Analytic germ of a GIT quotient at a fixed point Let $X$ be a smooth complex affine variety with an action of a complex reductive group $G$. Suppose that $x$ is a fixed point. Denote by $\varphi$ the GIT quotient $\varphi: X\to X//G$.
Question. How to prove that the analytic germ of $X//G$ at $\varphi(x)$ is isomorphic to the analytic germ at $0$ of the GIT quotient of the linear action of $G$ on $T_X(x)$? Is there some reference for this statement?
 A: The result that you need is Luna's slice theorem. You find a discussion in this thread with references to these lecture notes by Drezet.
To see that this is what you want, note that we have a reductive group action on a smooth, affine variety, so we are in the situation of Theorem 5.4 from the script. It says that there exists a locally closed subvariety $V$ of $X$ containing $x$ which is $G_x=G$-invariant (here we use $x$ is a fixed point). Moreover we have
$$G \times_G V \cong V \xrightarrow{\psi} X$$
is strongly étale. In this case, it is of course only an inclusion of $V$ as an open set in $X$. The important thing is that we also have an étale G-equivariant morphism $\phi: V \to T_x X$ , sending $x$ to $0$ with $T \phi = \mathrm{id}_{T_x X}$. Its image $W \subset T_x X$ is open and saturated and satisfies that $\phi: V \to W$ is a strongly étale $G$-morphism. Looking up the definition, this means that the induced map $V // G \to W //G$ is étale. But $V$ was an open set of $X$ and $W$ an open set of $T_x X$, so $\phi$ induces the isomorphism of analytic germs of the quotient spaces $X//G$ and $T_x X//G$.
