$\newcommand{\g}{\mathfrak{g}}$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.



Geometric properties of the adjoint action of a reductive group
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.


