Closed Orbits and uniform S-instability in Geometric Invariant Theory

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JournalTransactions of the AMS
DatePublished - Jul 2013
Issue number7
Volume365
Number of pages31
Pages (from-to)3643-3673
Original languageEnglish

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 how his notion of $G$-complete reducibility behaves 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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