolds
with the supercomplete set of vectors verifying the Parceval overcompletness
identity [17, 18, 19, 20]

$\displaystyle(\psi_{1},\psi_{2})_{{\mathcal{F}}_{K}}=\int_{M}(\psi_{1},e_{\bar{z}})(e_{\bar{z}},\psi_{2})\operatorname{d}\nu_{M}(z,\bar{z}),\qquad\psi_{1},\psi_{2}\in{{\mathfrak{H}}},$
(2.3)
$\displaystyle\operatorname{d}{\nu}_{M}(z,\bar{z})=\frac{\Omega_{M}(z,\bar{z})}{(e_{\bar{z}},e_{\bar{z}})},\qquad\Omega_{M}:=\frac{1}{n!}\underbrace{\omega\wedge\cdots\wedge\omega}_{\text{$n$
times}}.$

The reproducing kernel for the Hilbert space of holomorphic, square integrable
functions with respect to the scalar product of the type (2.3) was calculated
via CS as the scalar product $K_{M}(z,\bar{w})=(e_{\bar{z}},e_{\bar{w}})$ [5,
7, 9].

On the other side, let us consider the weighted Hilbert space
${{\mathfrak{H}}}_{f}$ of square integrable holomorphic functions on $M$, with
weight ${\mbox{\rm e}}^{-f}$ [34]

$\displaystyle{{\mathfrak{H}}}_{f}=\left\\{\phi\in\operatorname{hol}(M)\,|\int_{M}{\mbox{\rm
e}}^{-f}|\phi|^{2}\Omega_{M}<\infty\right\\}.$ (2.4)

In order to identify the Hilbert space ${{\mathfrak{H}}}_{f}$ defined by (2.4)
with the Hilbert ${{\mathcal{F}}_{K}}$ space with scalar product (2.3), we
have to consider the $\epsilon$-function [27, 66, 67]

$\displaystyle\epsilon(z)={\mbox{\rm e}}^{-f(z)}K_{M}(z,\bar{z}).$

If the Kähler metric on the complex manifold $M$ is obtained from the Kähler
potential via (1.3), (2.1) and (2.2) is such that $\epsilon(z)$ is a positive
constant, then the metric is called balanced. This denomination was firstly
used in [31] for compact manifolds, then it was used in [1] for noncompact
manifolds and also in [52] in the context of Berezin quantization on
homogeneous bounded domains.

The balanced hermitian metric of $M$ in local coordinates is

$\displaystyle\operatorname{d}s^{2}_{M}(z,\bar{z})=\sum_{\alpha,\beta=1}^{n}\frac{\partial^{2}}{\partial
z_{\alpha}\partial\bar{z}_{\beta}}\ln(K_{M}(z,\bar{z}))\operatorname{d}z_{\alpha}\otimes\operatorname{d}\bar{z}_{\beta},\qquad
K_{M}(z,\bar{z})=(e_{\bar{z}},e_{\bar{z}}).$ (2.5)

We recall that in [27, 66, 67] Berezin’s quantization via coherent states was
globalized and extended to non-homogeneous manifolds in the context of
geometric (pre-)quantization [47]. To the Kähler manifold $(M,\omega)$, it is
also attached the triple $\sigma=({{\mathcal{L}}},h,\nabla)$, where
${{\mathcal{L}}}$ is a holomorphic (prequantum) line bundle on $M$, $h$ is the
hermitian metric on ${{\mathcal{L}}}$ and $\nabla$ is a connection compatible
with metric and the Kähler structure [16]. The manifold is called quantizable
if the curvature of the connection [29]
$F(X,Y)=\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]}$ has the
property that $F=-{\rm i}\omega_{M}$, or
$\partial\bar{\partial}\log\hat{h}={\rm i}\omega_{M}$, where $\hat{h}$ is a
local representative of $h$. Then $\omega_{M}$ is integral, i.e.,
$c_{1}[{\mathcal{L}}]=[\omega_{M}]$. The reproducing (weighted Bergman) kernel
admits the series expansion

$\displaystyle
K_{M}(z,\bar{w})\equiv(e_{\bar{z}},e_{\bar{w}})=\sum_{i=0}^{\infty}\varphi_{i}(z)\bar{\varphi}_{i}(w),$
(2.6)

where $\Phi=(\varphi_{0},\varphi_{1},\dots)$ is an orthonormal base with
respect to the scalar product (2.3):

$\displaystyle\int_{M}\bar{\varphi}_{i}\varphi_{j}\frac{\Omega_{M}}{K_{M}}=\delta_{ij},\qquad
i,j\in{\use<EMAIL_ADDRESS>
The base $\Phi$ is finite-dimensional for compact manifolds $M$.

Let $\xi\colon{{\mathfrak{H}}}\setminus
0\rightarrow{\use@mathgroup\M@U\symAMSb{P}}({{\mathfrak{H}}})$ be the
canonical projection $\xi({\boldsymbol{z}})=[{\boldsymbol{z}}]$. The
Fubini–Study metric in the nonhomogeneous coordinates $[z]$ is the hermitian
metric on ${\use@mathgroup\M@U\symAMSb{CP}}^{\infty}$ (see [44] for details)

$\displaystyle\operatorname{d}s^{2}|_{\rm
FS}({\boldsymbol{[}}{z}])=\frac{(\operatorname{d}{\boldsymbol{z}},\operatorname{d}{\boldsymbol{z}})({\boldsymbol{z}},{\boldsymbol{z}})-(\operatorname{d}{\boldsymbol{z}},{\boldsymbol{z}})({\boldsymbol{z}},\operatorname{d}{\boldsymbol{z}})}{({\boldsymbol{z}},{\boldsymbol{z}})^{2}}.$
(2.7)

It was proved [67] (see also [12]) that:

######  Remark 2.1.

If $\epsilon(z)$ is constant on $M$, then the balanced Hermitian metric on $M$
is the pullback

$\displaystyle\operatorname{d}s^{2}_{M}(z)=\iota_{M}^{*}\operatorname{d}s^{2}_{\rm
FS}(z)=\operatorname{d}s^{2}_{\rm FS}(\iota_{M}(z))$ (2.8)

of the Fubini–Study metric (2.7) via the embedding

$\displaystyle\iota_{M}\colon\
M\hookrightarrow{\use@mathgroup\M@U\symAMSb{CP}}^{\infty},\qquad\iota_{M}(z)=[\varphi_{0}(z):\varphi_{1}(z):\dots].$
(2.9)

If $M$ is a compact manifold, the embedding (2.9) is the Kodaira embedding.

If the homogeneous Kähler manifold $M=G/H$ to which we associate the Hilbert
space of functions ${\mathcal{F}}_{K}$ with respect to the scalar product
(2.3) admits a holomorphic embedding $\iota_{M}\colon
M\hookrightarrow{{\use@mathgroup\M@U\symAMSb{P}}({\mathcal{H}^{\infty}})}$,
then $M$ is called a CS-orbit, and $G$ is called a CS-type group [4, 50, 62].

We denote the normalized Bergman kernel (the two-point function of $M$ [2,
16]) by

$\displaystyle\kappa_{M}(z,\bar{z}^{\prime}):=\frac{K_{M}(z,\bar{z}^{\prime})}{\sqrt{K_{M}(z)K_{M}(z^{\prime})}}=(\tilde{e}_{\bar{z}},\tilde{e}_{\bar{z}^{\prime}})=\frac{(e_{\bar{z}},e_{\bar{z}^{\prime}})}{\|e_{\bar{z}}\|\|e_{\bar{z}^{\prime}}\|}.$
(2.10)

The set $\Sigma_{z}:=\\{z^{\prime}\in
M\,|\,\kappa_{M}(z,\bar{z}^{\prime})=0\\}$ was called [2, 16] polar divisor
relative to $z\in M$, while a manifold for which $\Sigma_{z}=\varnothing$,
$\forall\,z\in M$ was called in [12] a Q.-K. Lu manifold, extending to
manifolds a denomination introduced for domains in
${\use@mathgroup\M@U\symAMSb{C}}^{n}$ [54]. Note that for a particular class
of compact homogeneous manifolds that includes the hermitian symmetric spaces,
$\Sigma_{z}$ is equal with the cut locus relative to $z\in M$ (see the
definition of the cut locus, e.g., [46, p. 100]), and $\Sigma_{z}$ is a
divisor in the sense of algebraic geometry [2, 16].

By Berezin kernel $b_{M}\colon M\times
M\rightarrow[0,1]\in{\use@mathgroup\M@U\symAMSb{R}}$, in this paper we mean:

$\displaystyle b_{M}(z,z^{\prime}):=|\kappa_{M}(z,\bar{z}^{\prime})|^{2}.$

The Calabi’s diastasis [28] expressed via the CS-vectors [27] reads

$\displaystyle D_{M}(z,z^{\prime}):=-\ln
b_{M}(z,z^{\prime})=-2\ln\left|(\tilde{e}_{\bar{z}},\tilde{e}_{\bar{z}^{\prime}})\right|.$
(2.11)

Generalizing a theorem proved for homogeneous bounded domains [52], in [53]
Loi and Mossa have proved:

######  Theorem 2.2.

Let $(M,\omega)$ be a simply-connected homogeneous Kähler manifold such that
the associated Kähler form $\omega$ is integral. Then there exists a constant
$\mu_{0}>0$ such that $M$ equipped with $\mu_{0}\omega$ is projectively
induced.

With Theorem 2.2 and the recalled definitions, we can formulate the

######  Remark 2.3.

Let $M=G/H$ be a simply-connected homogeneous Kähler manifold. Then the
following assertions are equivalent:

  1. A)

$M$ is a quantizable Kähler manifold,

  2. B)

$M$ admits a balanced metric,

  3. C)

$M$ is CS-type manifold and $G$ is a CS-type group,

  4. D)

$M$ is projectively induced and we have (2.8), (2.9).

Loi and Mossa have proved [53] the following

######  Theorem 2.4.

Let $(M,\omega)$ be a homogeneous Kähler manifold. Then the following are
equivalent:

  1. a)

$M$ is contractible,

  2. b)

$(M,\omega)$ admits a global Kähler potential,

  3. c)

$(M,\omega)$ admits a global diastasis $D_{M}\colon M\times
M\rightarrow{\use@mathgroup\M@U\symAMSb{R}}$,

  4. d)

$(M,\omega)$ admits a Berezin quantization.

The proof of the Theorem 2.4 is based on the fundamental conjecture for
homogeneous bounded domains [32] and the asymptotic of the Bergman kernel
[34].

Theorems 2.2, 2.4 and Remark 2.3 will be used in the proof of Proposition 3.3.

###  2.2 Elements of geometry of the Siegel upper half-plane and Siegel ball

Firstly we recall some standard facts about the symplectic group and its
algebra, see references and details in [7, 9]. Even if we will be concerned
mainly with the Jacobi group $G^{J}_{n}$, we recall some basic facts about
Cayley and partial Cayley transform which will appear in the paper.

####  2.2.1 The Cayley transform

If $X\in{\mathfrak{sp}}(n,{\use@mathgroup\M@U\symAMSb{R}})$, i.e.,
$X^{t}J+JX=0$, then

$\displaystyle X=\left(\begin{matrix}a&b\\\
c&-a^{t}\end{matrix}\right),\qquad\text{where}\quad b=b^{t},\quad
c=c^{t},\quad a,b,c\in M(n,{\use@mathgroup\M@U\symAMSb{R}}).$

If $g=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in{{\operatorname{Sp}}}(n,{\use@mathgroup\M@U\symAMSb{R}})$,
i.e., $g^{t}Jg=J$, then the matrices $a,b,c,d\in
M(n,{\use@mathgroup\M@U\symAMSb{R}})$ have the properties

$\displaystyle a^{t}c=c^{t}a,\\!\\!\qquad b^{t}d=d^{t}b,\\!\\!\qquad
a^{t}d-c^{t}b={\mathbbm{1}}_{n},\\!\\!\qquad ab^{t}=ba^{t},\\!\\!\qquad
cd^{t}=dc^{t},\\!\\!\qquad ad^{t}-bc^{t}={\mathbbm{1}}_{n}.$

We have the correspondence

$\displaystyle g=\left(\begin{matrix}a&b\\\ c&d\end{matrix}\right)\in
M(2n,{\use@mathgroup\M@U\symAMSb{R}})\leftrightarrow
g_{{\use@mathgroup\M@U\symAMSb{C}}}={\mathcal{C}}^{-1}g{\mathcal{C}}=\left(\begin{matrix}p&q\\\
\bar{q}&\bar{p}\end{matrix}\right),\qquad p,q\in
M(n,{\use@mathgroup\M@U\symAMSb{C}}),$ (2.12)

where

$\displaystyle{\mathcal{C}}=\left(\begin{matrix}{\rm i}{\mathbbm{1}}_{n}&{\rm
i}{\mathbbm{1}}_{n}\\\
-{\mathbbm{1}}_{n}&{\mathbbm{1}}_{n}\end{matrix}\right),\qquad{\mathcal{C}}^{-1}=\frac{1}{2}\left(\begin{array}[]{cc}-{\rm
i}{\mathbbm{1}}_{n}&-{\mathbbm{1}}_{n}\\\ -{\rm
i}{\mathbbm{1}}_{n}&{\mathbbm{1}}_{n}\end{array}\right),$ (2.15)
$\displaystyle 2a=p+q+\bar{p}+\bar{q},\qquad 2b={\rm
i}(\bar{p}-\bar{q}-p+q),\qquad 2c={\rm i}(p+q-\bar{p}-\bar{q}),$
$\displaystyle 2d=p-q+\bar{p}-\bar{q},\qquad 2p=a+d+{\rm i}(b-c),\qquad
2q=a-d-{\rm i}(b+c).$ (2.16)

To every $g\in\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})$, we
associate via (2.12) $g\mapsto
g_{{\use@mathgroup\M@U\symAMSb{C}}}\in\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}\equiv\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{C}})\cap{\rm
U}(n,n)$ as in (1.1), where the matrices $p,q\in
M(n,{\use@mathgroup\M@U\symAMSb{C}})$ have the properties

$\displaystyle pp^{*}-qq^{*}={\mathbbm{1}}_{n},\qquad pq^{t}=qp^{t},\qquad
p^{*}p-q^{t}\bar{q}={\mathbbm{1}}_{n},\qquad p^{t}\bar{q}=q^{*}p.$ (2.17)

Let us consider an element $h=(g,l)$ in
$G^{J}_{n}({\use@mathgroup\M@U\symAMSb{R}})_{0}$, i.e.,

$\displaystyle g=\left(\begin{matrix}a&b\\\
c&d\end{matrix}\right)\in{\operatorname{Sp}}(n,{\use@mathgroup\M@U\symAMSb{R}}),\qquad
l=(n,m)\in{\use@mathgroup\M@U\symAMSb{R}}^{2n},$ (2.18)

and $V\in{\mathcal{X}}_{n}$,
$u\in{\use@mathgroup\M@U\symAMSb{C}}^{n}\equiv{\use@mathgroup\M@U\symAMSb{R}}^{2n}$.

