PROPER HOLOMORPHIC MAPPINGS BETWEEN INVARIANT DOMAINS IN Cn

In the present paper, we prove the following result generalizing some well-known related results about biholomorphic or proper holomorphic mappings between some special domains in C-n. Let G(1) and G(2) be two compact Lie groups, which act linearly on C-n with O(C-n)(Gj) = C for j = 1, 2. Let 0. Oj be bounded Gj -invariant domains in C-n for j = 1, 2. If f : Omega(1) -> Omega(2) is a proper holomorphic mapping, then f extends holomorphically to an open neighborhood of (Omega) over bar (1), and in addition if f(-1)(0) = {0}, then f is a polynomial mapping. We also prove that if 0 is an element of Omega is a G(1)-invariant pseudoconvex domain in C-n with O(C-n)(G1) = C, then Omega is orbit convex. The second result is used to prove the first one.