Overgroups of Subsystem Subgroups in Exceptional Groups: Levels

Abstract

An embedding of root systems ∆ ⊆ Φ determines the corresponding regular embedding G(∆, R)≤ G(Φ, R) of Chevalley groups, over an arbitrary commutative ring R. Denote by E(∆, R) the elementary subgroup of G(∆, R). In the present paper we initiate the study of intermediate subgroups H, E∆, R) ≤ H ≤ G(Φ, R), provided that Φ=E6, E7, E8, F4 or G2, and there are no roots in Φ orthogonal to all of ∆. There are 72 such pairs (Φ, Δ)$. For F4 and G2 we assume, moreover, that 2 ∈ R * or 6 ∈ R *, respectively. For all such subsystems Δ we construct the levels of intermediate subgroups. We prove that these levels are determined by certain systems of ideals in R, one for each Δ-equivalence class of roots in Φ\∆, and calculate all relations among these ideals, in each case. Bibliography: 64 titles.