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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.

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    $\begingroup$ For the morphism $\psi:G\times X \to X\times X$, $(g,x)\mapsto (g\cdot x, x)$, the scheme $S=\psi^{-1}(\Delta(X))$ together with its induced morphism, $S\to X$, is the family of stabilizers. There is a maximal open subset of $X$ (possibly empty) over which this morphism has finite fibers by Chevalley's theorems. There is a stratification of this open into locally closed subsets based on the length of the stabilizer group. There are results about these strata, e.g., the answer by abx: mathoverflow.net/questions/203313/stabilisers-of-group-actions/… $\endgroup$
    – Jason Starr
    Commented Aug 3, 2017 at 15:50

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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.

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  • $\begingroup$ Thank you very much for the detailed answer. How Luna's Theorem enables us to reduce to the case that $G$ is finite? $\endgroup$
    – Ehud Meir
    Commented Aug 4, 2017 at 12:29

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