Hamiltonian Actions on Homogeneous Bounded Domains

Abstract: This paper examines Hamiltonian actions of non-compact Lie groups on homogeneous bounded domains $X$ in $\mathbb{C}^d$. In the main part, a Lie-theoretical condition for closed subgroups $H$ of the automorphism group of $X$ is described such that the symplectic reduction $\mu^{-1}(0)/H$ for the momentum map $\mu$ is a Stein manifold. Moreover, for another class of closed subgroups it is shown that the quotient $(H^\mathbb{C}\cdot X)/H^\mathbb{C}$ is a Stein manifold and the symplectic reduction $\mu^{-1}(0)/H$ is biholomorphic to this quotient.