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-Szeg\"{o} 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 $$ds^{2}=\sum\limits_{i,j=1}^{n}g_{ij}dz_{i}d\bar{z}_{j}, T(z,\bar{z})=(g_{ij})$$ and $$L(u)=T^{-1}(z,\bar{z}) [\frac{\partial^2u}{\partial z_i\partial\bar{z}_j}],$$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 $C^n$, $P=P(z,*)$ the Poisson kernel of $D$, then for any fixed $J(1\leq j \leq n)$ one has $L_j(P^{1/j})=0$ iff $D$ is a symmetric domain.