Density properties of complex Lie groups (Q581564)

The main result is the following complex Lie group version of Borel's density theorem. Theorem. Let G be a complex analytic group and H a subgroup of finite covolume in G. Let f be a holomorphic representation of G on a finite- dimensional complex vector space. Then every f(H)-invariant vector subspace is f(G)-invariant. The proof is analogous to Furstenberg's proof of Borel's density theorem.