RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS
Forum of Mathematics, Sigma, Tome 8 (2020)
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We study a relative variant of Serre’s notion of $G$-complete reducibility for a reductive algebraic group $G$. We let $K$ be a reductive subgroup of $G$, and consider subgroups of $G$ that normalize the identity component $K^{\circ }$. We show that such a subgroup is relatively $G$-completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$, as well as ‘rational’ versions over nonalgebraically closed fields.
@article{10_1017_fms_2020_25,
author = {MAIKE GRUCHOT and ALASTAIR LITTERICK and GERHARD R\"OHRLE},
title = {RELATIVE {COMPLETE} {REDUCIBILITY} {AND} {NORMALIZED} {SUBGROUPS}},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {8},
year = {2020},
doi = {10.1017/fms.2020.25},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2020.25/}
}
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%0 Journal Article %A MAIKE GRUCHOT %A ALASTAIR LITTERICK %A GERHARD RÖHRLE %T RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS %J Forum of Mathematics, Sigma %D 2020 %V 8 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.1017/fms.2020.25/ %R 10.1017/fms.2020.25 %G en %F 10_1017_fms_2020_25
MAIKE GRUCHOT; ALASTAIR LITTERICK; GERHARD RÖHRLE. RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS. Forum of Mathematics, Sigma, Tome 8 (2020). doi: 10.1017/fms.2020.25
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