Let
$g\in\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}$
be of the form (1.1), (2.17) and
$\alpha,z\in{\use@mathgroup\M@U\symAMSb{C}}^{n}$. The (transitive) action
$(g,\alpha)\times(W,z)=(W_{1},z_{1})$ of the Jacobi group $G^{J}_{n}$ on the
Siegel–Jacobi ball ${\mathcal{D}}^{J}_{n}$ is given by the formulas [7]

$\displaystyle
W_{1}=(pW+q)(\bar{q}W+\bar{p})^{-1}=(Wq^{*}+p^{*})^{-1}(q^{t}+Wp^{t}),$
$\displaystyle z_{1}=(Wq^{*}+p^{*})^{-1}(z+\alpha-W\bar{\alpha}).$ (2.19)

Now we consider the partial Cayley transform [7, 9]
$\Phi\colon{\mathcal{X}}^{J}_{n}\rightarrow{\mathcal{D}}_{n}^{J}$,
$\Phi(V,u)=(W,z)$

$\displaystyle W=(V-{\rm i}{\mathbbm{1}}_{n})(V+{\rm
i}{\mathbbm{1}}_{n})^{-1},\qquad z=2{\rm i}(V+{\rm
i}{\mathbbm{1}}_{n})^{-1}u,$ (2.20)

with the inverse partial Cayley transform
$\Phi^{-1}\colon{\mathcal{D}}^{J}_{n}\rightarrow{\mathcal{X}}^{J}_{n}$,
$\Phi^{-1}(W,z)=(V,u)$

$\displaystyle V={\rm i}({\mathbbm{1}}_{n}-W)^{-1}({\mathbbm{1}}_{n}+W),\qquad
u=({\mathbbm{1}}_{n}-W)^{-1}z.$ (2.21)

Let us define $\Theta\colon
G^{J}_{n}({\use@mathgroup\M@U\symAMSb{R}})_{0}\rightarrow(G^{J}_{n})_{0}$,
$\Theta(h)=h_{*}$, $h=(g,n,m)$,
$h_{*}=(g_{{\use@mathgroup\M@U\symAMSb{C}}},\alpha)$.

We have proved in [9, Proposition 2] (see also [15, 78])

######  Proposition 2.5.

$\Theta$ is an group isomorphism and the action of $(G^{J}_{n})_{0}$ on
${\mathcal{D}}^{J}_{n}$ is compatible with the action of
$G^{J}_{n}({\use@mathgroup\M@U\symAMSb{R}})_{0}$ on ${\mathcal{X}}^{J}_{n}$
through the biholomorphic partial Cayley transform (2.20), i.e., if
$\Theta(h)=h_{*}$, then $\Phi h=h_{*}\Phi$. More exactly, if the action of
$(G^{J}_{n})_{0}$ on ${\mathcal{D}}^{J}_{n}$ is given by (2.19), then the
action of $G^{J}_{n}({\use@mathgroup\M@U\symAMSb{R}})_{0}$ on
${\mathcal{X}}^{J}_{n}$ is given by
$(g,l)\times(V,u)\rightarrow(V_{1},u_{1})\in{\mathcal{X}}^{J}_{n}$, where

$\displaystyle V_{1}=(aV+b)(cV+d)^{-1}=(Vc^{t}+d^{t})^{-1}(Va^{t}+b^{t}),$
$\displaystyle u_{1}=(Vc^{t}+d^{t})^{-1}(u+Vn+m).$ (2.22)

The matrices $g$ in (2.18) and $g_{{\use@mathgroup\M@U\symAMSb{C}}}$ in (1.1)
are related by (2.12), (2.16), while $\alpha=m+{\rm i}n$,
$m,n\in{\use@mathgroup\M@U\symAMSb{R}}^{n}$.

####  2.2.2 The metric

Siegel has determined the metric on ${\mathcal{X}}_{n}$,
$\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})$-invariant to the action
(2.22) (see [71, equation (2)] or [37, Theorem 3, p. 644]):

$\displaystyle\operatorname{d}s^{2}_{{\mathcal{X}}_{n}}(R)=\operatorname{Tr}\big{(}R^{-1}\operatorname{d}VR^{-1}\operatorname{d}\bar{V}\big{)}.$
(2.23)

With the Cayley transform (2.21) and the relations

$\displaystyle\dot{V}=2{\rm i}U\dot{W}U,\qquad
U=({\mathbbm{1}}_{n}-W)^{-1},\qquad
2R=({\mathbbm{1}}_{n}+W)U+({\mathbbm{1}}_{n}+\bar{W})\bar{U},$ (2.24)

introduced in (2.23), it is obtained the metric on ${\mathcal{D}}_{n}$,
$\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}$-invariant
to the action (2.19):

$\displaystyle\operatorname{d}s^{2}_{{\mathcal{D}}_{n}}(W)=4\operatorname{Tr}(M\operatorname{d}W\bar{M}\operatorname{d}\bar{W}),\qquad
W\in{\mathcal{D}}_{n},\qquad M=({\mathbbm{1}}_{n}-W\bar{W})^{-1},$ (2.25)

associated with the Kähler two-form (3.26), modulo the factor $\frac{2}{k}$.

######  Remark 2.6.

The metric (2.25) can be written as in (2.27), (2.28).

###### Proof.

Equation (2.25) can be written down as

$\displaystyle\tfrac{1}{4}\operatorname{d}s^{2}_{{\mathcal{D}}_{n}}(W)=\sum_{p,q,m,n}H_{pq\bar{m}\bar{n}}\operatorname{d}w_{pq}\operatorname{d}\bar{w}_{mn},$
$\displaystyle H_{pq\bar{m}\bar{n}}:=M_{np}M_{mq},\qquad
M:=({\mathbbm{1}}_{n}-W\bar{W})^{-1}.$

We have

$\displaystyle\tfrac{1}{4}\operatorname{d}s^{2}_{{\mathcal{D}}_{n}}(W)=\sum_{p<q;\,m<n}(H_{pq\bar{m}\bar{n}}+H_{pq\bar{n}\bar{m}}+H_{qp\bar{m}\bar{n}}+H_{qp\bar{n}\bar{m}})\operatorname{d}w_{pq}\operatorname{d}\bar{w}_{mn}$
$\displaystyle\hphantom{\tfrac{1}{4}\operatorname{d}s^{2}_{{\mathcal{D}}_{n}}(W)=}{}+\sum_{p<q}(H_{pq\bar{m}\bar{m}}+H_{qp\bar{m}\bar{m}})\operatorname{d}w_{pq}\operatorname{d}\bar{w}_{mm}$
$\displaystyle\hphantom{\tfrac{1}{4}\operatorname{d}s^{2}_{{\mathcal{D}}_{n}}(W)=}{}+\sum_{m<n}(H_{pp\bar{m}\bar{n}}+H_{pp\bar{n}\bar{m}})\operatorname{d}w_{pp}\operatorname{d}\bar{w}_{mn}+\sum
H_{pp\bar{m}\bar{m}}\operatorname{d}w_{pp}\operatorname{d}\bar{w}_{mm}.$
(2.26)

For reasons which will be clarified further (see formulas (3.24), (3.18e)), we
write (2.26) as

$\displaystyle\tfrac{1}{4}\operatorname{d}s^{2}_{{\mathcal{D}}_{n}}(W)=\sum_{p\leq
q;\,m\leq n}h^{k}_{pqmn}\operatorname{d}w_{pq}\operatorname{d}\bar{w}_{mn},$
(2.27)

where

$\displaystyle
h^{k}_{pq\bar{m}\bar{n}}=2M_{mp}M_{nq}(1-\delta_{pq})+2M_{mq}M_{np}(1-\delta_{mn})+M^{2}_{mp}\delta_{pq}\delta_{mn},$
(2.28)

which proves the statement of the remark. ∎

In order to take into account the symmetry of the matrix
$W=(w_{ij})_{i,j=1,\dots,n}$, we introduce the notation (see [9, equation
(4.5)]):

$\displaystyle\Delta^{ij}_{pq}:=\frac{\partial w_{ij}}{\partial
w_{pq}}=\delta_{ip}\delta_{jq}+\delta_{iq}\delta_{jp}-\delta_{ij}\delta_{pq}\delta_{ip},\qquad
w_{ij}=w_{ji}.$ (2.29)

For calculation of the “inverse” $h^{-1}$ of the metric matrix $h$ of
${\mathcal{D}}^{J}_{n}$ – see Theorem 3.2 – we need the “inverse” of the
matrix (2.28), which we calculate in Proposition B.2, exploiting the formula
given in [57] for the Laplace–Beltrami operator (1.6) on ${\mathcal{X}}_{n}$
with the result (see (B.11)):

$\displaystyle
k_{mn\bar{u}\bar{v}}=\tfrac{1}{2}(N_{vn}\bar{N}_{mu}+N_{vm}\bar{N}_{nu}),\qquad
N={\mathbbm{1}}_{n}-W\bar{W}.$ (2.30)

We calculate

$\displaystyle E_{pq}^{uv}:=\sum_{m\leq
n}h^{k}_{pq\bar{m}\bar{n}}k_{mn\bar{u}\bar{v}}.$ (2.31)

We shall prove a relation which gives the sense of the “inverse” matrix of
(2.28), important in Theorem 3.2:

######  Lemma 2.7.

The “inverse” matrix of the metric matrix (2.28) on the Siegel ball
${\mathcal{D}}_{n}$ is the matrix (2.30) and we have

$\displaystyle E^{uv}_{pq}=\Delta^{uv}_{pq}.$ (2.32)

###### Proof.

a) Firstly, we consider the case $p=q$ in (2.31). We get successively

$\displaystyle E^{uv}_{pp}=\sum_{m\leq
n}(2-\delta_{mn})({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{mp}({\mathbbm{1}}_{n}-\bar{W}W)^{-1}_{pn}k_{mn\bar{u}\bar{v}}$
$\displaystyle\hphantom{E^{uv}_{pp}}{}=({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{mp}({\mathbbm{1}}_{n}-\bar{W}W)^{-1}_{pm}({\mathbbm{1}}_{n}-W\bar{W})_{vm}({\mathbbm{1}}_{n}-\bar{W}W)_{mu}$
$\displaystyle\hphantom{E^{uv}_{pp}=}{}+\sum_{m<n}({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{mp}({\mathbbm{1}}_{n}-\bar{W}W)^{-1}_{pn}$
$\displaystyle\hphantom{E^{uv}_{pp}=}{}\times[({\mathbbm{1}}_{n}-W\bar{W})_{vn}({\mathbbm{1}}_{n}-\bar{W}W)_{mu}+({\mathbbm{1}}_{n}-W\bar{W})_{vm}({\mathbbm{1}}_{n}-\bar{W}W)_{nu}]$
$\displaystyle\hphantom{E^{uv}_{pp}}{}=({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{mp}({\mathbbm{1}}_{n}-\bar{W}W)^{-1}_{pm}({\mathbbm{1}}_{n}-W\bar{W})_{vm}({\mathbbm{1}}_{n}-\bar{W}W)_{mu}$
$\displaystyle\hphantom{E^{uv}_{pp}=}{}+\sum_{m<n}({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{np}({\mathbbm{1}}_{n}-\bar{W}W)^{-1}_{pm}({\mathbbm{1}}_{n}-W\bar{W})_{vn}({\mathbbm{1}}_{n}-\bar{W}W)_{mu}$
$\displaystyle\hphantom{E^{uv}_{pp}=}{}+\sum_{n<m}({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{mp}({\mathbbm{1}}_{n}-\bar{W}W)^{-1}_{pm}({\mathbbm{1}}_{n}-W\bar{W})_{vn}({\mathbbm{1}}_{n}-\bar{W}W)_{mu}$
$\displaystyle\hphantom{E^{uv}_{pp}}{}=\delta_{up}\delta_{vp}.$ (2.33)

b) Now we consider the case $p\not=q$ in (2.31). We get successively

$\displaystyle E^{uv}_{pq}=2\\!\sum_{m\leq
n}\\!\\!\big{[}({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{mp}({\mathbbm{1}}_{n}-\bar{W}W)^{-1}_{qn}\\!+({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{mq}({\mathbbm{1}}_{n}-\bar{W}W)^{-1}_{pn}(1-\delta_{mn})\big{]}k_{mn\bar{u}\bar{v}}\\!$
$\displaystyle\hphantom{E^{uv}_{pq}}{}=2({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{mp}({\mathbbm{1}}_{n}-\bar{W}W)^{-1}_{qm}({\mathbbm{1}}_{n}-W\bar{W})_{vm}({\mathbbm{1}}_{n}-\bar{W}W)_{mu}$
$\displaystyle\hphantom{E^{uv}_{pq}=}{}+\sum_{m<n}\big{[}({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{mp}({\mathbbm{1}}_{n}-\bar{W}W)^{-1}_{qn}+({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{mq}({\mathbbm{1}}_{n}-\bar{W}W)^{-1}_{pn}\big{]}$
$\displaystyle\hphantom{E^{uv}_{pq}=}{}\times\big{[}({\mathbbm{1}}_{n}-W\bar{W})_{vn}({\mathbbm{1}}_{n}-\bar{W}W)_{vn}({\mathbbm{1}}_{n}-\bar{W}W)_{mu}+({\mathbbm{1}}_{n}-W\bar{W})_{vm}({\mathbbm{1}}_{n}-\bar{W}W)_{nu}\big{]}$
$\displaystyle\hphantom{E^{uv}_{pq}}{}=\delta_{pu}\delta_{vq}+\delta_{pv}\delta_{qu}.$
(2.34)

Putting together (2.33) and (2.34), with formula

$\displaystyle E^{pq}_{uv}=E^{pq}_{uv}(1-\delta_{pq})+E^{pq}_{uv}\delta_{pq},$

we have proved (2.32). ∎

###  2.3 Geometry of the Siegel–Jacobi disk

We recall some formulas describing the differential geometry of the
Siegel–Jacobi disk ${\mathcal{D}}^{J}_{1}$ [5, 10, 12].

The (transitive) action of the group
$G^{J}_{1}={\mathrm{H}}_{1}\rtimes\text{SU}(1,1)\ni(g,\alpha)\times(z,w)\rightarrow(z_{1},w_{1})\in{\mathcal{D}}^{J}_{1}$
on the Siegel disk is given by the formulas

$\displaystyle
w_{1}=\frac{aw+b}{\delta},\qquad\delta=\bar{b}w+\bar{a},\qquad\text{SU}(1,1)\ni
g=\left(\begin{matrix}a&b\\\ \bar{b}&\bar{a}\end{matrix}\right),$ (2.35)

where $|a|^{2}-|b|^{2}=1$,

$\displaystyle
z_{1}=\frac{\gamma}{\delta},\qquad\gamma=z+\alpha-\bar{\alpha}w.$ (2.36)

The balanced Kähler two-form is obtained with formulas (1.3), (2.1)

$\displaystyle-{\rm
i}\omega_{k\mu}(z,w)=h_{z\bar{z}}\operatorname{d}z\wedge\operatorname{d}\bar{z}+h_{z\bar{w}}\operatorname{d}z\wedge\operatorname{d}\bar{w}-h_{\bar{z}w}\operatorname{d}\bar{z}\wedge\operatorname{d}w+h_{w\bar{w}}\operatorname{d}w\wedge\operatorname{d}\bar{w},$

where the Kähler potential (2.2) is

$\displaystyle
f(z,w)=\mu\frac{2z\bar{z}+z^{2}\bar{w}+\bar{z}^{2}w}{2P}-2k\ln(P),\qquad
P=1-w\bar{w}.$ (2.37)

$k$ indexes the positive discrete series of $\text{SU}(1,1)$
($2k\in{\use@mathgroup\M@U\symAMSb{N}}$), while $\mu>0$ indexes the
representations of the Heisenberg group.

