Overgroups of subsystem subgroups in exceptional groups: nonideal levels

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.