Closed orbits and uniform $S$-instability in geometric invariant theory
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- by Michael Bate, Benjamin Martin, Gerhard Röhrle and Rudolf Tange PDF
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Abstract:
In this paper we consider various problems involving the action of a reductive group $G$ on an affine variety $V$. We prove some general rationality results about the $G$-orbits in $V$. In addition, we extend fundamental results of Kempf and Hesselink regarding optimal destabilizing parabolic subgroups of $G$ for such general $G$-actions.
We apply our general rationality results to answer a question of Serre concerning the behaviour of his notion of $G$-complete reducibility under separable field extensions. Applications of our new optimality results also include a construction which allows us to associate an optimal destabilizing parabolic subgroup of $G$ to any subgroup of $G$. Finally, we use these new optimality techniques to provide an answer to Tits’ Centre Conjecture in a special case.
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Additional Information
- Michael Bate
- Affiliation: Department of Mathematics, University of York, York YO10 5DD, United Kingdom
- Email: michael.bate@york.ac.uk
- Benjamin Martin
- Affiliation: Mathematics and Statistics Department, University of Canterbury, Private Bag 4800, Christchurch 1, New Zealand
- Address at time of publication: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 8140, New Zealand
- MR Author ID: 659870
- Email: B.Martin@math.canterbury.ac.nz, Ben.Martin@auckland.ac.nz
- Gerhard Röhrle
- Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
- MR Author ID: 329365
- Email: gerhard.roehrle@rub.de
- Rudolf Tange
- Affiliation: School of Mathematics, Trinity College Dublin, College Green, Dublin 2, Ireland
- Address at time of publication: College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter, EX4 4QF, United Kingdom
- Email: tanger@tcd.ie, R.Tange@exeter.ac.uk
- Received by editor(s): July 1, 2011
- Received by editor(s) in revised form: October 28, 2011
- Published electronically: December 27, 2012
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 3643-3673
- MSC (2010): Primary 20G15, 14L24, 20E42
- DOI: https://doi.org/10.1090/S0002-9947-2012-05739-4
- MathSciNet review: 3042598