On exponentiation and infinitesimal one-parameter subgroups of reductive groups

TL;DR

This paper investigates the structure of one-parameter subgroups in reductive algebraic groups over fields of positive characteristic, establishing conditions for their embeddings and describing the cohomological variety of Frobenius kernels.

Contribution

It proves that under certain nilpotence and characteristic conditions, embeddings of Frobenius kernels are contained in canonical one-parameter subgroups, and characterizes the cohomological variety of Frobenius kernels.

Findings

Embeddings of Frobenius kernels lie inside canonical one-parameter subgroups when nilpotence class is less than p.

The cohomological variety of G_{(r)} is homeomorphic to the variety of r-tuples of commuting nilpotent elements.

Results depend on the characteristic p being at least the Coxeter number of G.

Abstract

Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p>0$, and assume $p$ is good for $G$. Let $P$ be a parabolic subgroup with unipotent radical $U$. For $r \ge 1$, denote by $\mathbb{G}_{a(r)}$ the $r$-th Frobenius kernel of $\mathbb{G}_a$. We prove that if the nilpotence class of $U$ is less than $p$, then any embedding of $\mathbb{G}_{a(r)}$ in $U$ lies inside a one-parameter subgroup of $U$, and there is a canonical way in which to choose such a subgroup. Applying this result, we prove that if $p$ is at least as big as the Coxeter number of $G$, then the cohomological variety of $G_{(r)}$ is homeomorphic to the variety of $r$-tuples of commuting elements in $\mathcal{N}_1(\mathfrak{g})$, the $[p]$-nilpotent cone of $Lie(G)$.