Complete Reducibility and Geometric Invariant Theory

在本提案中，我们的目标是研究j.p。Serre用几何不变量理论（GIT）的工具提出的g -完全可约性（G-cr）的概念。在Bate， Martin和作者最近的一篇联合论文中，证明了Serre的G-cr概念等价于Richardson的强还原性概念。这种等价性使我们能够使用GIT中的方法来研究G-cr子群，例如Hilbert-Mumford定理，从而推导出G-cr子群的新准则。提出的研究的目的是扩展和深化这种几何调查。本工作的总体指导原则是全面研究G-cr子群在自然群理论操作下的行为，如取正规子群、取商、取集中子群、取规范化子群、形成半直积和应用群同态等。虽然通过早期的工作已经知道了一些结果，但还需要进行系统的研究。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，K、H、G和基场上的什么条件确保如果K是G-cr，那么K是H-cr，反之亦然？这涉及到扩展与贝特和马丁早期联合工作的几个结果。我们想进一步研究可约性和完全可约性之间的联系；这应该是对G-cr子组早期工作中的特征限制的有效替代。我们还想为一些相反的结果制定一些准则。此外，我们打算进一步研究G-cr子群的合理性性质，并将我们的结果推广到非连通约化群。在他最初的建筑理论方法的背景下。Serre观察到g -完全可约性的概念对半代数行为是有意义的。我们想把之前的结果扩展到这个场景。

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.