  Let G be complex linear-algebraic group, H a subgroup, which is dense in G in
the Zariski-topology. Assume that G/[G,G] is reductive and furthermore that (1)
G is solvable, or (2) the semisimple elements in G'=[G,G] are dense. Then every
H-invariant holomorphic function on G is constant. If G=G', furthermore every
H-invariant meromorphic or plurisubharmonic function is constant. Finally an
example of Margulis is used to show the existence of an algebraic group G with
G=G' such that there exists a Zariski-dense discrete subgroup without any
semisimple element.
