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.

33 pages, no figures, to appear in St.Petersburg Math Journal

References in corpus (5)

• Structure of Chevalley groups over rings via universal localization
• Groups normalized by the odd unitary group
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• Overgroups of subsystem subgroups in exceptional groups: nonideal levels
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Cited by in corpus (1)

• Overgroups of subsystem subgroups in exceptional groups: nonideal levels