Equivariant embeddings of Stein domains sitting inside of complex semigroups

In this paper we prove an equivariant version of Hormanders embedding theor em for Stein manifolds. More concretely, let G be a connected Lie group sit ting in its complexification G(C) and D subset of or equal to G(C) a G x G- invariant Stein domain. Under slight obstructions on D we construct a Hilbe rt space H equipped with a unitary G x G-action and a holomorphic equivaria nt closed embedding e: D --> H*\{0}.