Rational points and orbits on the variety of elementary subalgebras

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 g:=Lie(G) and the unipotent variety of G to study the projective variety of elementary subalgebras of g of rank r, denoted E(r,g). In the case that G is defined over Fp, we define the category of Fq-expressible subalgebras of g for q=pd, and prove that this category is isomorphic to a subcategory of Quillen's category of elementary abelian subgroups of the finite Chevalley group G(Fq). This isomorphism of categories leads to a correspondence between G-orbits of E(r,g) defined over Fq and G-conjugacy classes of certain elementary abelian subgroups of rank rd in G(Fq) which satisfy a closure property characterized by the Springer isomorphism. We use Magma to compute examples for G=GLn, n≤5.