The centralizer of a nilpotent section

Abstract

Let $F$ be an algebraically closed field and let $G$ be a semisimple $F$-algebraic group for which the characteristic of $F$ is very good. If $X \in \operatorname{Lie}(G) = \operatorname{Lie}(G)(F)$ is a nilpotent element in the Lie algebra of $G$, and if $C$ is the centralizer in $G$ of $X$, we show that (i) the root datum of a Levi factor of $C$, and (ii) the component group $C/C^{o}$ both depend only on the Bala-Carter label of $X$; i.e. both are independent of very good characteristic. The result in case (ii) depends on the known case when $G$ is (simple and) of adjoint type.

The proofs are achieved by studying the centralizer $\mathcal{C}$ of a nilpotent section $X$ in the Lie algebra of a suitable semisimple group scheme over a Noetherian, normal, local ring $\mathcal{A}$. When the centralizer of $X$ is equidimensional on $\operatorname{Spec}(\mathcal{A})$, a crucial result is that locally in the étale topology there is a smooth $\mathcal{A}$-subgroup scheme $L$ of $\mathcal{C}$ such that $L_{t}$ is a Levi factor of $\mathcal{C}_{t}$ for each $t \in \operatorname{Spec}(\mathcal{A})$.