Geometric properties of the adjoint action of a reductive group

$$\newcommand{\g}{\mathfrak{g}}$$Let $$G$$ be a reductive algebraic group over field $$k = \overline{k}$$ and consider the characteristic polynomial $$\g \to \g/\!/G := \operatorname{Spec} (k[\g]^G)$$ induced by the adjoint action. Assume $$|W|$$ is invertible in $$k$$, so Chevalley-Todd-Shepard implies the quotient is affine space.

I expect the following basic geometric properties to generalize from the $$\operatorname{GL}_n$$ case.

1. The $$k$$-points of $$\g/\!/G$$ are in bijection with Zariski-closed $$k$$-orbits.
2. Every $$k$$-rational orbit has a unique closed orbit in its closure, and the map on $$k$$-points is this identification.
3. The locus of regular elements (those $$X \in \g$$ with $$\dim G_X = \operatorname{rank} G$$, as small as possible) is exactly the smooth locus of the morphism.
4. The semisimple $$X \in \g$$ (in the sense of additive Jordan-Chevalley) are exactly those with closed $$k$$-orbits.

Do these hold in the setting above? If so, how much more general are they? The first two at least seem like general facts about actions on an affine $$k$$-scheme.

Edit: After some more reading and Ben's helpful answer, I was able to resolve most of this, though I haven't found anything that ensures surjectivity of the quotient morphism in the positive characteristic case: in characteristic zero the argument uses that for any $$A^G$$-algebra $$S$$, we have the identity $$S = (S \otimes_{A^G} A)^G$$. This doesn't work in positive characteristic (and GIT never claims that $$(\operatorname{Spec} A)/\!/G$$ is a "universal categorical quotient" unless char 0).

I'll leave this open for a bit in case Ben or anyone else can chip in with a substitute argument or counterexample.

Yes, all these things are true, and 1. and 2. hold for any action of a reductive group $$G$$ on an affine variety $$X$$. The key point is that if I have two closed orbits $$O_1,O_2$$, then the functions on their disjoint union is a quotient of functions on $$X$$ (because $$X$$ is affine), so there's a function on $$X$$ that is 1 on $$O_1$$ and 0 on $$O_2$$. Since the group is reductive, you can assume this function is $$G$$ invariant by projecting it to the trivial rep.

• Thanks so much, that's a big help. By the way, is the good-characteristic assumption enough to make this projection argument work in positive characteristic, where reductive doesn't imply complete reducibility of representations?
– C.D.
Jan 16, 2023 at 2:55
• Answering my own side-question: the argument works with no characteristic assumptions at all, see Corollary A.1.3 and the prior discussion on geometric reductivity in Appendix A to Mumford's GIT.
– C.D.
Jan 16, 2023 at 4:26
• Happy to accept this, though I was wondering whether you had any reference for (2) why/whether the quotient map is surjective in characteristic p -- it doesn't seem to follow from any of the general results I've found.
– C.D.
Jan 17, 2023 at 19:33
• It’s enough to check that a Cartan $\mathfrak{h}$ maps surjectively, and that’s true because the map is etale (and thus closed) on the Cartan. Jan 18, 2023 at 21:32
• This argument works because finite maps are closed, but by the way $$\mathfrak{h} \to \mathfrak{g}/\!/G \cong \mathfrak{h}/\!/W$$ is not etale except on the regular locus, right? (e.g. ramified over zero)
– C.D.
Jan 19, 2023 at 19:57