Serre's notion of complete reducibility and geometric invariant theory

在本研究中，我们打算进一步研究J-P。Serre的g -完全可约性概念用几何不变理论，集中在合理性和构建理论问题。具体的主要目标如下。首先，我们将Richardson关于Gn中闭Gorbits （G与自身的n倍笛卡尔积）的代数刻画推广到G在Gn上任意约化子群的共轭作用下。这反过来将导致Serre的g完全可约性概念的推广。其次，我们需要一个合适的最优性概念，将Kempf的不稳定概念与Hesselink的一致不稳定概念结合起来。除了独立的兴趣之外，这将用于解决建筑理论问题。特别地，在这里，我们追求一种统一的几何方法，以所谓的中心猜想，由于J. Tits。进一步，我们将用几何方法研究g -完全可约性的合理性问题，特别是在这里我们将讨论Serre提出的关于可分域扩展下g -完全可约性行为的一般问题。

In this proposed research we intend to further investigate J-P. Serre's notion of G-complete reducibility by means of geometric invariant theory, concentrating on rationality and building theoretic questions. The specific principal objectives are as follows. Firstly, we want to generalize Richardson's algebraic characterization of the closed Gorbits in Gn, the n-fold cartesian product of G with itself, under simultaneous conjugation to the action of an arbitrary reductive subgroup of G on Gn. This in turn will lead to a generalization of Serre's notion of G-complete reducibility. Secondly, we require a suitable concept of optimality by combining Kempf's instability notion with Hesselink's idea of uniform instability. Apart from being of independent interest, this will then be used to address building theoretic questions. In particular, here we pursue a uniform geometric approach towards the so called Center Conjecture due to J. Tits. Further, we will study rationality questions of G-complete reducibility by geometric means, specifically here we will address a general problem posed by Serre concerning the behavior of G-complete reducibility under separable field extensions.