######  Proposition 2.8.

The balanced Kähler two-form $\omega_{k\mu}$ on ${\mathcal{D}}^{J}_{1}$,
$G^{J}_{1}$-invariant to the action (2.35), (2.36), can be written as

$\displaystyle-{\rm
i}\omega_{k\mu}(z,w)=2k\frac{\operatorname{d}w\wedge\operatorname{d}\bar{w}}{P^{2}}+\mu\frac{{\mathcal{A}}\wedge\bar{{\mathcal{A}}}}{P},\qquad{\mathcal{A}}=\operatorname{d}z+\bar{\eta}(z,w)\operatorname{d}w,$
$\displaystyle
z=\eta-w\bar{\eta},\qquad\text{and}\qquad\eta=\eta(z,w):=\frac{z+\bar{z}w}{1-w\bar{w}}.$

The matrix of the balanced metric (2.5) $h=h(\varsigma)$,
$\varsigma:=(z,w)\in{\use@mathgroup\M@U\symAMSb{C}}\times{\mathcal{D}}_{1}$,
determined with the Kähler potential (2.37), reads

$\displaystyle h(\varsigma):=\left(\begin{matrix}h_{z\bar{z}}&h_{z\bar{w}}\\\
h_{w\bar{z}}=\bar{h}_{z\bar{w}}&h_{w\bar{w}}\end{matrix}\right)=\left(\begin{matrix}\frac{\mu}{P}&\mu\frac{\eta}{P}\\\
\mu\frac{\bar{\eta}}{P}&\frac{2k}{P^{2}}+\mu\frac{|\eta|^{2}}{P}\end{matrix}\right).$
(2.38)

The inverse of the matrix (2.38) reads

$\displaystyle
h^{-1}(\varsigma)=\left(\begin{matrix}h^{z\bar{z}}&h^{z\bar{w}}\\\
h^{w\bar{z}}&h^{w\bar{w}}\end{matrix}\right)=\left(\begin{matrix}\frac{P}{\mu}+\frac{P^{2}|\eta|^{2}}{2k}&-\frac{P^{2}\eta}{2k}\\\
-\frac{P^{2}\bar{\eta}}{2k}&\frac{P^{2}}{2k}\end{matrix}\right).$ (2.39)

If we introduce the notation

$\displaystyle{\mathcal{G}}_{M}(z):=\det(h_{\alpha\bar{\beta}})_{\alpha,\beta=1,\dots,n},$

then we find

$\displaystyle{\mathcal{G}}_{{\mathcal{D}}^{J}_{1}}(z,w)=\frac{2k\mu}{(1-w\bar{w})^{3}},\qquad
z\in{\use@mathgroup\M@U\symAMSb{C}},\qquad|w|<1.$ (2.40)

##  3 Elements of geometry of the Siegel–Jacobi ball

###  3.1 The balanced metric

The Jacobi algebra is the semi-direct sum
${\mathfrak{g}}^{J}_{n}:={\mathfrak{h}}_{n}\rtimes{\mathfrak{sp}}(n,{\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}$
[6, 7, 9]. The Heisenberg algebra ${\mathfrak{h}}_{n}$ is generated by the
boson creation (respectively, annihilation) operators ${a}_{i}^{\dagger}$
(${a}_{i}$), $i,j=1,\dots,n$, which verify the canonical commutation relations

$\displaystyle\big{[}a_{i},a^{\dagger}_{j}\big{]}=\delta_{ij},\qquad[a_{i},a_{j}]=\big{[}a_{i}^{\dagger},a_{j}^{\dagger}\big{]}=0.$
(3.1)

${\mathfrak{h}}_{n}$ is an ideal in ${\mathfrak{g}}^{J}_{n}$, i.e.,
$[{\mathfrak{h}}_{n},{\mathfrak{g}}^{J}_{n}]={\mathfrak{h}}_{n}$, determined
by the commutation relations:

$\displaystyle\big{[}a^{\dagger}_{k},K^{+}_{ij}\big{]}=[a_{k},K^{-}_{ij}]=0,$
(3.2a)
$\displaystyle[a_{i},K^{+}_{kj}]=\tfrac{1}{2}\delta_{ik}a^{\dagger}_{j}+\tfrac{1}{2}\delta_{ij}a^{\dagger}_{k},\qquad\big{[}K^{-}_{kj},a^{\dagger}_{i}\big{]}=\tfrac{1}{2}\delta_{ik}a_{j}+\tfrac{1}{2}\delta_{ij}a_{k},$
(3.2b)
$\displaystyle\big{[}K^{0}_{ij},a^{\dagger}_{k}\big{]}=\tfrac{1}{2}\delta_{jk}a^{\dagger}_{i},\qquad\big{[}a_{k},K^{0}_{ij}\big{]}=\tfrac{1}{2}\delta_{ik}a_{j}.$
(3.2c)

$K^{\pm,0}_{ij}$ are the generators of the
${\mathfrak{sp}}({\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}$
algebra:

$\displaystyle[K_{ij}^{-},K_{kl}^{-}]=[K_{ij}^{+},K_{kl}^{+}]=0,\qquad
2\big{[}K^{-}_{ij},K^{0}_{kl}\big{]}=K_{il}^{-}\delta_{kj}+K^{-}_{jl}\delta_{ki},$
(3.3a) $\displaystyle
2[K_{ij}^{-},K_{kl}^{+}]=K^{0}_{kj}\delta_{li}+K^{0}_{lj}\delta_{ki}+K^{0}_{ki}\delta_{lj}+K^{0}_{li}\delta_{kj},$
(3.3b) $\displaystyle
2\big{[}K^{+}_{ij},K^{0}_{kl}\big{]}=-K^{+}_{ik}\delta_{jl}-K^{+}_{jk}\delta_{li},\qquad
2\big{[}K^{0}_{ji},K^{0}_{kl}\big{]}=K^{0}_{jl}\delta_{ki}-K^{0}_{ki}\delta_{lj}.$
(3.3c)

Applying the same arguments as in the proof of Remark 3 in [10] (see also the
Appendix in [10], where the notion of homogeneous reductive space in the
meaning of Nomizu [63] is recalled; also see [11]) to the Jacobi algebra
${\mathfrak{g}}^{J}_{n}$ determined by (3.1), (3.2), (3.3), we make the

######  Remark 3.1.

The Jacobi group $G^{J}_{n}$ is an unimodular, non-reductive, algebraic group
of Harish-Chandra type. The Siegel–Jacobi domain ${\mathcal{D}}^{J}_{n}$ is a
reductive, non-symmetric manifold associated to the Jacobi group $G^{J}_{n}$
by the generalized Harish-Chandra embedding.

We have attached to the Jacobi group
$G^{J}_{n}:={\mathrm{H}}_{n}\rtimes\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}$
coherent states based on the Siegel–Jacobi ball ${\mathcal{D}}^{J}_{n}$ [7].
Perelomov’s CS vectors [64] associated to the group $G^{J}_{n}$ with Lie
algebra the Jacobi algebra ${\mathfrak{g}}^{J}_{n}$ based on the complex
$d$-dimensional ($d=n(n+3)/2$) manifold – the Siegel–Jacobi ball –
${\mathcal{D}}^{J}_{n}\approx{\mathrm{H}}_{n}/{\use@mathgroup\M@U\symAMSb{R}}\times\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}/\text{U}(n)={\use@mathgroup\M@U\symAMSb{C}}^{n}\times{\mathcal{D}}_{n}$
– have been defined as [7]

$\displaystyle
e_{z,W}=\exp({{\boldsymbol{X}}})e_{0},\qquad{\boldsymbol{X}}:=\sqrt{\mu}\sum_{i=1}^{n}z_{i}{\boldsymbol{a}}^{\dagger}_{i}+\sum_{i,j=1}^{n}w_{ij}{\boldsymbol{K}}^{+}_{ij},\qquad
z\in{\use@mathgroup\M@U\symAMSb{C}}^{n},\qquad W\in{\mathcal{D}}_{n}.$ (3.4)

The vector $e_{0}$ appearing in (3.4) verifies the relations

$\displaystyle{\boldsymbol{a}}_{i}e_{0}=0,\qquad i=1,\dots,n,$
$\displaystyle{\boldsymbol{K}}^{+}_{ij}e_{0}\not=0,\qquad{\boldsymbol{K}}^{-}_{ij}e_{0}=0,\qquad{\boldsymbol{K}}^{0}_{ij}e_{0}=\frac{k}{4}\delta_{ij}e_{0},\qquad
i,j=1,\dots,n.$ (3.5)

In (3.5) $e_{0}=e^{H}_{0}\otimes e^{K}_{0}$, where $e^{H}_{0}$ is the minimum
weight vector (vacuum) for the Heisenberg group ${\mathrm{H}}_{n}$, while
$e^{K}_{0}$ is the extremal weight vector for
$\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}$
corresponding to the weight $k$ in (3.5), and $\mu$ parametrizes the
Heisenberg group [7, 9, 15]. Holomorphic irreducible representations of the
Jacobi group based on Siegel–Jacobi domains have been studied in mathematics
[22, 23, 72, 73, 74]. In [5, 9, 15] we have compared our results on the
geometry of ${\mathcal{D}}^{J}_{n}$ and holomorphic representations of
$G^{J}_{n}$ obtained using CS with those of the mentioned mathematicians.

The scalar product
$K:{\mathcal{D}}^{J}_{n}\times{\mathcal{D}}^{J}_{n}\rightarrow{\use@mathgroup\M@U\symAMSb{C}}$,
$K(\bar{x},\bar{V};y,W)=(e_{x,V},e_{y,W})_{k\mu}$ is [7, 15]:

$\displaystyle(e_{x,V},e_{y,W})_{k\mu}=\det(U)^{k/2}\exp\mu
F(\bar{x},\bar{V};y,W),\qquad U=({\mathbbm{1}}_{n}-W\bar{V})^{-1},$
$\displaystyle 2F(\bar{x},\bar{V};y,W)=2(x,Uy)+(V\bar{y},Uy)+(x,UW\bar{x}).$
(3.6)

If we denote $\zeta=(x,V)$, $\zeta^{\prime}=(y,W)$, then we find for the
normalized Bergman kernel (2.10) of the Siegel–Jacobi ball the expression

$\displaystyle\kappa_{{\mathcal{D}}^{J}_{n}}(\zeta,\zeta^{\prime})=\kappa_{{\mathcal{D}}_{n}}(V,W)\exp\mu[2F(\zeta,\zeta^{\prime})-F(\zeta)-F(\zeta^{\prime})],$
(3.7)

where $\kappa_{{\mathcal{D}}_{n}}(V,W)$ is the normalized Bergman kernel of
the Siegel ball

$\displaystyle\kappa_{{\mathcal{D}}_{n}}(V,W)=\det{}^{k/2}\left[\frac{({\mathbbm{1}}_{n}-V\bar{V})({\mathbbm{1}}_{n}-W\bar{W})}{({\mathbbm{1}}_{n}-W\bar{V})^{2}}\right].$

We find for the Calabi diastasis (2.11) of the Siegel–Jacobi ball the
expression

$\displaystyle
D_{{\mathcal{D}}^{J}_{n}}(\zeta,\zeta^{\prime})=D_{{\mathcal{D}}_{n}}(V,W)+2\mu\big{[}\Re
F(\zeta)+\Re F(\zeta^{\prime})-2\Re F(\zeta,\zeta^{\prime})\big{]},$ (3.8)
$\displaystyle
D_{{\mathcal{D}}_{n}}(V,W)=-2\ln|\kappa_{{\mathcal{D}}_{n}}(V,W)|.$

In particular, the reproducing kernel $K(z,W)=(e_{z,W},e_{z,W})_{k\mu}$,
$z\in{\use@mathgroup\M@U\symAMSb{C}}^{n}$, $W\in{\mathcal{D}}_{n}$ is

$\displaystyle K(z,W)=\det(M)^{k/2}\exp\mu F,\qquad
M=({\mathbbm{1}}_{n}-W\bar{W})^{-1},$ (3.9) $\displaystyle
2F=2\bar{z}^{t}Mz+z^{t}\bar{W}Mz+\bar{z}^{t}MW\bar{z},\qquad
2F=2\bar{\eta}^{t}\eta-\eta^{t}\bar{W}\eta-\bar{\eta}^{t}W\bar{\eta},$
$\displaystyle\eta=M(z+W\bar{z}),\qquad z=\eta-W\bar{\eta}.$ (3.10)

The Hilbert space of holomorphic functions ${\mathcal{F}}_{K}$ associated to
the kernel $K$ given by (3.9) is endowed with the scalar product of the type
(2.3)

$\displaystyle(\phi,\psi)_{{\mathcal{F}}_{K}}=\Lambda_{n}\int_{z\in{\use@mathgroup\M@U\symAMSb{C}}^{n};\,W=W^{t};\,1-W\bar{W}>0}\bar{f}_{\phi}(z,W)f_{\psi}(z,W)QK^{-1}\operatorname{d}z\operatorname{d}W,$
$\displaystyle\operatorname{d}z=\prod_{i=1}^{n}\operatorname{d}\Re
z_{i}\operatorname{d}\Im z_{i},\qquad\operatorname{d}W=\prod_{1\leq i\leq
j\leq n}\operatorname{d}\Re w_{ij}\operatorname{d}\Im{w}_{ij},$

where the normalization constant $\Lambda_{n}$ is given by

$\displaystyle\Lambda_{n}=\mu^{n}\frac{k-3}{2\pi^{n(n+3)/2}}\prod_{i=1}^{n-1}\frac{\big{(}\frac{k-3}{2}-n+i\big{)}\Gamma(k+i-2)}{\Gamma[k+2(i-n-1)]},$

and the density of volume is [7] (see also Lemma A.2)

$\displaystyle Q(z,W)=\det({\mathbbm{1}}_{n}-W\bar{W})^{-(n+2)}.$ (3.11)

The manifold ${\mathcal{D}}^{J}_{n}$ has the Kähler potential (3.12), $f=\log
K$, with $K$ given by (3.9),

$\displaystyle f=-\tfrac{k}{2}\log\det({\mathbbm{1}}_{n}-W\bar{W})$
$\displaystyle\hphantom{f=}{}+\mu\big{\\{}\bar{z}^{t}({\mathbbm{1}}_{n}-W\bar{W})^{-1}{z}+\tfrac{1}{2}z^{t}\big{[}\bar{W}({\mathbbm{1}}_{n}-W\bar{W})^{-1}\big{]}z+\tfrac{1}{2}\bar{z}^{t}\big{[}({\mathbbm{1}}_{n}-W\bar{W})^{-1}W\big{]}\bar{z}\big{\\}}.$
(3.12)

Because the symmetry of the matrix $W\in{\mathcal{D}}_{n}$, we write down the
Kähler two-form on the Siegel–Jacobi ball as

$\displaystyle-{\rm
i}\omega_{{\mathcal{D}}^{J}_{n}}(z,W)=h_{i\bar{j}}\operatorname{d}z_{i}\wedge\operatorname{d}\bar{z}_{j}+\sum_{p\leq
q}(h_{i\bar{p}\bar{q}}\operatorname{d}z_{i}\wedge\operatorname{d}\bar{w}_{pq}-\bar{h}_{i\bar{p}\bar{q}}\operatorname{d}\bar{z}_{i}\wedge\operatorname{d}w_{pq})$
$\displaystyle\hphantom{-{\rm
i}\omega_{{\mathcal{D}}^{J}_{n}}(z,W)=}{}+\sum_{p\leq q;m\leq
n}h_{pq\bar{m}\bar{n}}\operatorname{d}w_{pq}\wedge\operatorname{d}\bar{w}_{mn}.$
(3.13)

We use the following notation for the matrix of the metric

$\displaystyle h=\left(\begin{matrix}h_{1}&h_{2}\\\
h_{3}&h_{4}\end{matrix}\right)=\left(\begin{matrix}h_{i\bar{j}}&h_{i\bar{p}\bar{q}}\\\
h_{pq\bar{i}}&h_{pq\bar{m}\bar{n}}\end{matrix}\right)\in
M(n(n+3)/2,{\use@mathgroup\M@U\symAMSb{C}}),$ (3.14) $\displaystyle p\leq
q,\qquad m\leq n,\qquad h=h^{*},$

where $h_{1}\in M(n,{\use@mathgroup\M@U\symAMSb{C}})$, $h_{2}\in M(n\times
n(n+1)/2,{\use@mathgroup\M@U\symAMSb{C}})$, $h_{3}\in M(n(n+1)/2\times
n,{\use@mathgroup\M@U\symAMSb{C}})$, $h_{4}\in
M(n(n+1)/2,{\use@mathgroup\M@U\symAMSb{C}})$.

