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

Gvozdevsky, P.

doi:10.1090/spmj/1682

Overgroups of subsystem subgroups in exceptional groups: A $2A_1$-proof
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by P. Gvozdevsky
Translated by: the author

St. Petersburg Math. J. 32 (2021), 1011-1031

DOI: https://doi.org/10.1090/spmj/1682

Published electronically: October 20, 2021

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Abstract:

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

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