Simply transitive groups and Kähler structures on homogeneous Siegel domains

Abstract:

We determine the Lie algebras of all simply transitive groups of automorphisms of a homogeneous Siegel domain $D$ as modifications of standard normal $j$-algebras. We show that the Lie algebra of all automorphisms of $D$ is a "complete isometry algebra in standard position". This implies that $D$ carries a riemannian metric $\tilde g$ with nonpositive sectional curvature satisfying Lie $\operatorname {Iso}(D,\tilde g) = \operatorname {Lie}\; \operatorname {Aut} \text {D}$. We determine all Kähler metrics $f$ on $D$ for which the group $\operatorname {Aut}(D,f)$ of holomorphic isometries acts transitively. We prove that in this case $\operatorname {Aut}(D,f)$ contains a simply transitive split solvable subgroup. The results of this paper are used to prove the fundamental conjecture for homogeneous Kähler manifolds admitting a solvable transitive group of automorphisms.