We use the following convention: indices of $z\in M(n\times
1,{\use@mathgroup\M@U\symAMSb{C}})$ are denoted with $i$, $j$, $k$, $l$;
indices of $W=W^{t}$, $W\in M(n,{\use@mathgroup\M@U\symAMSb{C}})$ are denoted
with $m$, $n$, $p$, $q$, $r$, $s$, $t$, $u$, $v$.

Let as write the “inverse” $h^{-1}$ of the matrix $h$ (3.14) as

$\displaystyle h^{-1}=\left(\begin{matrix}h^{1}&h^{2}\\\
h^{3}&h^{4}\end{matrix}\right)=\left(\begin{matrix}(h^{1})_{i\bar{j}}&(h^{2})_{i\bar{p}\bar{q}}\\\
(h^{3})_{pq\bar{i}}&(h^{4})_{pq\bar{m}\bar{n}}\end{matrix}\right),\qquad p\leq
q,\qquad m\leq n,$ (3.15)

where the matrices $h^{1}$, $h^{2}$, $h^{3}$, $h^{4}$ have the same dimensions
as the matrices $h_{1}$, $h_{2}$, $h_{3}$, respectively $h_{4}$.

In fact, we shall determine the “inverse” (3.15) of (3.14) such that

$\displaystyle\left(\begin{matrix}h_{1}&h_{2}\\\
h_{3}&h_{4}\end{matrix}\right)\left(\begin{matrix}h^{1}&h^{2}\\\
h^{3}&h^{4}\end{matrix}\right)=\left(\begin{matrix}{\mathbbm{1}}_{n}&0\\\
0&\Delta\end{matrix}\right),$ (3.16)

where $\Delta$ is defined in (2.29).

It is useful to introduce the notation

$\displaystyle f_{pq}:=1-\tfrac{1}{2}\delta_{pq}.$

We shall prove the following (partial results have been presented in [6,
Proposition 11], [7, Proposition 3.11] and [9, Proposition 1]):

######  Theorem 3.2.

The Kähler two-form $\omega_{{\mathcal{D}}^{J}_{n}}$, associated with the
balanced metric of the Siegel–Jacobi ball ${\mathcal{D}}^{J}_{n}$,
$G^{J}_{n}$-invariant to the action (2.19) is

$\displaystyle-{\rm
i}\omega_{{\mathcal{D}}^{J}_{n}}(z,W)=\tfrac{k}{2}\operatorname{Tr}({\mathcal{B}}\wedge\bar{{\mathcal{B}}})+\mu\operatorname{Tr}({\mathcal{A}}^{t}\bar{M}\wedge\bar{{\mathcal{A}}}),\qquad{\mathcal{A}}=\operatorname{d}z+\operatorname{d}W\bar{\eta},$
$\displaystyle{\mathcal{B}}=M\operatorname{d}W,\qquad
M=({\mathbbm{1}}_{n}-W\bar{W})^{-1}.$ (3.17)

The matrix (3.14) of the metric on ${\mathcal{D}}^{J}_{n}$ has the matrix
elements (3.18):

$\displaystyle h_{i\bar{j}}=\mu\bar{M}_{ij},$ (3.18a) $\displaystyle
h_{i\bar{p}\bar{q}}=\mu(\eta_{q}\bar{M}_{ip}+\eta_{p}\bar{M}_{iq})f_{pq},$
(3.18b) $\displaystyle
h_{pq\bar{i}}=\mu(\bar{\eta}_{q}\bar{M}_{pi}+\bar{\eta}_{p}\bar{M}_{qi})f_{pq},$
(3.18c) $\displaystyle
h_{pq\bar{m}\bar{n}}=\frac{k}{2}h^{k}_{pq\bar{m}\bar{n}}+\mu
h^{\mu}_{pq\bar{m}\bar{n}},$ (3.18d) $\displaystyle
h^{k}_{pq\bar{m}\bar{n}}=2(M_{mp}M_{nq}+M_{mq}M_{np})-2M_{mp}(M_{np}\delta_{pq}+M_{mq}\delta_{mn})+M^{2}_{mp}\delta_{pq}\delta_{mn}$
$\displaystyle\hphantom{h^{k}_{pq\bar{m}\bar{n}}}{}=2M_{mp}M_{nq}(1-\delta_{pq})+2M_{mq}M_{np}(1-\delta_{mn})+M^{2}_{mp}\delta_{pq}\delta_{mn},$
(3.18e) $\displaystyle
h^{\mu}_{pq\bar{m}\bar{n}}=\bar{\eta}_{p}(\eta_{n}M_{mq}+\eta_{m}M_{nq})+\bar{\eta}_{q}({\eta}_{n}M_{mp}+\eta_{m}M_{np})-\bar{\eta}_{p}(\eta_{n}M_{mp}$
$\displaystyle\hphantom{h^{\mu}_{pq\bar{m}\bar{n}}=}{}+\eta_{m}M_{np})\delta_{pq}-{\eta}_{m}(\bar{\eta}_{p}M_{mq}+\bar{\eta}_{q}M_{mp})\delta_{mn}+\bar{\eta}_{p}\eta_{m}M_{mp}\delta_{pq}\delta_{mn}$
$\displaystyle\hphantom{h^{\mu}_{pq\bar{m}\bar{n}}}{}=[\bar{\eta}_{p}(\eta_{n}\bar{M}_{qm}+\eta_{m}\bar{M}_{qn})+\bar{\eta}_{q}(\eta_{n}\bar{M}_{pm}+\eta_{m}\bar{M}_{pn})]f_{pq}f_{mn}.$
(3.18f)

The “inverse” $h^{-1}$ of the metric matrix $h$ which verifies (3.16), with
components (3.18), obtained with the inverse (2.30) of $h^{k}$ (3.18e) has the
elements $h^{1}$–$h^{4}$ given by

$\displaystyle\big{(}h^{1}\big{)}_{ij}=\theta\bar{M}^{-1}_{ij},\qquad\theta=\frac{1}{\mu}+\alpha\frac{2}{k},\qquad\alpha=\eta^{t}\bar{M}^{-1}\bar{\eta}=\bar{\eta}_{n}S_{n},\qquad
S_{n}=\sum\eta_{q}\bar{M}^{-1}_{qn},\\!\\!$ (3.19a)
$\displaystyle\big{(}h^{2}\big{)}_{i\bar{m}\bar{n}}=-\frac{1}{k}\big{(}S_{n}\bar{M}^{-1}_{im}+S_{m}\bar{M}^{-1}_{in}\big{)},$
(3.19b)
$\displaystyle\big{(}h^{3}\big{)}_{mn\bar{i}}=-\frac{1}{k}\big{(}\bar{S}_{n}\bar{M}^{-1}_{mi}+\bar{S}_{m}\bar{M}^{-1}_{ni}\big{)},$
(3.19c)
$\displaystyle\big{(}h^{4}\big{)}_{pq\bar{m}\bar{n}}=\frac{2}{k}(h^{k})^{-1}_{pq\bar{m}\bar{n}}=\frac{1}{k}\big{(}\bar{M}^{-1}_{qn}\bar{M}^{-1}_{pm}+\bar{M}^{-1}_{pn}\bar{M}^{-1}_{qm}\big{)}.$
(3.19d)

The determinant of the metric matrix $h$ is

$\displaystyle{\mathcal{G}}_{{\mathcal{D}}^{J}_{n}}(z,W)=\det
h_{{\mathcal{D}}^{J}_{n}}(z,W)=\left(\frac{k}{2}\right)^{\frac{n(n+1)}{2}}\mu^{n}\det({\mathbbm{1}}_{n}-W\bar{W})^{-(n+2)}.$
(3.20)

In the case $n=1$ formulas (3.18), (3.19), (3.20) became the formulas (2.38),
(2.39), respectively (2.40), with $2k\leftrightarrow\frac{k}{2}$.

###### Proof.

Firstly, we determine the matrix elements of the metric $h$ on the
Siegel–Jacobi disk ${\mathcal{D}}^{J}_{n}$ applying formula (2.1) to the
Kähler potential (3.12) as in (3.13).

We use the formulas (see [9, equations (4.23)])

$\displaystyle\frac{\partial M_{ab}}{\partial
w_{ik}}=M_{ai}X_{kb}+M_{ak}X_{ib}-M_{ai}X_{ib}\delta_{ik},\qquad\text{where}\quad
X=X^{t}=\bar{W}M=\bar{M}\bar{W},$ (3.21a) $\displaystyle\frac{\partial
X_{ab}}{\partial w_{ik}}=X_{ai}X_{bk}+X_{ak}X_{ib}-X_{ai}X_{ib}\delta_{ik},$
(3.21b) $\displaystyle\frac{\partial\bar{X}_{ab}}{\partial
w_{ik}}=M_{ai}M_{bk}+M_{ak}M_{bi}-M_{ai}M_{bk}\delta_{ik}.$ (3.21c)

From (3.10), we get

$\displaystyle\frac{\partial\eta_{q}}{\partial z_{l}}=M_{ql},$ (3.22a)
$\displaystyle\frac{\partial\bar{\eta}_{q}}{\partial z_{j}}={X}_{qj},$ (3.22b)
$\displaystyle\frac{\partial\eta_{t}}{\partial
w_{pq}}=M_{tp}\bar{\eta}_{q}+M_{tq}\bar{\eta}_{p}-M_{tp}\bar{\eta}_{p}\delta_{pq},$
(3.22c) $\displaystyle\frac{\partial\bar{\eta}_{n}}{\partial
w_{pq}}=\bar{\eta}_{p}X_{qn}+\bar{\eta}_{q}X_{pn}-\bar{\eta}_{q}X_{pn}\delta_{pq}.$
(3.22d)

With (2.29), (3.21) and (3.22), we find the matrix elements (3.18) of the
metric of the Siegel–Jacobi ball.

Now we prove (3.18e). We calculate

$\displaystyle
h^{k}_{mn\bar{p}\bar{q}}=\frac{\partial}{\partial\bar{w}_{pq}}J_{mn},\qquad
J_{mn}=\frac{\partial}{\partial w_{mn}}\ln\det M.$ (3.23)

With formula

$\displaystyle\frac{\operatorname{d}(\det A)}{\operatorname{d}t}=\det
A\operatorname{Tr}\left(A^{-1}\frac{\partial A}{\partial t}\right),$

we have

$\displaystyle J_{mn}=M^{-1}_{ij}\frac{\partial M_{ji}}{\partial w_{mn}}.$

We find

$\displaystyle J_{mn}=X_{nm}+X_{mn}-X_{mn}\delta_{mn}=2X_{mn}f_{mn}.$

In order to obtain $h^{k}_{mn\bar{p}\bar{q}}$, we calculate

$\displaystyle\frac{\partial
X_{nm}}{\partial\bar{w}_{pq}}=\frac{\partial\bar{w}_{nk}}{\partial\bar{w}_{pq}}M_{kn}+w_{nk}\frac{\partial
M_{kb}}{\partial\bar{w}_{pq}}.$

But with

$\displaystyle\frac{\partial
M_{km}}{\partial\bar{w}_{pq}}=\overline{\left(\frac{M_{mk}}{\partial
w_{pq}}\right)},\qquad\text{and
with}\qquad{\mathbbm{1}}_{n}+\bar{W}\bar{X}=\bar{M},$

we find

$\displaystyle\frac{\partial
X_{nm}}{\partial\bar{w}_{pq}}=2(M_{pm}\bar{M}_{nq}+M_{qm}\bar{M}_{np}-M_{pm}\bar{M}_{nq}\delta_{pq})-(2M_{pm}\bar{M}_{mq}-M_{pm}\bar{M}_{mq}\delta_{pq})\delta_{mn},$

i.e., we reobtain formula (3.18e).

