Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. Let $\mathcal{U}(G)$ be the closed subvariety of unipotent elements of $G$, i.e., all elements whose order is some power of $p$. Then $G$ acts by conjugation on $\mathcal{U}(G)$, and there is a unique open, dense orbit, called the regular orbit, which we will denote $\mathcal{O}_{\text{reg}}(G)$.
The situation for $\mathfrak{g}=\mathrm{Lie}(G)$ is similar. There is a closed subvariety $\mathcal{N}(\mathfrak{g})$ of nilpotent elements on which $G$ acts via the adjoint action. Under this action, there again is a unique open, dense orbit, called the regular orbit, which we will denote $\mathcal{O}_{\text{reg}}(\mathfrak{g})$.
Take $u\in \mathcal{O}_{\text{reg}}(G)$, and consider $C_G(u)$, the centralizer of $u$ in $G$. Let $\mathfrak{c}_G(u)\subset\mathfrak{g}$ be the lie algebra of $C_G(u)$. I have two related questions:
- Must $\mathfrak{c}_G(u)\cap\mathcal{O}_{\text{reg}}(\mathfrak{g})$ be non-empty? It seems so, because $\mathcal{O}_{\text{reg}}(\mathfrak{g})$ is dense, but I don't know much about $\mathfrak{c}_G(u)$.
- If there is some $X\in\mathfrak{c}_G(u)\cap\mathcal{O}_{\text{reg}}(\mathfrak{g})$, then must $C_G(u)=C_G(X)$, where $C_G(X)$ is the centralizer of $X$ in $G$ under the adjoint action? I also believe the answer to this question is yes, but my knowledge of algebraic groups is not enough to see why.