Explicit computations of Bergman kernel on Cartan-Egg domains of second and third types (Q2751187)

Let \(R_{II}(p)\) be the second type of classical domain, i.e. NEWLINE\[NEWLINER_{II}(p)=\{Z : Z \;\text{is a} \;p\times p \;\text{symmetric complex matrix,} \;(I_p-ZZ^T)>0\},NEWLINE\]NEWLINE where \(A^T\) is the conjugate transpose of \(A\). Let \(R_{III}(p)\) be the third type of classical domain, i.e. NEWLINE\[NEWLINER_{III}(p)=\{Z : Z \;\text{is a} \;p\times p \;\text{skew symmetric complex matrix,} \;(I_p-ZZ^T)>0\}.NEWLINE\]NEWLINE Then the Cartan-Egg domains of second and third type are defined as NEWLINE\[NEWLINECE_{A}(n,m,p)=\{(W_1,W_2,Z)\in \mathbb{C}^n\times \mathbb{C}^m\times R_{A} : |W_1|^{2}+|W_2|^{2K}<\det (I_p-ZZ^T)\},NEWLINE\]NEWLINE where \(K>0\) and \(A=II\;\text{or} \;III\), respectively. NEWLINENEWLINENEWLINEThe main result of this paper is to give an explicit formula for the Bergman kernel function \(K(W_1,W_2,Z)\) of \(CE_{A}(n,m,p)\). Since \(CE_{A}(n,m,p)\) is Reinhardt in \((W_1,W_2)\) and circular in \(Z\), there is a complete orthogonal system of \(L^2\)-holomorphic functions on \(CE_{A}(n,m,p)\), \(\{\varphi_{(\alpha,l,i)}(W_1,W_2,Z)=W^\alpha p^l_{i,\alpha}(Z)\}\) where \(p^l_{i,\alpha}(Z)\) is a polynomial of degree \(l\) and the kernel function \(K(W_1,W_2,Z)\) is of the form \(L(|W_1|^2,|W_2|^2,Z)\). In general it is hard to get the orthonormal system and hence a closed form of \(K(W_1,W_2,Z)\). In the first step, the authors work on the case \(n=1=m\) and normalize \(\{\varphi^0_{1,\alpha}(W_1,W_2,Z)\}\). So using the properties of the \(\Gamma\) function, the authors obtain a closed form of the kernel function in this case at the points \((W_1,W_2,0)\). For general points \((W_1,W_2,_Z)\), using the relation on the Bergman kernel function NEWLINE\[NEWLINEK(W_1,W_2,Z)=K(F(W_1,W_2,Z))|J_F(W_1,W_2,Z)|^2NEWLINE\]NEWLINE for every automorphism \(F\) of \(CE_{A}(n,m,p)\), and the automorphism NEWLINE\[NEWLINE\begin{aligned} W_1^* &=W_1\det (I_p-Z_0 Z_0^T)^{\frac 1 2}\det (I_p-Z Z_0^T)^{-1},\\ W_2^* &=W_2\det (I_p-Z_0 Z_0^T)^{\frac 1 {2K}}\det (I_p-Z Z_0^T)^{-\frac 1 K},\\ Z^* &=A(Z-Z_0)(I_p-Z_0^T Z)^{-1}D^{-1},\end{aligned} NEWLINE\]NEWLINE which maps \((W_1,W_2,Z_0)\) to \((W_1^*,W_2^*,0)\), and with NEWLINE\[NEWLINE|J_F(W_1,W_2,Z_0)|^2=\det(I_p-Z_0Z_0^T)^{-p-2-\frac 1 K},NEWLINE\]NEWLINE the authors derive a closed form of \(K(W_1,W_2,Z)\) in this case. The second step is to apply the ``Principle of Inflation'' to obtain a closed form of \(K(W_1,W_2,Z)\) in general case.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00026].