With (3.18e), we have

$\displaystyle h^{k}_{pp\bar{m}\bar{m}}=M^{2}_{mp},$ $\displaystyle
h^{k}_{pp\bar{m}\bar{n}}=2M_{mp}M_{nq},\qquad m<n,$ $\displaystyle
h^{k}_{pq\bar{m}\bar{m}}=2M_{mp}M_{mq},\qquad p<q,$ $\displaystyle
h^{k}_{pq\bar{m}\bar{n}}=2(M_{mp}M_{nq}+M_{mq}M_{np}),\qquad p<q,\qquad m<n.$
(3.24)

In order to find $-{\rm
i}\frac{2}{k}\operatorname{d}\omega_{{\mathcal{D}}_{n}}(W)$, we make the
summation

$\displaystyle\sum
h^{k}_{pp\bar{m}\bar{m}}\operatorname{d}w_{pp}\wedge\operatorname{d}\bar{w}_{mm}+\sum_{m<n}h^{k}_{pp\bar{m}\bar{n}}\operatorname{d}w_{pp}\wedge\operatorname{d}\bar{w}_{mn}$
$\displaystyle\qquad\quad{}+\sum_{p<q}h^{k}_{pq\bar{m}\bar{m}}\operatorname{d}w_{pq}\wedge\operatorname{d}\bar{w}_{mm}+\sum_{p<q,\,m<n}h^{k}_{pq\bar{m}\bar{n}}\operatorname{d}w_{pq}\wedge\operatorname{d}\bar{w}_{mn}$
$\displaystyle\qquad{}=\sum
M_{mp}\bar{M}_{pm}\operatorname{d}w_{pp}\wedge\operatorname{d}\bar{w}_{mm}+2\sum_{m<n}M_{mp}M_{pn}\operatorname{d}w_{pp}\wedge\operatorname{d}\bar{w}_{mn}$
$\displaystyle\qquad\quad{}+2\sum_{p<q}M_{mp}M_{mq}+2\sum_{p<q,\,m<n}(M_{mp}M_{nq}+M_{mq}M_{np})\operatorname{d}w_{pq}\wedge\operatorname{d}\bar{w}_{mn}$
$\displaystyle\qquad{}=\sum
M_{mp}\bar{M}_{pm}\operatorname{d}w_{pp}\wedge\operatorname{d}\bar{w}_{mm}+\sum_{m\not=n}M_{mp}M_{pn}\operatorname{d}w_{pp}\wedge\operatorname{d}\bar{w}_{mn}$
$\displaystyle\qquad\quad{}+\sum_{p\not=q}M_{mp}M_{nq}\operatorname{d}w_{pq}\wedge\operatorname{d}\bar{w}_{mm}+\sum_{p\not=q,\,m\not=n}M_{mp}M_{nq}\operatorname{d}w_{pq}\wedge\operatorname{d}\bar{w}_{mn}$
$\displaystyle\qquad{}=(M\operatorname{d}W)_{mq}\wedge(\bar{M}\operatorname{d}\bar{W})_{qm}.$
(3.25)

We write down (3.25) as

$\displaystyle-{\rm
i}\frac{2}{k}\operatorname{d}\omega_{{\mathcal{D}}_{n}}(W)=\operatorname{Tr}(M\operatorname{d}W\wedge\bar{M}\operatorname{d}\bar{W}).$
(3.26)

Now we make the summation

$\displaystyle\sum_{p\leq
q}h^{\mu}_{i\bar{p}\bar{q}}\operatorname{d}z_{i}\wedge\operatorname{d}\bar{w}_{pq}=\sum
h^{\mu}_{i\bar{p}\bar{p}}\operatorname{d}z_{i}\wedge\operatorname{d}\bar{w}_{pp}+\sum_{p<q}h^{\mu}_{i\bar{p}\bar{q}}\operatorname{d}z_{i}\wedge\operatorname{d}\bar{w}_{pq}$
$\displaystyle\hphantom{\sum_{p\leq
q}h^{\mu}_{i\bar{p}\bar{q}}\operatorname{d}z_{i}\wedge\operatorname{d}\bar{w}_{pq}}{}=\bar{M}_{ip}\eta_{p}\operatorname{d}z_{i}\wedge\operatorname{d}\bar{w}_{pp}+\sum_{p<q}(\bar{M}_{ip}\eta_{q}+\bar{M}_{iq}\eta_{p})\operatorname{d}z_{i}\wedge\operatorname{d}\bar{w}_{pq}$
$\displaystyle\hphantom{\sum_{p\leq
q}h^{\mu}_{i\bar{p}\bar{q}}\operatorname{d}z_{i}\wedge\operatorname{d}\bar{w}_{pq}}{}=(M\operatorname{d}z)_{p}\wedge(\operatorname{d}\bar{W}\eta)_{p}.$

We calculate the sum

$\displaystyle H^{\mu}:=\sum_{p\leq q,m\leq
n}h^{\mu}_{pq\bar{m}\bar{n}}\operatorname{d}w_{pq}\wedge\operatorname{d}\bar{w}_{mn}.$

We have

$\displaystyle H^{\mu}=\sum
h^{\mu}_{pp\bar{m}\bar{m}}\operatorname{d}w_{pp}\wedge\bar{w}_{mm}+\sum_{m<n}h^{\mu}_{pp\bar{m}\bar{n}}\operatorname{d}w_{pp}\wedge\bar{w}_{mn}$
$\displaystyle\hphantom{H^{\mu}=}{}+\sum_{p<q}h^{\mu}_{pq\bar{m}\bar{m}}\operatorname{d}w_{pq}\wedge\bar{w}_{mm}+\sum_{p<q,\,m<n}h^{\mu}_{pq\bar{m}\bar{n}}\operatorname{d}w_{pq}\wedge\bar{w}_{mn},$

and finally we get

$\displaystyle
H^{\mu}=M_{mq}\operatorname{d}w_{qp}\wedge\operatorname{d}\bar{w}_{mn}\eta_{n}=(M\operatorname{d}W\bar{\eta})_{m}\wedge(\operatorname{d}\bar{W}\eta)_{m}.$
(3.27)

Putting together (3.26), (3.25) and (3.27), we have proved (3.17).

In order to check out the homogeneity of the Kähler two-form
$\omega_{{\mathcal{D}}^{J}_{n}}$ (3.17), we use the formulas

$\displaystyle\operatorname{d}W_{1}=(Wq^{*}+p^{*})^{-1}\operatorname{d}W(\bar{q}W+\bar{p})^{-1},$
(3.28) $\displaystyle M_{1}=(q\bar{W}+p)M(Wq^{*}+p^{*}),$ (3.29)
$\displaystyle\operatorname{d}z_{1}=(Wq^{*}+p^{*})^{-1}(\operatorname{d}WV+\operatorname{d}z),\qquad
V=-q^{*}(Wq^{*}+p^{*})^{-1}[\eta+\alpha-W(\bar{\eta}+\bar{\alpha})]-\bar{\alpha},$
$\displaystyle\eta_{1}=p(\alpha+\eta)+q(\bar{\alpha}+\bar{\eta}),\qquad{\mathcal{A}}_{1}=(Wq^{*}+p^{*})^{-1}{\mathcal{A}}.$

Now we calculate the “inverse” (3.15) $h^{-1}$ of the matrix $h$ whose
elements are given by (3.18).

We have (see, e.g., in [56, (1) and (2), p. 30]): if the matrices $h_{1}$,
$h_{4}-h_{3}h^{-1}_{1}h_{2}$, $h_{4}$ and respectively
$h_{1}-h_{2}h^{-1}_{4}h_{3}$ are nonsingular, then

$\displaystyle
h^{1}=\big{(}h_{1}-h_{2}h^{-1}_{4}h_{3}\big{)}^{-1}=h_{1}^{-1}+h_{1}^{-1}h_{2}h^{4}h_{3}h_{1}^{-1},$
(3.30a) $\displaystyle h^{2}=-h^{-1}_{1}h_{2}h^{4},$ (3.30b) $\displaystyle
h^{3}=-h^{4}h_{3}h_{1}^{-1},$ (3.30c) $\displaystyle
h^{4}=(h_{4}-h_{3}h^{-1}_{1}h_{2})^{-1}.$ (3.30d)

In order to calculate $h^{4}$, we write (3.30d) as

$\displaystyle(h^{4})^{-1}=h_{4}-\mu h^{I},\qquad\text{where}\quad
h^{I}:=\frac{1}{\mu}h_{3}h_{1}^{-1}h_{2}.$ (3.31)

With formulas (3.18a)–(3.18c), we get

$\displaystyle\big{(}h^{I}\big{)}_{pq\bar{m}\bar{n}}=h_{pq\bar{i}}(h_{1})^{-1}_{ij}h_{j\bar{m}\bar{n}}=(\bar{\eta}_{q}\bar{M}_{pi}+\bar{\eta}_{p}\bar{M}_{qi})\bar{M}^{-1}_{ij}(\eta_{n}\bar{M}_{jm}+\eta_{m}\bar{M}_{jn})f_{pq}f_{mn}$
$\displaystyle\hphantom{\big{(}h^{I}\big{)}_{pq\bar{m}\bar{n}}}{}=(\bar{\eta}_{q}\bar{M}_{pi}+\bar{\eta}_{p}\bar{M}_{qi})(\eta_{n}\delta_{im}+\eta_{m}\delta_{im})f_{pq}f_{mn}$
$\displaystyle\hphantom{\big{(}h^{I}\big{)}_{pq\bar{m}\bar{n}}}{}=[\bar{\eta}_{q}(\eta_{n}\bar{M}_{pm}+\eta_{m}\bar{M}_{pn})+\bar{\eta}_{p}(\eta_{n}\bar{M}_{qm}+\eta_{m}\bar{M}_{qn})]f_{pq}f_{mn},$

and with equation (3.18f) we get

$\displaystyle h^{I}=h^{\mu}.$

With (3.31), we obtain

$\displaystyle\big{(}h^{4}\big{)}^{-1}=\frac{k}{2}h^{k},$ (3.32)

and (3.19d), where $h^{k}$ has the expression (3.18e), while $(h^{k})^{-1}$
has the expression (2.30).

In order to calculate $h^{2}$, taking into account (3.32), we write (3.30b) as

$\displaystyle h^{2}=-(h_{1})^{-1}h^{J},\qquad\text{where}\quad
h^{J}:=\frac{2}{k}h_{2}\big{(}h^{k}\big{)}^{-1}.$ (3.33)

With formulas (3.18b), (B.11), we have

$\displaystyle\big{(}h^{J}\big{)}_{j\bar{m}\bar{n}}=2\frac{\mu}{k}\sum_{p\leq
q}f_{pq}(\eta_{q}\bar{M}_{jp}+\eta_{p}\bar{M}_{jq})k_{pq\bar{m}\bar{n}}$
$\displaystyle\hphantom{\big{(}h^{J}\big{)}_{j\bar{m}\bar{n}}}{}=2\frac{\mu}{k}\left[\sum_{p}\eta_{p}({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{pj}({\mathbbm{1}}_{n}-W\bar{W})_{np}({\mathbbm{1}}_{n}-\bar{W}W)_{pm}\right]$
$\displaystyle\hphantom{\big{(}h^{J}\big{)}_{j\bar{m}\bar{n}}=}{}+\frac{\mu}{k}\sum_{p<q}\big{[}\eta_{q}({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{pj}+\eta_{p}({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{pj}\big{]}$
$\displaystyle\hphantom{\big{(}h^{J}\big{)}_{j\bar{m}\bar{n}}=}{}\times\big{[}({\mathbbm{1}}_{n}-W\bar{W})_{nq}({\mathbbm{1}}_{n}-\bar{W}W)_{pm}+({\mathbbm{1}}_{n}-W\bar{w})_{np}({\mathbbm{1}}_{n}-\bar{W}W)_{qm}\big{]}$
$\displaystyle\hphantom{\big{(}h^{J}\big{)}_{j\bar{m}\bar{n}}}{}=\frac{\mu}{k}\left[\sum_{p\not=q}(\eta_{q}({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{pj}({\mathbbm{1}}_{n}-W\bar{W})_{nq}({\mathbbm{1}}_{n}-\bar{W}W)_{pm}\right.$
$\displaystyle\hphantom{\big{(}h^{J}\big{)}_{j\bar{m}\bar{n}}=}{}+\sum_{p\not=q}\eta_{p}({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{qj}({\mathbbm{1}}_{n}-W\bar{W})_{nq}({\mathbbm{1}}_{n}-\bar{W}{W})_{pm}$
$\displaystyle\left.\hphantom{\big{(}h^{J}\big{)}_{j\bar{m}\bar{n}}=}{}+2\sum_{p}\eta_{p}({\mathbbm{1}}_{n}-W\bar{W})^{-1}_{pj}({\mathbbm{1}}_{n}-W\bar{W})_{np}({\mathbbm{1}}_{n}-\bar{W}W)_{pm}\right]$
$\displaystyle\hphantom{\big{(}h^{J}\big{)}_{j\bar{m}\bar{n}}=}{}=\frac{\mu}{k}(S_{n}\delta_{jm}+S_{m}\delta_{jn}),\qquad\text{where}\quad
S_{n}:=\sum_{q}\eta_{q}\bar{M}^{-1}_{qn}.$

With (3.33), we get (3.19b) and similarly for (3.19c).

In order to calculate $h^{1}$, we write (3.30a) as

$\displaystyle h^{1}=h^{-1}_{1}-h^{K}h^{-1}_{1},\qquad\text{where}\quad
h^{K}:=h^{2}h_{3}.$

With formulas (3.19b) and (3.18c), we get

$\displaystyle\big{(}h^{K}\big{)}_{ik}=-\frac{\mu}{k}\sum_{m\leq
n}f_{mn}\big{(}S_{n}\bar{M}^{-1}_{im}+S_{m}\bar{M}^{-1}_{in}\big{)}(\bar{\eta}_{n}\bar{M}_{mk}+\bar{\eta}_{m}\bar{M}_{nk})$
$\displaystyle\hphantom{\big{(}h^{K}\big{)}_{ik}}{}=-\frac{\mu}{k}\left[\frac{1}{2}\sum_{m}4S_{m}\bar{\eta}_{m}\bar{M}^{-1}_{im}\bar{M}_{mk}+\sum_{m<n}\big{(}S_{n}\bar{M}^{-1}_{im}+S_{m}\bar{M}^{-1}_{in}\big{)}\big{(}\bar{\eta}_{n}\bar{M}_{mk}+\bar{\eta}_{m}\bar{M}_{nk}\big{)}\right]$
$\displaystyle\hphantom{\big{(}h^{K}\big{)}_{ik}}{}=-\frac{2}{k}\mu\alpha\delta_{ik},$

where

$\displaystyle\alpha:=\eta^{t}\bar{M}^{-1}\bar{\eta}=\eta^{*}M^{-1}\eta=\bar{\eta}_{n}S_{n}.$

With (3.18a), we get (3.19a).

In order to prove (3.20), we use the relation

$\displaystyle\det h=\det\left(\begin{matrix}h_{1}&h_{2}\\\
h_{3}&h_{4}\end{matrix}\right)=\det
h_{1}\det\big{(}h_{4}-h_{3}h^{-1}_{1}h_{2}\big{)}.$

With (3.30d) and (3.32), we get

$\displaystyle\det h=\left(\frac{k}{2}\right)^{n}\det h_{1}\det h^{k}.$

(3.20) is obtained introducing in the above relation the expressions (3.18a)
and (A.4). ∎

Now we formulate several geometric properties of the Siegel–Jacobi ball
relevant for the Berezin quantization of this manifold:

######  Proposition 3.3.

  1. $i)$

The homogeneous Kähler manifold ${\mathcal{D}}^{J}_{n}$ is contractible.

  2. $ii)$

The Kähler potential of the Siegel–Jacobi ball (3.12) is global.
${\mathcal{D}}^{J}_{n}$ is a Q.-K. Lu manifold, with normalized Bergman kernel
(3.7) and nowhere vanishing diastasis (3.8).

  3. $iii)$

The manifold ${\mathcal{D}}^{J}_{n}$ is a quantizable manifold.

  4. $iv)$

${\mathcal{D}}^{J}_{n}$ is projectively induced and the Jacobi group
$G^{J}_{n}$ is a CS-type group.

  5. $v)$

${\mathcal{D}}^{J}_{n}$ is a homogeneous reductive space.

