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

TL;DR

This paper establishes a classification framework for overgroups of subsystem subgroups in simply laced Chevalley groups, linking them to nets of ideals and stabilizers, advancing understanding of subgroup structures.

Contribution

It proves a weak sandwich classification for overgroups of subsystem subgroups in simply laced Chevalley groups, connecting overgroups to nets of ideals and stabilizers.

Findings

Existence of a unique net of ideals for each overgroup

Overgroups are bounded between elementary subgroups and stabilizers

Provides a classification framework for subgroup structures in Chevalley groups

Abstract

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