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RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

Part of: Algebraic groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  26 May 2020

MAIKE GRUCHOT ,
ALASTAIR LITTERICK Open the ORCID record for ALASTAIR LITTERICK [Opens in a new window]  and
GERHARD RÖHRLE
MAIKE GRUCHOT
Affiliation:
Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen University, D-52062Aachen, Germany; maike.gruchot@rwth-aachen.de
ALASTAIR LITTERICK
Affiliation:
Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, EssexCO4 3SQ, UK; a.litterick@essex.ac.uk
GERHARD RÖHRLE
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780Bochum, Germany; gerhard.roehrle@rub.de

Abstract

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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.

MSC classification

Primary: 20G15: Linear algebraic groups over arbitrary fields
Secondary: 14L24: Geometric invariant theory
Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e30
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

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