Cartan's Lemma on a disjoint union of $g$-translates

Question

Let $X$ be a complex manifold equipped with an action of a finite group $G$ by automorphisms of $X$. Assume that $x$ is a point on $X$ with stabilizer ${\rm Stab}(x)=:H$. Let $U$ be an $H$-invariant neighbourhood of $x$ such that $U\cap gU=\emptyset$ for all $g\in G/H$. What is the correct way of applying Cartan's Lemma on the disjoint union $\cup_{g\in G/H}gU$ of translates of $U$ simultaneously? Is $\cup_{g\in G/H}gU$ biholomorphic to a disjoint union $\cup_{g\in G/H}gV=G\times_HV\subset G\times_HT_xX$, where $V\subset T_xX$ is an open $H$-invariant subset? More concretely, what I mean is, $\cup_{g\in G/H}gU$ is not biholomorphic to a set in the induced representation ${\rm ind}_H^GT_xX:=\mathbb CG\otimes_{\mathbb CH}T_xX$ of $T_xX$, but in $G\times_HT_xX$?

Answer 1

Suppose that $U\subset X$ is an $H$-invariant neighborhood of $x$ in $X$ such that:

1. $gU\cap U\ne \emptyset \iff g\in H$.

2. There exists a neighborhood $V\subset T_xX$ of $0$ and a biholomorphic $H$-equivariant map $h: U\to V$.

I will equip $GV\subset TX$ with the induced complex structure. Then:

Lemma. There exists an equivariant biholomorphic map $f: GU\to GV$.

Proof. Given $z\in gU$, define $f(z)= dg \circ h \circ g^{-1}(z)$. Equivariance of $h$ shows that this formula defines a map $f: GU\to GV$. This map is a composition of three holomorphic maps, hence, is holomorphic. The inverse to this map is given by $g\circ h \circ dg^{-1}$, which is well-defined and holomorphic for the same reason as above. qed