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.
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.
| Original language | English |
|---|---|
| Pages (from-to) | 3643-3673 |
| Number of pages | 31 |
| Journal | Transactions of the AMS |
| Volume | 365 |
| Issue number | 7 |
| DOIs | |
| Publication status | Published - Jul 2013 |