The computation of the Bergman kernel (Q2751195)

Let \(R_{II}(p)\) be the second type of classical domain, i.e. NEWLINE\[NEWLINER_{II}(p)=\{Z : Z \;\text{is a} \;p\times p \;\text{symmetric complex matrix,} (I_p-Z\overline Z)>0\}.NEWLINE\]NEWLINE Then the Cartan-Egg domain of second type is defined as NEWLINE\[NEWLINED_{II}(n,p)=\{(w,Z)\in\mathbb{C}^n\times R_{II} : |w_1|^{2p_1}+\cdots +|w_n|^{2p_n}<\det (I_p-Z\overline Z)\},NEWLINE\]NEWLINE where \(\frac 1 {p_1},\frac 1 {p_2}, \cdots ,\frac 1 {p_{n-1}}\) are positive integers and \(p_n >0\). NEWLINENEWLINENEWLINEThe main result of this paper is to give an explicit formula for the Bergman kernel function \(K(w,Z)\) of \(D_{II}(n,p)\). Since \(D_{II}(n,p)\) is Reinhardt in \(w\) and circular in \(Z\), there is a complete orthogonal system of \(L^2\)-holomorphic functions on \(D_{II}(n,p)\), \(\{\phi_{(\alpha,m,i)}(w,Z)=w^\alpha p^m_{i,\alpha}(Z)\}\) where \(p^m_{i,\alpha}(Z)\) is a polynomial of degree \(m\). NEWLINENEWLINENEWLINEIn general it is hard to get the orthonormal system and hence a closed form of \(K(w,Z)\). The authors are able to normalize \(\{\phi^0_{1,\alpha}(w,Z)\}\), so using the properties of the \(\Gamma\) function, they obtain a closed form of the kernel function at the points \((w,0)\). For general points \((w,Z)\), using the relation on the Bergman kernel function NEWLINE\[NEWLINEK(w,Z)=K(F(w,Z))|J_F(w,Z)|^2NEWLINE\]NEWLINE for every automorphism \(F\) of \(D_{II}(n,p)\), and the automorphism NEWLINE\[NEWLINE\begin{aligned} w^*_j &=w_j\det (I_n-Z_0\overline Z_0)^{\frac 1 {2p_j}}\det (I_p-Z\overline Z_0)^{\frac 1 {p_j}}, \quad j=1,\dots ,n,\\ Z^* &=A(Z-Z_0)(I_p-\overline Z_0Z)^{-1}\overline A^{-1}.\end{aligned} NEWLINE\]NEWLINE which maps \((w,Z_0)\) to \((w^*,0)\), with NEWLINE\[NEWLINE|J_F(w,Z_0)|^2=\det(I_p-Z_0\overline Z_0)^{\sum_1^n -\frac 1 {p_j}-p-1},NEWLINE\]NEWLINE the authors derive a closed form of \(K(w,Z)\).NEWLINENEWLINEFor the entire collection see [Zbl 0966.00026].