###### Proof.

$i)$ We have proved in Theorem 3.2 (see also [6, 9]) that
${\mathcal{D}}^{J}_{n}$ is a homogeneous Kähler manifold. We have shown in [9,
Proposition 4.1] that the ${\rm FC}$-transform ${\rm
FC}\colon(\eta,W)\rightarrow(z,W)$ expressed by (3.10) is a homogeneous Kähler
diffeomorphism
${\mathcal{D}}^{J}_{n}\approx{\mathcal{D}}_{n}\times{\use@mathgroup\M@U\symAMSb{C}}^{n}$,
where the Siegel ball admits the realization (1.2). But ${\mathcal{D}}_{n}$
can be achieved as the open ball ${\mathcal{B}}_{n}$ (see, e.g., [62, p. 502])

$\displaystyle{\mathcal{B}}_{n}=\big{\\{}W\in
M(n,{\use@mathgroup\M@U\symAMSb{C}}),\qquad W=W^{t}\,|\,||W||<1\big{\\}}.$

So, ${\mathcal{D}}^{J}_{n}$ is diffeomorphic with the product of the
contractible spaces ${\mathcal{D}}_{n}$ and
${\use@mathgroup\M@U\symAMSb{C}}^{n}$, and consequently, it is itself
contractible.

$ii)$ We have for (3.9) $K(z,W)>0$, $\forall\,(z,W)\in{\mathcal{D}}^{J}_{n}$.
The explicit expressions of the Kähler potential, normalized Bergman kernel
and diastasis imply the assertions of $ii)$, but they could be also derived
from the Theorem 2.4, once we have proved $i)$. Even more, Theorem 2.4 asserts
that ${\mathcal{D}}^{J}_{n}$ admits a Berezin quantization.

$iii)$, $iv)$ We have observed in Theorem 3.2 that the Kähler two-form (3.17)
is associated with the balanced metric on ${\mathcal{D}}^{J}_{n}$, and we
apply Remark 2.3.

$v)$ This assertion was already mentioned in Remark 3.1, where it was used the
explicit form (3.1), (3.2), (3.3) of the Jacobi algebra
${\mathfrak{g}}^{J}_{n}$. We mention it again here because in [13] (see Remark
1 there for a more precise formulation) we have proved that the CS-manifolds
are reductive spaces. ∎

The fact that the Jacobi algebra ${\mathfrak{g}}^{J}_{n}$ is a CS-algebra is
well known, see, e.g., [50, Theorem 5.2] or [62, Example VII.2.3 , p. 294].
Using the explicit form of the base $\Phi(z,w)$ of orthonormal polynomials in
which the Bergman kernel $K_{{\mathcal{D}}^{J}_{1}}(z,w)$ is developed [5], we
have proved in [12, Proposition 2] that the Siegel–Jacobi disk is a CS-
manifold. In [15] we have determined the base $\Phi(z,W)$ on
${\mathcal{D}}^{J}_{n}$ in which the reproducing kernel (3.6) admits a series
expansion as in (2.6), but in Proposition 3.3 we have used Theorems 2.2, 2.4
and Remark 2.3 to prove directly Proposition 3.3.

###  3.2 Ricci form and scalar curvature

The Ricci form associated to the Kählerian two-form $\omega_{M}$ (1.3) is (see
[61, p. 90])

$\displaystyle\rho_{M}(z):={\rm
i}\sum_{\alpha,\beta=1}^{n}\text{Ric}_{\alpha\bar{\beta}}(z)\operatorname{d}z_{\alpha}\wedge\operatorname{d}\bar{z}_{\beta},\qquad\text{Ric}_{\alpha\bar{\beta}}(z)=-\frac{\partial^{2}}{\partial
z_{\alpha}\partial\bar{z}_{\beta}}\ln{\mathcal{G}}_{M}(z).$ (3.34)

The scalar curvature at a point $p\in M$ of coordinates $z$ is (see [45, p.
294] or [41, p. 144])

$\displaystyle
s_{M}(p):=\sum_{\alpha,\beta=1}^{n}(h_{\bar{\alpha}{\beta}})^{-1}\text{Ric}_{\alpha\bar{\beta}}(z).$
(3.35)

The Bergman metric corresponds to the Kähler two-form (see references in [12])

$\displaystyle\omega^{1}_{M}={\rm i}\partial\bar{\partial}\ln{\mathcal{G}}.$
(3.36)

Q.-K. Lu [55] has introduced for a bounded domain the positive definite
(1,1)-form

$\displaystyle\tilde{\omega}_{M}(z):={\rm
i}\sum_{\alpha,\beta=1}^{n}\tilde{h}_{\alpha\bar{\beta}}(z)\operatorname{d}z_{\alpha}\wedge\operatorname{d}\bar{z}_{\beta},\qquad\tilde{h}_{\alpha\bar{\beta}}(z):=(n+1)h_{\alpha\bar{\beta}}(z)-\text{Ric}_{\alpha,\beta}(z),$
(3.37)

which is Kähler, corresponding to the Kähler potential
$\tilde{f}=\ln(K_{M}(z)^{n+1}{\mathcal{G}}(z))$.

Now we prove

######  Proposition 3.4.

  1. $i)$

The Siegel–Jacobi ball ${\mathcal{D}}^{J}_{n}$ is not an Einstein manifold
with respect to the homogeneous Kähler metric attached to the Kähler two-form
(3.17), but it is one with respect to the Bergman metric corresponding to the
Bergman Kähler two-form (3.36).

  2. $ii)$

The Ricci form (3.34) on the Siegel–Jacobi ball associated to the homogeneous
Kählerian two-form (3.17) has the expression

$\displaystyle\rho_{{\mathcal{D}}^{J}_{n}}(z,W)=-\omega^{1}_{{\mathcal{D}}^{J}_{n}}(z,W)=-{\rm
i}(n+2)\operatorname{Tr}(M\operatorname{d}W\wedge\bar{M}\operatorname{d}\bar{W}),$

while

$\displaystyle\rho_{{\mathcal{D}}_{n}}(W)=-\omega^{1}_{{\mathcal{D}}_{n}}(W)=-{\rm
i}(n+1)\operatorname{Tr}(M\operatorname{d}W\wedge\bar{M}\operatorname{d}\bar{W}).$

  3. $iii)$

The scalar curvature (3.35) is constant and negative

$\displaystyle
s_{{\mathcal{D}}^{J}_{n}}(z,W)=-\frac{2}{k}n\frac{(n+1)(n+2)}{2}.$

  4. $iv)$

The Q.-K. Lu Kähler two-form (3.37) has the expression

$\displaystyle\tilde{\omega}_{{\mathcal{D}}^{J}_{n}}(z,W)=\frac{(n+1)(n+2)}{2}\omega_{{\mathcal{D}}^{J}_{n}}(z,W)-\rho_{{\mathcal{D}}^{J}_{n}}(W).$

###### Proof.

With formula (3.20) (or (3.11)) and (3.23), we find that the only nonzero
components of the Ricci tensor are

$\displaystyle(\text{Ric}_{mn\bar{p}\bar{q}})_{{\mathcal{D}}^{J}_{n}}(z,W)=-(n+2)h^{k}_{mn\bar{p}\bar{q}}(W).$
(3.38)

With calculation (3.25), we find

$\displaystyle\rho_{{\mathcal{D}}^{J}_{n}}(z,W)=-{\rm
i}(n+2)\bar{M}_{pi}\bar{M}_{jq}\operatorname{d}w_{pj}\wedge\operatorname{d}\bar{w}_{qi}.$

Applying (3.35) for ${\mathcal{D}}^{J}_{n}$, with formulas (3.19d), (3.38), we
get

$\displaystyle
s_{{\mathcal{D}}^{J}_{n}}(z,W):=\sum_{\alpha,\beta=1}^{n}(h)^{-1}_{\bar{\alpha}{\beta}}\text{Ric}_{\alpha\bar{\beta}}(z,W)=\sum_{p\leq
q;\,m\leq
n}(h)^{-1}_{mn\bar{p}\bar{q}}(\text{Ric}_{pq\bar{m}\bar{n}})_{{\mathcal{D}}^{J}_{n}}$
$\displaystyle\hphantom{s_{{\mathcal{D}}^{J}_{n}}(z,W)}{}=-\frac{2}{k}(n+2)\sum_{p\leq
q;\,m\leq
n}(h^{k})^{-1}_{mn\bar{p}\bar{q}}(h^{k})_{pq\bar{m}\bar{n}}=-\frac{2}{k}(n+2)\sum_{m\leq
n}\Delta^{mn}_{mn}$
$\displaystyle\hphantom{s_{{\mathcal{D}}^{J}_{n}}(z,W)}{}=-\frac{2}{k}(n+2)\frac{n(n+1)}{2}.$
∎

The scalar curvature of the Siegel–Jacobi disk was calculated in [10, 12, 80].

###  3.3 Laplace–Beltrami operator

We introduce the formulas (3.19) into the expression (1.6) of the
Laplace–Beltrami operator and we use Remark A.1. In the formula (3.40) of the
Laplace–Beltrami operator on ${\mathcal{D}}^{J}_{n}$, we use the expression
given in Proposition B.2 for the Laplace–Beltrami operator on
${\mathcal{D}}_{n}$. We introduce the notation (no summation!)

$\displaystyle(D_{z})_{\mu\nu}=\left(e_{\mu\nu}\frac{\partial}{\partial
z_{\mu\nu}}\right),\qquad
e_{\mu\nu}=\frac{1+\delta_{\mu\nu}}{2},~{}z_{\mu\nu}=z_{\nu\mu}.$ (3.39)

We obtain:

######  Proposition 3.5.

The Laplace–Beltrami operator on the Siegel–Jacobi ball
${\mathcal{D}}^{J}_{n}$, invariant to the action (2.19) of the Jacobi group
$G^{J}_{n}$, has the expression

$\displaystyle\Delta_{{\mathcal{D}}^{J}_{n}}(z,W)=\theta\frac{\partial}{\partial
z^{t}}N\frac{\partial}{\partial\bar{z}}-\frac{2}{k}\left\\{\operatorname{Tr}\left[SD_{W}N\frac{\partial}{\partial\bar{z}}\right]+cc\right\\}+\frac{2}{k}\Delta_{{\mathcal{D}}_{n}}(W),$
(3.40) $\displaystyle N={\mathbbm{1}}_{n}-W\bar{W}.$

We have also the relations

$\displaystyle\Delta_{{\mathcal{D}}^{J}_{n}}(\ln{\mathcal{G}})=-s_{{\mathcal{D}}^{J}_{n}}=\frac{2}{k}n\frac{(n+1)(n+2)}{2}.$
(3.41)

###### Proof.

We apply formula (1.6) with the matrix elements (3.19) and we use Remark A.1.
∎

We recall that Theorem 2.5 in Berezin’s paper [18] asserts essentially that

$\displaystyle\Delta_{M}(z)(\ln({\mathcal{G}}(z)))=ct$

for the balanced metric plus other 3 conditions. The first equation in (3.41)

$\displaystyle s=-\Delta\ln\det h$

is a general recipe for calculation of scalar curvature, see, e.g., [41,
equation (5.2.24), p. 253].

Also we recall that the Laplace operator on the Siegel–Jacobi ball was
calculated in [79] in other coordinates. We have determined the
Laplace–Beltrami operator on the Siegel–Jacobi disk in [12].

##  Appendix A Appendix

###  A.1 A remark

######  Remark A.1.

Let $\omega_{M}(z)$ be the non-degenerate two-form (1.3) on a complex manifold
$M$, invariant to a invertible holomorphic transformation
$z^{\prime}=z^{\prime}(z)$. Then the differential operator $\Delta_{M}$ given
by (1.6) is also invariant to the transformation $z\rightarrow z^{\prime}$. In
particular, $M$ may be a homogeneous Kähler manifold and $\Delta_{M}$ the
corresponding Laplace–Beltrami operator.

###### Proof.

Let us consider the action $G\times M\rightarrow M$: $g\times z=z^{\prime}$.
Then we have

$\displaystyle-{\rm
i}\omega_{M}(z^{\prime})=h^{\prime}_{\lambda\bar{\gamma}}(z^{\prime})\operatorname{d}z^{\prime}_{\lambda}\wedge\operatorname{d}\bar{z}_{\gamma}=h^{\prime}_{\lambda\bar{\gamma}}(z^{\prime}(z))a_{\lambda\alpha}\bar{a}_{\gamma\beta}\operatorname{d}z_{\alpha}\wedge\operatorname{d}\bar{z}_{\beta},\qquad\text{where}\quad
a_{\alpha\lambda}=\frac{\partial z^{\prime}_{\alpha}}{\partial z_{\lambda}}.$

If $\omega_{M}(z^{\prime})=\omega_{M}(z)$, then

$\displaystyle
h_{\alpha\bar{\beta}}(z)=h^{\prime}_{\lambda\gamma}(z^{\prime}(z))a_{\lambda\alpha}\bar{a}_{\gamma\beta}.$
(A.1)

Now we consider the Laplace–Beltrami operator (1.6) in the point $z^{\prime}$.
We have

$\displaystyle\frac{\partial}{\partial
z^{\prime}_{\alpha}}=b_{\gamma\alpha}\frac{\partial}{\partial
z_{\gamma}},\qquad\text{where}\quad b_{\lambda\beta}=\frac{\partial
z_{\lambda}}{\partial z^{\prime}_{\beta}},$

and

$\displaystyle a_{\alpha\lambda}b_{\lambda\beta}=\delta_{\alpha\beta}.$

We have

$\displaystyle\Delta_{M}(z^{\prime})=(h^{\prime}(z^{\prime}))^{-1}_{\alpha\bar{\beta}}\bar{b}_{\gamma\alpha}b_{\rho\beta}\frac{\partial^{2}}{\partial\bar{z}_{\gamma}\partial
z_{\rho}}.$

If $\Delta_{M}(z^{\prime})=\Delta_{M}(z)$, then

$\displaystyle
h^{-1}_{\gamma\bar{\rho}}(z)=(h^{\prime}(z^{\prime}(z)))^{-1}_{\alpha\bar{\beta}}\bar{b}_{\gamma\alpha}b_{\rho\beta}.$
(A.2)

We write (A.1) as

$\displaystyle h=a^{t}h^{\prime}\bar{a},$

which implies

$\displaystyle
h^{-1}=(\bar{a})^{-1}(h^{\prime})^{-1}(a^{t})^{-1}=\bar{b}(h^{\prime})^{-1}b^{t},$

which is exactly (A.2). ∎

###  A.2 A lemma

Formula (3.20), modulo the numerical factor, was obtained in [7]. We recall
the method to determine the Liouville form used in [7]. We have applied the
following technique (see [38, Chapter IV]):

######  Lemma A.2.

Let $z^{\prime}=f(g,z)$ denote the action of the group $G$ on the circular
domain $M$. Let us determine the element $g\in G$ such that
$z^{\prime}(z_{1})=0$ Then the density of the volume form is $Q=|J|^{2}$,
where $J$ is the Jacobian $J=\frac{\partial z^{\prime}}{\partial z}$.

