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I would be glad to have some guidance in the following.

Let $k$ be an algebraically closed field. Let $G$ be a connected reductive group over $k$. Denote by $\mathfrak{c}$ the Zariski spectrum of the algebra of $G$-invariant functions on $\mathfrak{g}$.

In $\mathfrak{g}$, we have an open $G$-invariant subset $\mathfrak{g}^{\circ}$, consisting of regular semisimple elements. Its image $\mathfrak{c}^{\circ}$ in $\mathfrak{c}$ is open.

I want to claim that $\mathfrak{c}^{\circ}$ is the fppf quotient of $\mathfrak{g}^{\circ}$ by the action of $G$; i.e., it represents the fppf-sheafification of the functor $R \mapsto G(R) \backslash \mathfrak{g}^{\circ}(R)$. How to see this?

As a second side question, I would like to ask whether something around these issues (the Lie algebra, Chevalley restriction theorem, something else...) breaks down in characteristic p?

Thank you, Sasha

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    $\begingroup$ Xuhua He and I did a GIT analysis of the stable and semistable locus for the conjugation action on the wonderful compactification of $G$. I realize that $\mathfrak{g}$ is different, but probably the same techniques apply. In particular, I believe that the regular locus is contained in the GIT stable locus, which should imply that the GIT quotient is a uniform geometric quotient on the regular locus. $\endgroup$ Feb 3, 2016 at 19:12
  • $\begingroup$ @JasonStarr: Thank you. I think that I will try to learn a bit about GIT and then consider this again. $\endgroup$
    – Sasha
    Feb 4, 2016 at 9:14

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