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 $⋙$. In this paper, we investigate the relation between $F$-stable restricted modules of $⋙$ and closed conical subvarieties defined over $\bbf_q$ in the null cone $\cn(\ggg)$ of $⋙$. Furthermore, we clearly investigate the $\bbf_q$-rational structure for all nilpotent orbits in $⋙$ 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 ${⋙}^{\prime }$ is classical simple Lie algebra in the sense of \cite{Sel} and $G^{\prime }$ is the adjoint group of ${⋙}^{\prime }$.

A Preliminary Version