On orbit connectedness, orbit convexity and envelopes of holomorphy

We are concerned with the univalence and discription of the envelope of holomorphy $E(D)$ for a domain $D$ having a compact Lie group action. Our main result is the following:
Let $X$ be a holomorphic Stein $K^C$-manifold, $D\subset X$ a $K$-invariant orbit connected domain. Then $E(D)$ is schlicht and orbit convex if and only if $E(K^C\cdot D)$ is schlicht. Moreover, in this case, $E(K^C\cdot D)=K^C\cdot e(d)$.