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Semisimplification for subgroups of reductive algebraic groups
Published online by Cambridge University Press: 09 November 2020
Abstract
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
MSC classification
- Type
- Algebra
- Information
- Creative Commons
- 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
References
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