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 $E(\Delta,R)$ of the Chevalley group $G(\Phi,R)$ for the three types of pair $(\Phi,\Delta)$ (the root system and its subsystem) such that the group $G(\Delta,R)$ 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 $\sigma$ of the ring $R$ such that $E(\Phi,\Delta,R,\sigma)\le H\le N_{G(\Phi,R)}(E(\Phi,\Delta,R,\sigma))$.