Overgroups of subsystem subgroups in exceptional groups: 2A1-proof

Abstract

In the present paper we prove a weak form of sandwich classification for the overgroups of the subsystem subgroup $E(Δ,R)$ of the Chevalley group $G(Φ,R)$ where $Φ$ is a simply laced root sysetem and $Δ$ is its sufficiently large subsystem. Namely we show that for any such an overgroup $H$ there exists a unique net of ideals $σ$ of the ring $R$ such that $E(Φ,Δ,R,σ)\le H\le {\mathop{\mathrm{Stab}}\nolimits}_{G(Φ,R)}(L(σ))$ where $E(Φ,Δ,R,σ)$ is an elementary subgroup associated with the net and $L(σ)$ is a corresponding subalgebra of the Chevalley Lie algebra.