An Analogue of the Jacobson-Morozov Theorem for Lie Algebras of Reductive Groups of Good Characteristics

Abstract

Let $\mathfrak {g}$ be the Lie algebra of a connected reductive group $G$ over an algebraically closed field of characteristic $p > 0$. Suppose that ${G^{(1)}}$ is simply connected and $p$ is good for the root system of $G$. Given a one-dimensional torus $\lambda \subset G$ let $\mathfrak {g}(\lambda ,1)$ denote the weight component of ${\text {Ad(}}\lambda {\text {)}}$ corresponding to weight $i \in X(\lambda ) \cong \mathbb {Z}$. It is proved in the paper that, for any nonzero nilpotent element $e \in \mathfrak {g}$, there is a one-dimentional torus ${\lambda _e} \subset G$ such that $e \in \mathfrak {g}({\lambda _e},2)$ and ${\text {Ker}}{\text {ad}}e \subseteq { \oplus _{i \geqslant 0}}\mathfrak {g}({\lambda _e},i)$.