Poisson kernel and Cauchy formula of a non-symmetric transitive domain

Abstract

In 1965, Lu Yu-Qian discovered that the Poisson kernel of the homogenous domain

$$ \mathfrak{S}_{m,p,q} \left\{ {Z \in \mathbb{C}^{m \times m} ,Z_1 \mathbb{C}^{m \times p} ,Z_2 \in \mathbb{C}^{q \times m} \left| {\frac{1} {{2i}}(Z - Z^\dag ) - Z_1 \overline {Z_1 } ^\prime - \overline {Z_2 } ^\prime - Z_2 > 0} \right.} \right\} $$

does not satisfy the Laplace-Beltrami equation associated with the Bergman metric when $\mathfrak{S}_{m,p,q}$ is not symmetric. However the map

$$ T_0 :Z \mapsto Z,Z_1 \mapsto Z_1 ,Z_2 \mapsto \overline {Z_2 } $$

transforms $\mathfrak{S}_{m,p,q}$ into a domain $\mathfrak{S}$ _I (m, m+p+q) which can be mapped by the Cayley transformation into the classical domains $\Re$ _I (m,m + p + q). The pull back of the Bergman metric of $\Re$ _I (m,m + p + q) to $\mathfrak{S}$ _m,p,q is a Riemann metric ds ² which is not a Kähler metric and even not a Hermitian metric in general. It is proved that the Laplace-Beltrami operator Δ associated with the metric ds ² when it acts on the Poisson kernel of $\mathfrak{S}$ _m,p,q equals 0. Consequently, the Cauchy formula of $\mathfrak{S}$ _m,p,q can be obtained from the Poisson formula.