The transformation with the desired properties is

$\displaystyle z^{\prime}=U(1-W_{1}\bar{W}_{1})^{1/2}(1-W\bar{W}_{1})^{-1}$
$\displaystyle\hphantom{z^{\prime}=}{}\times\big{[}z-(1-W\bar{W}_{1})(1-W_{1}\bar{W_{1}})^{-1}z_{1}+(W-W_{1})(1-\bar{W}_{1}W_{1})^{-1}\bar{z}_{1}\big{]},$
$\displaystyle
W^{\prime}=U(1-W_{1}\bar{W}_{1})^{-1/2}(W-W_{1})(1-\bar{W}_{1}W)^{-1}(1-\bar{W}_{1}W_{1})^{1/2}U^{t},$

where $U$ is an unitary matrix. We find that

$\displaystyle\frac{\partial z^{\prime}}{\partial
z}=U(1-W_{1}\bar{W}_{1})^{1/2}(1-W\bar{W}_{1})^{-1},$
$\displaystyle\operatorname{d}W^{\prime}=A\operatorname{d}WA^{t},\qquad
A=U(1-W_{1}\bar{W}_{1})^{1/2}(1-W\bar{W}_{1})^{-1}.$ (A.3)

In order to calculate the Jacobian of the transformation (A.3), we use the
following property extracted from Berezin’s paper [20, p. 398]:

Let $A$ be a matrix and $L_{A}$ the transformation of a matrix of the same
order $n$, $L_{A}\xi=A\xi A^{t}$. If the matrices $A$ and $\xi$ are symmetric,
then $\det L_{A}=(\det A)^{n+1}$. We find as in [7], in accord with formula
before Theorem 4.3.2 in [38]

$\displaystyle
Q_{{\mathcal{D}}_{n}}=\det({\mathbbm{1}}_{n}-W\bar{W})^{-(n+1)},$ (A.4)

while $Q_{{\mathcal{D}}^{J}_{n}}(z,W)$ has the expression (3.11).

##  Appendix B Appendix
Laplace–Beltrami operator on the Siegel upper half-plan
and Siegel ball

The Laplace–Beltrami operator on ${\mathcal{X}}_{n}$ has the expression (see
[57, equation (19)])

$\displaystyle\Delta_{{\mathcal{X}}_{n}}=-\operatorname{Tr}(Z-W)((Z-W)D_{w})^{t}D_{z},\qquad
Z=X+{\rm i}Y\in{\mathcal{X}}_{n},\qquad W=\bar{Z},$ (B.1)

where we use the notation (3.39). Formula (43) in [57] reads

$\displaystyle D_{w}(Z-W)=-\frac{n+1}{2}{\mathbbm{1}}_{n}+((Z-W)D_{w})^{t},$
(B.2)

where

$\displaystyle D_{w}(Z-W)=\sum_{\rho=1}^{n}{\mbox{\rm
e}}_{\mu\rho}\frac{\partial}{\partial w_{\mu\rho}}(z_{\rho\nu}-w_{\rho\nu})$
$\displaystyle\hphantom{D_{w}(Z-W)}{}=\left(\sum_{\rho=1}^{n}(z_{\mu\rho}-w_{\mu\rho})e_{\rho\nu}\frac{\partial}{\partial
w_{\rho\nu}}\right)^{t}-\left(\delta_{\mu\nu}\sum_{\rho=1}^{n}e_{\mu\rho}\right).$
(B.3)

With formula (2.29), introducing in (B.3), it is obtained

$\displaystyle((Z-W)D_{w})^{t}_{\mu\nu}(f)=\left[\frac{\partial f}{\partial
W}(Z-W)\right]_{\mu\nu}=(z_{\rho\nu}-w_{\rho\nu})e_{\mu\rho}\frac{\partial
f}{\partial w_{\mu\rho}}.$

Equation (54) of the Cayley transform [57] in the notation (B.2) reads

$\displaystyle\tilde{Z}=(Z-{\rm i}{\mathbbm{1}}_{n})(Z+{\rm
i}{\mathbbm{1}}_{n})^{-1}.$

Equation (55) in [57] asserts that

$\displaystyle D_{z}=-\frac{{\rm
i}}{2}(\tilde{Z}-{\mathbbm{1}}_{n})((\tilde{Z}-{\mathbbm{1}}_{n})D_{\tilde{z}})^{t},$

i.e., if now $W$ describes a point (1.2) in ${\mathcal{D}}_{n}$, we have the
formula

######  Lemma B.1.

$\displaystyle(D_{z})_{\alpha\beta}:=e_{\alpha\beta}\frac{\partial}{\partial
z_{\alpha\beta}}=-\frac{{\rm
i}}{2}\big{[}({\mathbbm{1}}_{n}-W)(({\mathbbm{1}}_{n}-W)D_{w})^{t}\big{]}_{\alpha\beta}$
$\displaystyle\hphantom{(D_{z})_{\alpha\beta}}{}=-\frac{{\rm
i}}{2}({\mathbbm{1}}_{n}-W)_{\alpha\gamma}({\mathbbm{1}}_{n}-W)_{\beta\lambda}e_{\lambda\gamma}\frac{\partial}{\partial
w_{\lambda\gamma}}.$ (B.4)

###### Proof.

We write (2.24) as

$\displaystyle 2{\rm i}\operatorname{d}W=A\operatorname{d}ZA,\qquad
A={\mathbbm{1}}_{n}-W,\qquad W=W^{t},\qquad Z=Z^{t},$

i.e.,

$\displaystyle 2{\rm
i}\operatorname{d}w_{pq}=A_{pm}\operatorname{d}z_{mn}A_{nq}.$

With formula (2.29), we get

$\displaystyle 2{\rm i}\frac{\partial w_{pq}}{\partial
z_{\alpha\beta}}=A_{pm}A_{nq}\frac{\partial z_{mn}}{\partial
z_{\alpha\beta}}=A_{pm}A_{nq}(\delta_{m\alpha}\delta_{n\beta}+\delta_{m\beta}\delta_{n\alpha}-\delta_{mn}\delta_{\alpha\beta}\delta_{m\alpha})$
$\displaystyle\hphantom{2{\rm i}\frac{\partial w_{pq}}{\partial
z_{\alpha\beta}}}{}=A_{p\alpha}A_{\beta q}+A_{p\beta}A_{\alpha
q}-A_{p\alpha}A_{\beta q}\delta_{\alpha\beta}.$ (B.5)

We have the formula

$\displaystyle\frac{\partial f}{\partial z_{\alpha\beta}}=\sum_{p\leq
q}\frac{\partial f}{\partial w_{pq}}\frac{\partial w_{pq}}{\partial
z_{\alpha\beta}}.$

For $\alpha\not=\beta$, we get from (B.5):

$\displaystyle 2{\rm i}\frac{\partial f}{\partial
z_{\alpha\beta}}=\sum_{p<q}(A_{p\alpha}A_{\beta q}+A_{p\beta}A_{\alpha
q})\frac{\partial f}{\partial w_{pq}}+\sum_{p=q}(A_{p\alpha}A_{\beta
p}+A_{p\beta}A_{\alpha p})\frac{\partial f}{\partial w_{pq}}$
$\displaystyle\hphantom{2{\rm i}\frac{\partial f}{\partial
z_{\alpha\beta}}}{}=\sum_{p<q}A_{p\alpha}A_{\beta q}\frac{\partial f}{\partial
w_{pq}}+\sum_{q<p}A_{q\beta}A_{\alpha p}\frac{\partial f}{\partial
w_{pq}}+2\sum_{p=q}A_{p\alpha}A_{\beta q}\frac{\partial f}{\partial
w_{pq}}\delta_{pq}$ $\displaystyle\hphantom{2{\rm i}\frac{\partial f}{\partial
z_{\alpha\beta}}}{}=\sum_{pq}A_{p\alpha}A_{\beta
q}(1+\delta_{pq})\frac{\partial f}{\partial w_{pq}}.$ (B.6)

Similarly, for $\alpha=\beta$, we have

$\displaystyle 2{\rm i}\frac{\partial f}{\partial z_{\alpha\beta}}=\sum_{p\leq
q}A_{p\alpha}A_{\beta q}\frac{\partial f}{\partial w_{pq}}$
$\displaystyle\hphantom{2{\rm i}\frac{\partial f}{\partial
z_{\alpha\beta}}}{}=\frac{1}{2}\sum_{p<q}A_{p\alpha}A_{\beta q}\frac{\partial
f}{\partial w_{pq}}+\frac{1}{2}\sum_{q<p}A_{q\alpha}A_{\beta p}\frac{\partial
f}{\partial w_{qp}}\frac{1}{2}+\sum_{p=q}A_{p\alpha}A_{\beta q}\frac{\partial
f}{\partial w_{pq}}$ $\displaystyle\hphantom{2{\rm i}\frac{\partial
f}{\partial z_{\alpha\beta}}}{}=\frac{1}{2}\sum_{p\not=q}A_{p\alpha}A_{\beta
q}\frac{\partial f}{\partial w_{pq}}+\sum_{p=q}A_{p\alpha}A_{\beta
q}\frac{\partial f}{\partial w_{pq}}=\sum_{pq}A_{p\alpha}A_{\beta
q}\frac{1+\delta_{pq}}{2}\frac{\partial f}{\partial w_{pq}}.$ (B.7)

We put together (B.6) and (B.7) as in (B.4). ∎

With Lemma B.1, it is obtained the Laplace–Beltrami operator on the Siegel
ball (see [57, equation (56)])

######  Proposition B.2.

The Laplace–Beltrami operator on the Siegel ball ${\mathcal{D}}_{n}$ has the
expression

$\displaystyle\Delta_{{\mathcal{D}}_{n}}(W)=\operatorname{Tr}\big{[}N(ND_{\bar{W}})^{t}D_{W}\big{]}=\sum_{p,q,m,n}{\use@mathgroup\M@U\symAMSb{K}}_{pq\bar{m}\bar{n}}\frac{\partial^{2}}{\partial\bar{w}_{mn}\partial
w_{pq}},\qquad N:={\mathbbm{1}}_{n}-W\bar{W},$ (B.8)
$\displaystyle{\use@mathgroup\M@U\symAMSb{K}}_{pq\bar{m}\bar{n}}=e_{mn}e_{pq}K_{pq\bar{m}\bar{n}},\qquad
K_{pq\bar{m}\bar{n}}=N_{qn}\bar{N}_{mp},$ (B.9)

which is
$\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})_{{\use@mathgroup\M@U\symAMSb{C}}}$-invariant
to the action (2.19). We write down the sum (B.8) as

$\displaystyle\Delta_{{\mathcal{D}}_{n}}(W)=\sum_{m\leq n;\,p\leq
q}k_{mn\bar{p}\bar{q}}(W)\frac{\partial^{2}}{\partial\bar{w}_{mn}\partial
w_{pq}},$

where, with the expression (B.9), we have

$\displaystyle
k_{pq\bar{m}\bar{n}}=\frac{1}{4}(K_{pq\bar{m}\bar{n}}+K_{qp\bar{m}\bar{n}}+K_{pq\bar{n}\bar{m}}+K_{qp\bar{n}\bar{m}}),\qquad
m<n,\qquad p<q,$ $\displaystyle
k_{pq\bar{m}\bar{m}}=\frac{1}{2}(K_{pq\bar{m}\bar{m}}+K_{qp\bar{m}\bar{m}}),\qquad
p<q,\qquad n=m,$ $\displaystyle
k_{pp\bar{m}\bar{n}}=\frac{1}{2}(K_{pp\bar{m}\bar{n}}+K_{pp\bar{n}\bar{m}}),\qquad
p=q,\qquad m<n,$ $\displaystyle k_{pp\bar{m}\bar{m}}=K_{pp\bar{m}\bar{m}}.$
(B.10)

(B.10) can be written down as

$\displaystyle
k_{mn\bar{p}\bar{q}}=e_{mn}(1-\delta_{pq})N_{pm}\bar{N}_{nq}+e_{pq}(1-\delta_{mn})N_{qm}\bar{N}_{np}+\delta_{pq}\delta_{mn}N_{pm}\bar{N}_{nq}$
$\displaystyle\hphantom{k_{mn\bar{p}\bar{q}}}{}=\frac{1}{2}(N_{qn}\bar{N}_{mp}+N_{qm}\bar{N}_{np}).$
(B.11)

###### Proof.

With equation (B.4) introduced in (B.1), it is obtained (B.8).

In order to directly (i.e., without Remark A.1) prove the invariance of
$\Delta_{{\mathcal{D}}_{n}}$ to the group action $g\times W=W_{1}$,
$g\in\operatorname{Sp}(n,{\use@mathgroup\M@U\symAMSb{R}})_{C}$,
$W,W_{1}\in{\mathcal{D}}_{n}$, we use the formulas (3.28), (3.29). We write
(3.28) as

$\displaystyle\operatorname{d}w_{pq}=A_{pm}(\operatorname{d}w_{1})_{mn}A_{qn},\qquad\text{where}\quad
A=Wq^{*}+p^{*}.$

With formula (2.29), we have successively

$\displaystyle\frac{\partial
w_{pq}}{\partial(w_{1})_{\alpha\beta}}=A_{pm}\frac{\partial(w_{1})_{mn}}{\partial(w_{1})_{\alpha\beta}}A_{nq}=A_{pm}A_{nq}\Delta^{mn}_{\alpha\beta}$
$\displaystyle\hphantom{\frac{\partial
w_{pq}}{\partial(w_{1})_{\alpha\beta}}}{}=A_{p\alpha}A_{q\beta}+A_{p\beta}A_{q\alpha}-\delta_{\alpha\beta}A_{p\alpha}A_{q\beta}.$
(B.12)

With (B.12), we calculate $\frac{\partial f}{\partial(w_{1})_{\alpha\beta}}$.

a) Firstly, we consider the case $\alpha\not=\beta$. We have

$\displaystyle\frac{\partial
f}{\partial(w_{1})_{\alpha\beta}}=\left(\sum_{p<q}A_{p\alpha}A_{q\beta}+\sum_{q<p}A_{q\alpha}A_{p\beta}+2\sum_{p=q}A_{p\alpha}A_{q\beta}\right)\frac{\partial
f}{\partial w_{pq}}$ $\displaystyle\hphantom{\frac{\partial
f}{\partial(w_{1})_{\alpha\beta}}}{}=2\sum
A_{p\alpha}A_{q\beta}e_{pq}\frac{\partial f}{\partial w_{pq}}.$ (B.13)

b) Now we consider the case $\alpha=\beta$. We have

$\displaystyle\frac{\partial
f}{\partial(w_{1})_{\alpha\beta}}=2f_{\alpha\beta}\sum_{p,q}A_{p\alpha}A_{q\beta}e_{pq}.$
(B.14)

(B.13) and (B.14) are written together as

$\displaystyle D_{W_{1}}=A^{t}D_{W}A.$ (B.15)

Also we write (3.29) as

$\displaystyle M_{1}=\bar{A}^{t}MA.$ (B.16)

With (B.15) and (B.16) introduced in formula (B.8), we prove

$\displaystyle\Delta_{{\mathcal{D}}_{n}}(W_{1})=\Delta_{{\mathcal{D}}_{n}}(W),\qquad\text{where}\quad
W_{1}=g\times W.$ ∎

######  Remark B.3.

In the notation of [39]

