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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Closed orbits and uniform $ S$-instability in geometric invariant theory


Authors: Michael Bate, Benjamin Martin, Gerhard Röhrle and Rudolf Tange
Journal: Trans. Amer. Math. Soc. 365 (2013), 3643-3673
MSC (2010): Primary 20G15, 14L24, 20E42
Published electronically: December 27, 2012
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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
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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05739-4
PII: S 0002-9947(2012)05739-4
Keywords: Reductive group, $G$-variety, closed orbit, uniform instability, optimal cocharacter, Centre Conjecture
Received by editor(s): July 1, 2011
Received by editor(s) in revised form: October 28, 2011
Published electronically: December 27, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.