On Kaehlerian Homogeneous Spaces of Unimodular Lie Groups

Abstract

Recently the results of E. Cartan on Hermitian symmetric space have been extended to the case of Kaehlerian homogeneous spaces by several authors. In particular the structure of compact Kaehlerian homogeneous spaces has been fully exploited. As for the non-conmpact case, there are few known results except in the case where the groups are semi-simple or reductive. A. Borel [1] and J. L. Koszul [6] have studied the structure of Kaehlerian homogeneous spaces of semi-simple Lie groups and have shown, independently of each other, an interesting result that a bounded domain in a unitary space admitting a transitive semi-simple Lie group of complex analytic homeomorphisms is symmetric. This result gives a partial answer to the problem of E. Cartan. In his imiportant paper in 1935 [2], E. Cartan raised the question whether a bounded homogeneous domain is always a symmetric bounded domain. Y. Matsushima [8] has st-udied the structure of Kaehlerian homogeneous spaces of reductive Lie groups and has shown that these spaces are the direct produict of a locally flat Kaehlerian space and a Kaehlerian homogeneous space of a semi-simple Lie group. The purpose of the present paper is to study the structure of EKaehlerian homogeneous spaces of a more general class of Lie groups, that is, those of unimodular Lie groups. Specifically, we shall deal with the following two cases. First we shall consider the case where the isotropy group is semisimple and obtain the following two theorems: