Assume that $G$ is an affine reductive algebraic group (I am mostly interested in the case $GL\_n$) over an algebraically closed field $K$ of characteristic zero. Assume also that $G$ acts on an affine variety $X$ with finite stabilizers. I would like to ask how does the isomorphism type $Stab\_G(x)$ varies with respect to $X$. For example, for every finite group $H$ we can define $$X\_H=\{x\in X| Stab\_G(x)\cong H\}.$$
What can we say about the sets $X\_H$? are they open? closed? constructible?
Is there something that can be said about $X\_H$ if for example $|H|$ is maximal or minimal among all stabilizers?


If we were talking about the dimension of $Stab\_G(x)$ instead of the cardinality, then there are some semi-continuity results: the set of points $x$ for which $dim(Stab\_G(x))$ is $\geq n$ is closed for example (where $n$ is some natural number). 
I wonder what can be said if we consider actions in which all the stabilizers are finite. 



variations of finite stabilizer in the action of an algebraic group on an affine variety
If all stabilizers are finite then all orbits are closed. Hence Luna's slice theorem applies to every orbit which means that one can reduce to the case that $G$ is finite. This yields a lot of regularity:


* The action is proper.
* All sets $X\_H$ are locally closed.
* There is a unique $H$ (the generic isotropy group) such that $X\_H$ is open.
* If $x$ is contained in the closure of $X\_H$ then $Stab\_G(x)$ contains $H$ as a subgroup.
* Equivalently: All stabilizers in a neighborhood of $x$ are (isomorphic to) subgroups of $Stab\_G(x)$.


By the way, usually one considers a finer decomposition $X'\_H$ where $H$ is a conjugacy class of subgroups and $x\in X'\_H$ if $Stab\_G(x)$ is conjugate to $H$.


If $X$ is smooth one can say even more:


* All $X\_H$ are smooth.
* The irreducible components of all $X\_H$ form a stratification of $X$, i.e. closures of strata are unions of strata.


NB: All this requires that $G$ is reductive and $X$ is affine. Otherwise, $X\_H$ may be only constructible.


