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
