Relative complete reducibility and normalised 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 \(G\) which normalise the identity component \(K^{\circ}\). We show that such a subgroup is relatively \(G...

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Bibliographic Details
Published in:arXiv.org 2020-04
Main Authors: Gruchot, Maike, Litterick, Alastair, Roehrle, Gerhard
Format: Article
Language:English
Subjects:
Algebra
Automorphisms
Group theory
Lie groups
Subgroups
Online Access:Get full text
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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 \(G\) 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.
ISSN:2331-8422