Proper holomorphic mappings between invariant domains in $\mathbb {C}^n$

Abstract:

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 $\mathbb {C}^n$. Let $G_1$ and $G_2$ be two compact Lie groups, which act linearly on $\mathbb {C}^n$ with $\mathcal {O}(\mathbb {C}^n)^{G_j}=\mathbb {C}$ for $j=1,2$. Let $0\in \Omega _j$ be bounded $G_j$-invariant domains in $\mathbb {C}^n$ for $j=1,2$. If $f:\Omega _1\rightarrow \Omega _2$ is a proper holomorphic mapping, then $f$ extends holomorphically to an open neighborhood of $\overline {\Omega }_1$, and in addition if $f^{-1}(0)=\{0\}$, then $f$ is a polynomial mapping. We also prove that if $0\in \Omega$ is a $G_1$-invariant pseudoconvex domain in $\mathbb {C}^n$ with $\mathcal {O}(\mathbb {C}^n)^{G_1}=\mathbb {C}$, then $\Omega$ is orbit convex. The second result is used to prove the first one.