R. Tange | Searcharxiv

arXiv2011

Closed Orbits and uniform S-instability in Geometric Invariant Theory

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.

math.AG↗

arXiv2010

Complete Reducibility and Conjugacy classes of tuples in Algebraic Groups and Lie algebras

Let H be a reductive subgroup of a reductive group G over an algebraically closed field k. We consider the action of H on G^n, the n-fold Cartesian product of G with itself, by simultaneous conjugation. We give a purely algebraic characterization of the closed H-orbits in G^n, generalizing work of Richardson which treats the case H = G. This characterization turns out to be a natural generalization of Serre's notion of G-complete reducibility. This concept appears to be new, even in characteristic zero. We discuss how to extend some key results on G-complete reducibility in this framework. We also consider some rationality questions.

math.GR↗