Serre's notion of complete reducibility and geometric invariant theory

在这项拟议的研究中，我们打算进一步调查J-P。Serre的G-完全可约性概念，借助几何不变量理论，集中于合理性，构建理论问题。具体的主要目标如下。首先，我们要推广Richardson对Gn中闭Gorbit的代数刻画，Gn是G与其自身的n重笛卡尔积，在G的任意约化子群在Gn上的作用的同时共轭下。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的不稳定性概念与Hesselink的一致不稳定性概念结合起来。除了独立的利益，这将被用来解决建筑理论问题。特别地，在这里，我们对J.Tits提出的所谓的中心猜想采取统一的几何方法。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，具体地说，这里我们将讨论Serre提出的关于G-完全可约性在可分域扩张下的行为的一般问题。

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.