TITLE: Question on transversal slice of Lie group
QUESTION [6 upvotes]: Assume we have action of Lie group $G$ on a manifold $X$. Fix some orbit $\mathcal{O}$, it is known there exist transversal slice $S$ with respect to this orbit. Fix some point $x$ in $\mathcal{O}$, and let $G_x$ be the stabilizer of $x$.
My question is, can we find a transversal slice which is $G_x$-stable?
How about in the algebraic situation?

REPLY [2 votes]: Assume the base field is of characteristic zero. If $G$ is an affine algebraic group with reductive connected component which acts by morphisms on an affine variety $X$, then Luna has shown that there exists a slice étale at $x$ for each closed orbit $G\cdot x$ in $X$. This means there exists a $G_x$-invariant locally closed affine subvariety $S$ of $X$ containing $x$ such that the morphism $\psi: G*_{G_x}S\to X$, $[g,s]\mapsto gs$ is excellent. In particular, the image of $\psi$ is a saturated open subset $V$ of $X$ and 
$\psi: G*_{G_x}S\to V$ is étale. Here $G*_{G_x}S$ denotes the homogeneous fiber bundle 
$(G\times S)//G_x$. A good, easily available reference are the notes by 
Drézet.