Overgroups of subsystem subgroups in exceptional groups: nonideal levels

Gvozdevsky, P.

doi:10.1090/spmj/1733

Overgroups of subsystem subgroups in exceptional groups: nonideal levels
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by P. Gvozdevsky
Translated by: the author

St. Petersburg Math. J. 33 (2022), 897-925

DOI: https://doi.org/10.1090/spmj/1733

Published electronically: October 31, 2022

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Abstract:

In the present paper, a description of overgroups for the subsystem subgroups $E(\Delta ,R)$ of the Chevalley groups $G(\Phi ,R)$ over the ring $R$, where $\Phi$ is a simply laced root system and $\Delta$ is its sufficiently large subsystem, is almost entirely finished. Namely, objects called levels are defined and it is shown that for any such overgroup $H$ there exists a unique level $\sigma$ with $E(\sigma )\le H\le \operatorname {Stab}_{G(\Phi ,R)}(L_{\max }(\sigma ))$, where $E(\sigma )$ is an elementary subgroup associated with the level $\sigma$ and $L_{\max }(\sigma )$ is the corresponding subalgebra of the Chevalley algebra. Unlike the previous papers, here levels can be more complicated than nets of ideals.

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