On $\bbf_q$-rational structure of nilpotent orbits

Abstract

Let $G$ be a simple algebraic group and $\ggg=\Lie(G)$ over $k=\bar\bbf_q$ where $q$ is a power of the prime characteristic of $k$, and $F$ a Frobenius morphism on $G$ which can be defined naturally on $\ggg$. In this paper, we investigate the relation between $F$-stable restricted modules of $\ggg$ and closed conical subvarieties defined over $\bbf_q$ in the null cone $\cn(\ggg)$ of $\ggg$. Furthermore, we clearly investigate the $\bbf_q$-rational structure for all nilpotent orbits in $\ggg$ under the adjoint action of $G$ when the characteristic of $k$ is good for $G$ and bigger than 3. The arguments are also valid when $\ggg'$ is classical simple Lie algebra in the sense of \cite{Sel} and $G'$ is the adjoint group of $\ggg'$.