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

TL;DR

This paper establishes a classification of overgroups of certain Levi subgroups within Chevalley groups, specifically those with abelian unipotent radicals, using ideal-based subgroup structures.

Contribution

It provides a unique ideal-based sandwich classification for overgroups of subsystem subgroups in Chevalley groups with abelian unipotent radicals.

Findings

Overgroups are classified by pairs of ideals in the ring R.

Existence of a unique ideal pair for each overgroup.

The classification applies to specific root system pairs (Φ, Δ).

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 $E(\Phi,\Delta,R,\sigma)\le H\le N_{G(\Phi,R)}(E(\Phi,\Delta,R,\sigma))$.