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Relative complete reducibility and normalised subgroups

Gruchot, Maike and Litterick, Alastair and Roehrle, Gerhard (2020) 'Relative complete reducibility and normalised subgroups.' Forum of Mathematics, Sigma, 8. ISSN 2050-5094

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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 $G$ which normalise 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 generalise 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 non-algebraically closed fields.

Item Type: Article
Additional Information: 33 pages
Uncontrolled Keywords: math.GR; math.RT
Divisions: Faculty of Science and Health
Faculty of Science and Health > Mathematical Sciences, Department of
SWORD Depositor: Elements
Depositing User: Elements
Date Deposited: 30 Apr 2020 13:23
Last Modified: 06 Jan 2022 14:04

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