Overgroups of Levi subgroups I. The case of abelian unipotent radical

Abstract

In the present paper we prove sandwich classification for the overgroups of the subsystem subgroup $E(Δ,R)$ of the Chevalley group $G(Φ,R)$ for the three types of pair $(Φ,Δ)$ (the root system and its subsystem) such that the group $G(Δ,R)$ is (up to torus) a Levi subgroup of the parabolic subgroup with abelian unipotent radical. Namely we show that for any such an overgroup $H$ there exists a unique pair of ideals $σ$ of the ring $R$ such that $E(Φ,Δ,R,σ)\le H\le N_{G(Φ,R)}(E(Φ,Δ,R,σ))$.