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Let a group $G$ act on a (not necessarily irreducible) algebraic variety over ${\bf C}$.

It seems to be well-known that the quotient in the sense of geometric invariant theory (i.e., the categorical quotient in the category of algebraic varieties) agrees with the largest Hausdorff quotient (i.e., the categorical quotient in the category of Hausdorff spaces).

The reference which I find cited for this fact in the literature are two papers of Luna from the 70s: paper 1, paper 2. For example, this reference says (on page 6): It can be shown that the categorical quotient in the category of affine varieties (over ${\bf C}$ and with respect to a reductive group action) is also the categorical quotient for Hausdorff spaces or complex analytic varieties [Lun75, Lun 76].

My question is just how the claim about the categorical Hausdorff quotient follows from Luna's papers.

I understand that Luna is proving that not only (by definition) the invariant polynomials correspond 1-1 to polynomial functions on the GIT-quotient, but that also the same holds true for smooth or analytic functions instead of polynomials. Does this imply by some known results that the GIT-quotient must be the largest Hausdorff quotient?

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    $\begingroup$ What do you assume on $G$? reductive? if $\Gamma$ is an additive lattice in the complex plane, the topological quotient is not algebraic. $\endgroup$ – YCor Sep 18 '17 at 20:08
  • $\begingroup$ The case I am interested in is the action of an algebraic group (on the variety of representations of some finitely generated group into the algebraic group). $\endgroup$ – ThiKu Sep 18 '17 at 20:32
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    $\begingroup$ Maybe you could specify the statement you claim to be "well-known". $\endgroup$ – YCor Sep 18 '17 at 21:04
  • $\begingroup$ I have edited the question to include a literal citation from the literature. You were right that the group should be reductive. $\endgroup$ – ThiKu Sep 19 '17 at 0:01
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I recommend reading the very nice exposition here (which is where I learned this fact):

Schwarz, Gerald W. The topology of algebraic quotients. Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988), 135–151, Progr. Math., 80, Birkhäuser Boston, Boston, MA, 1989.

In particular, see the proof starting at the bottom of Page 141.

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