On exponentiation and infinitesimal one-parameter subgroups of reductive groups

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 $Lie(G)$.