Two Problems on Cartan Domains

Abstract

Firstly, we consider the unitary geometry of two exceptional Cartan domains ${\Re }_V(16)$ and ${\Re }_{VI}(27)$. We obtain the explicit formulas of Bergman kernal funtion, Cauchy-Szegö kernel, Poinsson kernel and Bergman metric for ${\Re }_V(16)$ and ${\Re }_{VI}(27)$. Secondly, we give a class of invariant differential operators for Cartan domain $\Re$ of dimension n: If the Bergman metric of $\Re$ is $$d{s}^2=\sum _{i,j=1}^n{g}_{ij}d{z}_{i}d{\bar{z}}_j,T(z,\bar{z})=(g_{ij})$$ and $$L(u)=T^{-1}(z,\bar{z})[\frac{{\partial }^{2}u}{\partial z_i\partial {\bar{z}}_j}],$$then $$L_j(u)=\{\mbox {The sum of all prinipal minors of degree} j {for} L(u)\}$$ is invariant under the biholomorphic mapping of $\Re$. Let $D$ be the irreducible bounded homogeneous domain in $C^n$, $P=P(z,\ast )$ the Poisson kernel of $D$, then for any fixed $J(1\le j\le n)$ one has $L_j(P^{1/j})=0$ iff $D$ is a symmetric domain.

11 pages