Homogeneous K\"ahler and Hamiltonian manifolds

We consider actions of reductive complex Lie groups $G=K^C$ on K\"ahler manifolds $X$ such that the $K$--action is Hamiltonian and prove then that the closures of the $G$--orbits are complex-analytic in $X$. This is used to characterize reductive homogeneous K\"ahler manifolds in terms of their isotropy subgroups. Moreover we show that such manifolds admit $K$--moment maps if and only if their isotropy groups are algebraic.