RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

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...

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Bibliographic Details
Published in:Forum of mathematics. Sigma 2020, Vol.8, Article e30
Main Authors: GRUCHOT, MAIKE, LITTERICK, ALASTAIR, RÖHRLE, GERHARD
Format: Article
Language:English
Subjects:
14L24
20G15
Automorphisms
Group theory
Lie groups
Subgroups
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Summary: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.
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2020.25