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Semisimplification for subgroups of reductive algebraic groups

Part of: Algebraic groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  09 November 2020

Michael Bate ,
Benjamin Martin  and
Gerhard Röhrle
Michael Bate
Affiliation:
Department of Mathematics, University of York, YorkYO10 5DD, United Kingdom; E-mail: michael.bate@york.ac.uk
Benjamin Martin
Affiliation:
Department of Mathematics, University of Aberdeen, King’s College, Fraser Noble Building, AberdeenAB24 3UE, United Kingdom; E-mail: b.martin@abdn.ac.uk
Gerhard Röhrle
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, Universitätsstraße 150, D-44780Bochum, Germany; E-mail: gerhard.roehrle@rub.de

Abstract

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Let G be a reductive algebraic group—possibly non-connected—over a field k, and let H be a subgroup of G. If $G= {GL }_n$ , then there is a degeneration process for obtaining from H a completely reducible subgroup $H'$ of G; one takes a limit of H along a cocharacter of G in an appropriate sense. We generalise this idea to arbitrary reductive G using the notion of G-complete reducibility and results from geometric invariant theory over non-algebraically closed fields due to the authors and Herpel. Our construction produces a G-completely reducible subgroup $H'$ of G, unique up to $G(k)$ -conjugacy, which we call a k-semisimplification of H. This gives a single unifying construction that extends various special cases in the literature (in particular, it agrees with the usual notion for $G= GL _n$ and with Serre’s ‘G-analogue’ of semisimplification for subgroups of $G(k)$ from [19]). We also show that under some extra hypotheses, one can pick $H'$ in a more canonical way using the Tits Centre Conjecture for spherical buildings and/or the theory of optimal destabilising cocharacters introduced by Hesselink, Kempf, and Rousseau.

Keywords

semisimplificationG-complete reducibilitygeometric invariant theoryrationalitycocharacter-closed orbitsdegeneration of G-orbits

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 , e43
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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. Published by Cambridge University Press

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