$\mathbb{F}_p$-expressible subalgebras and orbits of $\mathbb{E}(r,\mathfrak{g})$

Abstract

For $G$ a connected, reductive group over an algebraically closed field $k$ of large characteristic, we use the canonical Springer isomorphism between the nilpotent variety of $\mathfrak{g}:=\mathrm{Lie}(G)$ and the unipotent variety of $G$ to study the projective variety of elementary subalgebras of rank $r$, denoted $\mathbb{E}(r,\mathfrak{g})$. In the case that $G$ is defined over $\mathbb{F}_p$, we define the category of $\mathbb{F}_p$-expressible subalgebras of $\mathfrak{g}$, and prove that this category is isomorphic to Quillen's category of elementary abelian subgroups of the finite Chevalley group $G(\mathbb{F}_p)$. This isomorphism of categories leads to a correspondence between $G$-orbits of $\mathbb{E}(r,\mathfrak{g})$ defined over $\mathbb{F}_p$ and $G$-conjugacy classes of elementary abelian subgroups of rank $r$ in $G(\mathbb{F}_p)$. We use Magma to compute examples for $G=\mathrm{GL}_n$, $n\le 5$.