Serre's notion of complete reducibility and geometric invariant theory

塞尔的完全可约性概念和几何不变量理论

• 批准号: 125049979
• 负责人: Professor Dr. Gerhard Röhrle
• 金额: --
• 依托单位: Lehrstuhl Algebra/Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2009
• 资助国家: 德国
• 起止时间: 2008-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/125049979?language=en
• 关键词: Serre; notion; complete; reducibility; geometric

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.

在本研究中，我们打算进一步研究J-P. Serre的概念的G-完全约化的几何不变理论的手段，集中在合理性和建设的理论问题。具体主要目标如下。首先，我们要推广Richardson的代数特征的封闭Gorbits在Gn，的n重carbohydrate产品的G与本身，同时共轭的作用下，G的任意约化子群在Gn上。这反过来又会导致塞尔的G-完全约化概念的推广。其次，我们需要一个合适的概念，结合肯普夫的不稳定性的概念与Hesselink的一致不稳定性的想法。除了是独立的利益，这将被用来解决建筑理论问题。特别是，在这里，我们追求一个统一的几何方法对所谓的中心猜想由于J。此外，我们将研究的合理性问题的G-完全约化的几何手段，特别是在这里，我们将解决一个一般性的问题所提出的塞尔有关的行为G-完全约化下的可分域扩张。