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 D as modifications of standard normal j j -algebras. We show that the Lie algebra of all automorphisms of D D is a "complete isometry algebra in standard position". This implies that D D carries a riemannian metric g ~ \tilde g with nonpositive sectional curvature satisfying Lie Iso ⁡ ( D , g ~ ) = Lie Aut D \operatorname {Iso}(D,\tilde g) = \operatorname {Lie}\; \operatorname {Aut}\, \text {D} . We determine all Kähler metrics f f on D D for which the group Aut ⁡ ( D , f ) \operatorname {Aut}(D,f) of holomorphic isometries acts transitively. We prove that in this case Aut ⁡ ( D , f ) \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.