A Remark on Poisson Kernels

Abstract

It is well known that for any symmetric domain D in ℂn Prof. Hua Lao-Keng (L.K. Hua) has computed the Cauchy kernel H(Z,U) [l]. He and Prof. Lu Qi-Keng [2] defined the Poisson kernel P(Z,U) in terms of H(Z,U) by $$ P(Z,U) = \frac{{\left| {\left. {H(Z,U)} \right|^2 } \right.}}{{H(Z,\overline Z )}}, $$ (1) where U belongs to te Silov boundary of D and proved that the Poisson kernel P(Z,U) was annihilated by the Laplace-Beltrami operator of D with respect to the Bergman metric. For some nonsymmetric homogeneous Siegel domains, Prof. Lu Ru-Qian [3] in 1964 has shown that the Poisson kernels were not annihilated by the Laplace-Beltrami operator and pointed out that perhaps this fact was the characteristic property of nonsymmetric homogeneous domains. In this note I want to discuss the following question: If D is a nonsymmetric homogeneous Siegel domain and Aut(D) denotes the group of analytic automorphisms of D, does there exist another differential operator Δ invariant under Aut(D) such that the Poisson kernel can be annihilated by Δ?