Overgroups of subsystem subgroups in exceptional groups: nonideal levels

TL;DR

This paper completes the classification of overgroups of subsystem subgroups in simply laced Chevalley groups over rings by introducing levels, which generalize ideals, to describe overgroup structures precisely.

Contribution

It introduces the concept of levels as complex objects generalizing ideals, providing a complete description of overgroups of subsystem subgroups in exceptional groups.

Findings

Existence of a unique level for each overgroup

Overgroups are bounded between elementary subgroups and stabilizers of Lie subalgebras

Levels can be more complex than just nets of ideals

Abstract

In the present paper, we practicaly complete the solution of the problem on the description of overgroups of the subsystem subgroup $E(\Delta,R)$ in the Chevalley group $G(\Phi,R)$ over the ring $R$, where $\Phi$ is a simply laced root system, and $\Delta$ is its large enough subsystem. Namely we define objects called levels, and show that for any such an overgroup $H$ there exists a unique level $\sigma$ such that $E(\sigma)\le H\le \mathrm{Stab}_{G(\Phi,R)}(L_{\max}(\sigma))$, where $E(\sigma)$ is an elementary subgroup defined by the level $\sigma$, and $L_{\max}(\sigma)$ is the corresponding Lie subalgebra in the Chevalley algebra. Unlike all the previous papers, now levels can be more complicated objects that the nets of ideals.