Complete Reducibility and Geometric Invariant Theory

在这个项目中，我们的目标是研究J。P. Serre的G-完全约化（G-cr）概念使用几何不变理论（GIT）的工具。在最近的一份联合文件的贝特，马丁和作者表明，塞尔的概念的G-铬是等价于理查森的概念强还原性。这种等价性使我们能够使用GIT的方法来研究约化代数群G的G-cr子群，如希尔伯特-芒福德定理，以获得G-cr子群的新准则。本文的目的是扩展和深化这一几何研究.本文工作的总的指导原则是对G-cr子群在自然群论运算下的行为进行全面的研究，这些自然群论运算包括取正规子群，取同分群，取中心化子，取正规化子，形成半直积和应用群同态等.虽然在早期的工作中已有一些结果，我们要解决的另一个一般性问题是：设K，H和G是约化群，K包含在H中，H包含在G中。关于K、H、G和基场的什么条件能保证如果K是G-cr，则K是H-cr，反之亦然？这涉及到扩展的几个结果从早期的联合工作与Bate和Martin.We希望进一步研究之间的联系约化巴黎和完全reductionary;这应该是一个有效的替代特征限制在早期的工作G-cr子群。此外，我们还打算进一步研究G-cr子群的合理性，并将所得结果推广到非连通约化群，在J. - P. Serre观察到G-完全归约的概念对于半代数作用是有意义的。我们想把我们以前的结果推广到这种情况。

In this proposal we aim to study J.-P. Serre's notion of G-complete reducibility (G-cr) using tools from geometric invariant theory (GIT). In a recent joint paper by Bate, Martin and the author it was shown that Serre's concept of G-cr is equivalent to Richardson's notion of strong reductivity. This equivalence allowed us to use methods from GIT in the study of G-cr subgroups of reductive algebraic groups G such as the Hilbert-Mumford Theorem to derive new criteria for G-cr subgroups. The aim of the proposed research is to extend and deepen this geometric investigation.The general guiding principle of this work is to undertake a comprehensive study of the behaviour of G-cr subgroups under natural group-theoretic operations, such as taking normal subgroups, taking quotients, taking centralisers, taking normalisers, forming semi-direct products and applying group homomorphisms, etc. Although by earlier work some results are known, a systematic study is needed.Another general question we aim to address is the following: Let K,H, and G be reductive groups with K contained in H and H contained G. What conditions on K,H,G and the ground field that ensure that if K is G-cr, then K is H-cr, and vice versa? This involves extending several results form earlier joint work with Bate and Martin.We want to further investigate the connection between reductive paris and complete reducibility; this should be an effective replacement for characteristic restrictions in earlier work on G-cr subgroups. Also we want to develop some criteria for some converse results.Moreover, we intend to study further rationality properties of G-cr subgroups and generalisations of our results to non-connected reductive groups.In the context of his original building-theoretic approach J.-P. Serre observed that the notion of G-complete reducibility makes sense for semi-algebraic actions. We want to extend our earlier results to this setting.