Hamiltonian Actions on Homogeneous Bounded Domains

Abstract

We investigate Hamiltonian actions of non-compact Lie groups on a homogeneous bounded domain $X$. As a main result, we point out a Lie-theoretical condition for a closed Lie group $H$ of the automorphism group of $X$ which ensures that the symplectic reduction $μ^{-1}(0)/H$ with respect to the momentum map $μ$ at hand, is a Stein manifold. Moreover, for the class of connected subgroups of translations the quotient $(H^\mathbb{C}\cdot X)/H^\mathbb{C}$ is realized as a Siegel domain and we show that the symplectic reduction $μ^{-1}(0)/H$ is biholomorphic to such a Stein quotient.