$\displaystyle Z=\left(\begin{matrix}\sqrt{2}z_{11}&z_{12}&\dots&z_{1n}\\\
z_{12}&\sqrt{2}z_{22}&\dots&z_{2n}\\\ \dots&\dots&\dots&\dots\\\
z_{1n}&z_{2n}&\dots&\sqrt{2}z_{nn}\end{matrix}\right),\qquad\partial_{Z}=\left(\begin{matrix}\sqrt{2}\frac{\partial}{\partial
z_{11}}&\frac{\partial}{\partial z_{12}}&\dots&\frac{\partial}{\partial
z_{1n}}\vspace{1mm}\\\ \frac{\partial}{\partial
z_{12}}&\sqrt{2}\frac{\partial}{\partial
z_{22}}&\dots&\frac{\partial}{\partial z_{2n}}\\\ \dots&\dots&\dots&\dots\\\
\frac{\partial}{\partial z_{1n}}&\frac{\partial}{\partial
z_{2n}}&\dots&\sqrt{2}\frac{\partial}{\partial z_{nn}}\end{matrix}\right),$

the Laplace–Beltrami operator reads (see [39, equation (2.6.4)])

$\displaystyle\Delta_{{\mathcal{D}}_{n}}(W)=\operatorname{Tr}(({\mathbbm{1}}_{n}-\bar{W}W)\partial_{W}({\mathbbm{1}}_{n}-W\bar{W})\bar{\partial}_{W}),$

i.e.,

$\displaystyle\Delta_{{\mathcal{D}}_{n}}(W)=\sum_{\alpha\beta\lambda\mu=1}^{n}(\delta_{\lambda\mu}-\sum_{\sigma=1}^{n}P_{\lambda\sigma}P_{\mu\sigma}w_{\lambda\sigma}\bar{w}_{\mu\sigma})$
$\displaystyle\hphantom{\Delta_{{\mathcal{D}}_{n}}(W)=}{}\times(\delta_{\alpha\beta}-\sum_{\gamma=1}^{n}P_{\alpha\gamma}P_{\beta\gamma}w_{\alpha\gamma}\bar{w}_{\beta\gamma})P_{\lambda\alpha}P_{\mu\beta}\frac{\partial^{2}}{\partial
w_{\lambda\alpha}\partial\bar{w}_{\mu\beta}},$

where

$\displaystyle P_{\alpha\beta}=1+(\sqrt{2}-1)\delta_{\alpha\beta}.$

“$P$” is related with the symbol “$e$” in (3.39) by the relation

$\displaystyle
P_{\alpha\beta}=\sqrt{2}(\sqrt{2}-1)(1+\sqrt{2}e_{\alpha\beta}).$

### Acknowledgements

I am grateful to Daniel Beltita for clarifying some aspects of contractibility
related with this paper. This research was conducted in the framework of the
ANCS project program PN 16 42 01 01/2016 and UEFISCDI-Romania program PN-II-
PCE-55/05.10.2011.

## References

  * [1] Arezzo C., Loi A., Moment maps, scalar curvature and quantization of Kähler manifolds, Comm. Math. Phys. 246 (2004), 543–559.
  * [2] Berceanu S., Coherent states and geodesics: cut locus and conjugate locus, J. Geom. Phys. 21 (1997), 149–168, dg-ga/9502007.
  * [3] Berceanu S., On the geometry of complex Grassmann manifold, its noncompact dual and coherent states, Bull. Belg. Math. Soc. Simon Stevin 4 (1997), 205–243, dg-ga/9509002.
  * [4] Berceanu S., Realization of coherent state Lie algebras by differential operators, in Advances in Operator Algebras and Mathematical Physics, Theta Ser. Adv. Math., Vol. 5, Editors F. Boca, O. Bratteli, R. Longo, H. Siedentop, Theta, Bucharest, 2005, 1–24, math.DG/0504053.
  * [5] Berceanu S., A holomorphic representation of the Jacobi algebra, Rev. Math. Phys. 18 (2006), 163–199, Errata, Rev. Math. Phys. 24 (2012), 1292001, 2 pages, math.DG/0408219.
  * [6] Berceanu S., A holomorphic representation of the semidirect sum of symplectic and Heisenberg Lie algebras, J. Geom. Symmetry Phys. 5 (2006), 5–13.
  * [7] Berceanu S., A holomorphic representation of the multidimensional Jacobi algebra, in Perspectives in Operator Algebras and Mathematical Physics, Theta Ser. Adv. Math., Vol. 8, Editors F.-P. Boca, R. Purice, S. Stratila, Theta, Bucharest, 2008, 1–25, math.DG/0604381.
  * [8] Berceanu S., The Jacobi group and the squeezed states – some comments, AIP Conf. Proc. 1191 (2009), 21–29, arXiv:0910.5563.
  * [9] Berceanu S., A convenient coordinatization of Siegel-Jacobi domains, Rev. Math. Phys. 24 (2012), 1250024, 38 pages, arXiv:1204.5610.
  * [10] Berceanu S., Coherent states and geometry on the Siegel–Jacobi disk, Int. J. Geom. Methods Mod. Phys. 11 (2014), 1450035, 25 pages, arXiv:1307.4219.
  * [11] Berceanu S., Quantum mechanics and geometry on the Siegel–Jacobi disk, in Geometric Methods in Physics, Editors P. Kielanowski, P. Bieliavsky, A. Odesski, A. Odzijewicz, M. Schlichenmaier, T. Voronov, Trends in Mathematics, Birkhäuser, Basel, 2014, 89–98.
  * [12] Berceanu S., Bergman representative coordinates on the Siegel–Jacobi disk, Romanian J. Phys. 60 (2015), 867–896, arXiv:1409.0368.
  * [13] Berceanu S., Gheorghe A., Differential operators on orbits of coherent states, Romanian J. Phys. 48 (2003), 545–556, math.DG/0211054.
  * [14] Berceanu S., Gheorghe A., Applications of the Jacobi group to quantum mechanics, Romanian J. Phys. 53 (2008), 1013–1021, arXiv:0812.0448.
  * [15] Berceanu S., Gheorghe A., On the geometry of Siegel–Jacobi domains, Int. J. Geom. Methods Mod. Phys. 8 (2011), 1783–1798, arXiv:1011.3317.
  * [16] Berceanu S., Schlichenmaier M., Coherent state embeddings, polar divisors and Cauchy formulas, J. Geom. Phys. 34 (2000), 336–358, math.DG/9903105.
  * [17] Berezin F.A., Quantization in complex bounded domains, Dokl. Akad. Nauk SSSR 211 (1973), 1263–1266.
  * [18] Berezin F.A., Quantization, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1116–1175.
  * [19] Berezin F.A., General concept of quantization, Comm. Math. Phys. 40 (1975), 153–174.
  * [20] Berezin F.A., Quantization in complex symmetric spaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 363–402.
  * [21] Berndt R., Sur l’arithmétique du corps des fonctions elliptiques de niveau $N$, in Seminar on Number Theory, Paris 1982–83 (Paris, 1982/1983), Progr. Math., Vol. 51, Birkhäuser Boston, Boston, MA, 1984, 21–32.
  * [22] Berndt R., Böcherer S., Jacobi forms and discrete series representations of the Jacobi group, Math. Z. 204 (1990), 13–44.
  * [23] Berndt R., Schmidt R., Elements of the representation theory of the Jacobi group, Progr. Math., Vol. 163, Birkhäuser Verlag, Basel, 1998.
  * [24] Borel A., Kählerian coset spaces of semisimple Lie groups, Proc. Nat. Acad. Sci. USA 40 (1954), 1147–1151.
  * [25] Cahen B., Global parametrization of scalar holomorphic coadjoint orbits of a quasi-Hermitian Lie group, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 52 (2013), 35–48.
  * [26] Cahen B., Stratonovich–Weyl correspondence for the Jacobi group, Commun. Math. 22 (2014), 31–48.
  * [27] Cahen M., Gutt S., Rawnsley J., Quantization of Kähler manifolds. II, Trans. Amer. Math. Soc. 337 (1993), 73–98.
  * [28] Calabi E., Isometric imbedding of complex manifolds, Ann. of Math. 58 (1953), 1–23.
  * [29] Chern S.S., Complex manifolds without potential theory, 2nd ed., Universitext, Springer-Verlag, New York – Heidelberg, 1979.
  * [30] Chiribella G., Adesso G., Quantum benchmarks for pure single-mode Gaussian states, Phys. Rev. Lett. 112 (2014), 010501, 6 pages, arXiv:1308.2146.
  * [31] Donaldson S.K., Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (2001), 479–522.
  * [32] Dorfmeister J., Nakajima K., The fundamental conjecture for homogeneous Kähler manifolds, Acta Math. 161 (1988), 23–70.
  * [33] Eichler M., Zagier D., The theory of Jacobi forms, Progr. Math., Vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985.
  * [34] Engliš M., Berezin quantization and reproducing kernels on complex domains, Trans. Amer. Math. Soc. 348 (1996), 411–479.
  * [35] Gay-Balmaz F., Tronci C., Vlasov moment flows and geodesics on the Jacobi group, J. Math. Phys. 53 (2012), 123502, 36 pages, arXiv:1105.1734.
  * [36] Helgason S., Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, Vol. 80, Academic Press, Inc., New York – London, 1978.
  * [37] Hua L.-K., On the theory of automorphic functions of a matrix level. I. Geometrical basis, Amer. J. Math. 66 (1944), 470–488.
  * [38] Hua L.-K., Harmonic analysis of functions of several complex variables in the classical domains, Amer. Math. Soc., Providence, R.I., 1963.
  * [39] Hua L.-K., Look K.H., Theory of harmonic functions in classical domains, Scientia Sinica 8 (1959), 743–806.
  * [40] Hunziker M., Sepanski M.R., Stanke R.J., Global Lie symmetries of a system of Schrödinger equations and the oscillator representation, Miskolc Math. Notes 14 (2013), 647–657.
  * [41] Jost J., Riemannian geometry and geometric analysis, 5th ed., Universitext, Springer-Verlag, Berlin, 2008.
  * [42] Kähler E., Raum-Zeit-Individuum, Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 16 (1992), 115–177.
  * [43] Kähler E., Mathematische Werke/Mathematical works, Walter de Gruyter & Co., Berlin, 2003.
  * [44] Kobayashi S., Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1959), 267–290.
  * [45] Kobayashi S., Nomizu K., Foundations of differential geometry. Vol. I, Interscience Publishers, New York – London, 1963.
  * [46] Kobayashi S., Nomizu K., Foundations of differential geometry. Vol. II, Interscience Publishers, New York – London – Sydney, 1969.
  * [47] Kostant B., Quantization and unitary representations. I. Prequantization, in Lectures in Modern Analysis and Applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, 87–208.
  * [48] Kramer P., Saraceno M., Semicoherent states and the group ${\rm ISp}(2,{\bf R})$, Phys. A 114 (1982), 448–453.
  * [49] Lee M.H., Theta functions on Hermitian symmetric domains and Fock representations, J. Aust. Math. Soc. 74 (2003), 201–234.
  * [50] Lisiecki W., Coherent state representations. A survey, Rep. Math. Phys. 35 (1995), 327–358.
  * [51] Loi A., Regular quantizations of Kähler manifolds and constant scalar curvature metrics, J. Geom. Phys. 53 (2005), 354–364.
  * [52] Loi A., Mossa R., Berezin quantization of homogeneous bounded domains, Geom. Dedicata 161 (2012), 119–128, arXiv:1106.2510.
  * [53] Loi A., Mossa R., Some remarks on homogeneous Kähler manifolds, Geom. Dedicata 179 (2015), 377–383, arXiv:1502.00011.
  * [54] Lu Q.-K., On Kähler manifolds with constant curvature, Acta. Math. Sin. 66 (1966), 269–281.
  * [55] Lu Q.-K., Holomorphic invariant forms of a bounded domain, Sci. China Ser. A 51 (2008), 1945–1964.
  * [56] Lütkepohl H., Handbook of matrices, John Wiley & Sons, Ltd., Chichester, 1996\.
  * [57] Maass H., Die Differentialgleichungen in der Theorie der Siegelschen Modulfunktionen, Math. Ann. 126 (1953), 44–68.
  * [58] Mandel L., Wolf E., Optical coherence and quantum optics, Cambridge University Press, Cambridge, 1995.
  * [59] Marmo G., Michor P.W., Neretin Yu.A., The Lagrangian Radon transform and the Weil representation, J. Fourier Anal. Appl. 20 (2014), 321–361, arXiv:1212.4610.
  * [60] Molitor M., Gaussian distributions, Jacobi group, and Siegel–Jacobi space, J. Math. Phys. 55 (2014), 122102, 40 pages, arXiv:1409.7917.
  * [61] Moroianu A., Lectures on Kähler geometry, London Mathematical Society Student Texts, Vol. 69, Cambridge University Press, Cambridge, 2007.
  * [62] Neeb K.-H., Holomorphy and convexity in Lie theory, de Gruyter Expositions in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin, 2000.
  * [63] Nomizu K., Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33–65.
  * [64] Perelomov A., Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986.
  * [65] Quesne C., Vector coherent state theory of the semidirect sum Lie algebras ${\rm wsp}(2N,{\bf R})$, J. Phys. A: Math. Gen. 23 (1990), 847–862.
  * [66] Rawnsley J., Cahen M., Gutt S., Quantization of Kähler manifolds. I. Geometric interpretation of Berezin’s quantization, J. Geom. Phys. 7 (1990), 45–62.
  * [67] Rawnsley J.H., Coherent states and Kähler manifolds, Quart. J. Math. Oxford Ser. (2) 28 (1977), 403–415.
  * [68] Satake I., Algebraic structures of symmetric domains, Kanô Memorial Lectures, Vol. 4, Iwanami Shoten, Tokyo, Princeton University Press, Princeton, N.J., 1980.
  * [69] Sepanski M.R., Stanke R.J., Global Lie symmetries of the heat and Schrödinger equation, J. Lie Theory 20 (2010), 543–580.
  * [70] Shuman K.L., Complete signal processing bases and the Jacobi group, J. Math. Anal. Appl. 278 (2003), 203–213.
  * [71] Siegel C.L., Symplectic geometry, Amer. J. Math. 65 (1943), 1–86.
  * [72] Takase K., A note on automorphic forms, J. Reine Angew. Math. 409 (1990), 138–171.
  * [73] Takase K., On unitary representations of Jacobi groups, J. Reine Angew. Math. 430 (1992), 139–149.
  * [74] Takase K., On Siegel modular forms of half-integral weights and Jacobi forms, Trans. Amer. Math. Soc. 351 (1999), 735–780.
  * [75] Vinberg E.B., Gindikin S.G., Kähler manifolds admitting a transitive solvable group of automorphisms, Math. USSR Sb. 3 (1967), 333–351.
  * [76] Yang J.-H., The method of orbits for real Lie groups, Kyungpook Math. J. 42 (2002), 199–272, math.RT/0602056.
  * [77] Yang J.-H., Invariant metrics and Laplacians on Siegel–Jacobi space, J. Number Theory 127 (2007), 83–102, math.NT/0507215.
  * [78] Yang J.-H., A partial Cayley transform of Siegel–Jacobi disk, J. Korean Math. Soc. 45 (2008), 781–794, math.NT/0507216.
  * [79] Yang J.-H., Invariant metrics and Laplacians on Siegel–Jacobi disk, Chin. Ann. Math. Ser. B 31 (2010), 85–100, math.NT/0507217.
  * [80] Yang J.-H., Yong Y.-H., Huh S.-N., Shin J.-H., Min G.-H., Sectional survatures of the Siegel–Jacobi space, Bull. Korean Math. Soc. 50 (2013), 787–799.