# Intuition for Luna's Étale Slice Theorem

I am trying to understand the intuition for Luna's Étale Slice Theorem in the affine setting over $$\mathbb{C}$$.

Here is the setup. Let $$X$$ be an affine algebraic variety and $$G$$ a reductive group acting on $$X$$. Moreover, let $$x\in X$$ be a closed point and $$G_{x}$$ its stabilizer under the $$G$$ action where $$G_{x}$$ is also a reductive group, and $$V\subseteq X$$ a $$G_{x}$$ invariant locally closed affine variety containing $$x$$.

The group action $$G\times V\to X$$ by $$(g,v)\mapsto g\cdot v$$ is $$G_{x}$$ invariant if the action of $$G_{x}$$ on $$G\times V$$ is by $$h\cdot (g,v)\mapsto (gh^{-1},hv)$$. Indeed, one can check that $$gh^{-1}\cdot h\cdot v=gh^{-1}h\cdot v=g\cdot v\in X$$ but if $$g\in G_{x}$$ then $$g\cdot v\in V$$ since $$V$$ is $$G_{x}$$ invariant.

What is the geometric intuition for the quotient $$G\times_{G_{x}}V= (G\times V)// G_{x}$$?

Formally, it seems we are identifying points $$(g,v)\sim (g',v')$$ if there exists some $$h\in G_{x}$$ such that $$gh^{-1}\cdot hv$$ and $$g'\cdot v'$$ map to the same point in $$X$$. Namely $$v,v'\in V$$ lie in the same $$G$$ orbit. So $$G\times_{G_{x}} V$$ is just the set of $$G$$-orbits in $$V$$?

This induces a morphism $$\psi: G\times_{G_{x}} V\to X$$. Say $$V$$ is an étale slice if $$\psi$$ is an étale morphism. Luna's theorem shows the existence of such étale slices.

What is the structure of $$G\times_{G_{x}} V$$ as a variety?

What is the correct way to think about the morphism $$\psi$$?

Moduli spaces (say of semi-stable vector bundles on a smooth projective curve) may be constructed by taking the geometric quotient of a Quot-scheme by a reductive group $$G$$. The stabilizer of a point $$x$$ is usually included in $$\mathrm{Hom}(E_x, E_x)$$, where $$E_x$$ the vector bundle corresponding to $$x$$. In case $$E_x$$ is semi-stable, there are many techniques available to compute $$\mathrm{Hom}(E_x, E_x)$$. If you are in a favorable situation, the Quot-scheme is smooth at $$x$$ and Luna Étale's Slice tells you that the singularity of the moduli space at $$x$$ is locally isomorphic to $$\mathbb{C}^n/G_x$$.