Special algebraic properties of Käehler algebras

Abstract

Homogeneous Kahler manifolds M are frequently investigated via Kahler algebras (g, k, j, p), where g denotes a Lie algebra of infinitesimal automorphisms of M and k the isotropy subalgebra of some point in M. Moreover, / corresponds to the complex structure tensor and p to the Kahler form. In particular, Kahler algebras have been used intensively in the proof of the geometric Fundamental Conjecture for homogeneous Kahler manifolds: Every homogeneous Kahler manifold is a holomorphic fiberbundle over a homogeneous bounded domain in which the fiberis (with the induced Kahler metric) the product of a flat homogeneous Kahler manifold and a compact simply connected homogeneous Kahler manifold. Two additional properties of Kahler algebras have proven to be particularly useful. One is that g or ad g is an algebraic Lie algebra. The second one is the assumption that p is the differentialof a leftinvariant 1-form, p=d<o. This is the case of /-algebras. It has been investigated intensively by Gindikin, Piatetskii-Shapiro,Vinberg and others. The proof of the Fundamental Conjecture for homogenous Kahler manifolds is much shorter for /-algebras than for general Kahler algebras. This is due to some extent to the fact that one can embed a /-algebra into an algebraic /-algebra. The purpose of this note is threefold. First we want to prove that for the Lie algebra gM of all infinitesimal automorphisms of an arbitrary homogeneous Kahler manifold M, the Lie algebra z.dgM is algebraic. Secondly, we decompose gM into the orthogonal sum of /-invariant subalgebras. This decomposition will be of importance for a forthcoming publicationin which we give a detailed description of kM and the Kahler form p. The orthogonal decomposition in question has a simple geometric interpretation. It is essentially induced by a representation of the base domain (occuring in the Fundamental Conjecture) as a Siegel domain of type three.