Rational points and orbits on the variety of elementary subalgebras

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 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 F p , we define the category of F q -expressible subalgebras of g for q = p d , and prove that this category is isomorphic to a subcategory of Quillen's category of elementary abelian subgroups of the finite Chevalley group G(F q ). This isomorphism of categories leads to a correspondence between G-orbits of E(r, g) defined over F q and G-conjugacy classes of certain elementary abelian subgroups of rank rd in G(F q ) which satisfy a closure property characterized by the Springer isomorphism. We use Magma to compute examples for G = GL n , n ≤ 5. Keywords: restricted Lie algebras, Springer isomorphisms 2010 MSC: 17B45, 20G40In [2], J. Carlson, E. Friedlander, and J. Pevtsova initiated the study of E(r, g), the projective variety of rank r elementary subalgebras of a restricted lie algebra g. The authors demonstrate that the study of E(r, g) informs the representation theory and cohomology of g. This is all reminiscent of the case of a finite group G, where the elementary abelian p-subgroups play a significant role in the story of the representation theory and cohomology of G, as first explored by Quillen in [11] and [12].In this paper, we further explore the structure of E(r, g) and its relationship with elementary abelian subgroups. Theorem 3.3 shows in the case that g is the Lie algebra of a connected, reductive group G defined over F p , the category of F q -expressible subalgebras (Definitions 2.2 and 3.2) is isomorphic to a subcategory of Quillen's category of elementary abelian p-subgroups of G(F q ), where q = p d . Specifically, we introduce the notion of an F q -linear subgroup (Definition 3.5), and we show in Corollary 3.9 that the F q -expressible subalgebras of rank r are in bijection with the F q -linear elementary abelian subgroups of rank rd in G(F q ). This bijection leads to Corollary 3.11, which allows us to compute the largest integer R = R(g) such that E(R, g) is nonempty for a simple Lie algebra g. These values are presented in Table 1.The results and definitions in §3 rely on the canonical Springer isomorphism σ : N (g) → U(G), which has been shown to exist under the hypotheses we assume in this paper, as detailed in [13], [3], [16], and [7]. Together with Lang's theorem, Theorem 3.3 implies Theorem 4.3, which establishes a natural bijection between the G-orbits of E(r, g) defined over F q and the G-conjugacy classes of F q -linear elementary abelian subgroups of rank rd in G(F q ). Example 4.8, due to R. Guralnick, shows that E(r, g) may be an infinite union of G-orbits (in fact this is usually the case). However, Proposition 4.4 demonstrates that E(R(g), g) is a finite union of orbits for all connected, reductive G such that (G, G) is an almost-d...

Varieties of elementary subalgebras of submaximal rank in type A

“…We list the results of E(rk p (g) − 1, g) for A n (n ≤ 4) in the following. The reference we give is the paper [14] of Warner, in which the author discuss the irreducibility of E(r, gl n ) in section 5.…”