Geometric and Cohomological Methods in Representations of p-adic Groups.

p-adic 群表示中的几何和上同调方法。

• 批准号: 2426296
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2426296
• 关键词: Geometric; Cohomological; Methods; Representations; adic

The complex character theory of the general linear group over a finite field was described completely by Green using combinatorial means [1]. This was generalised into a geometric setting by the work of Deligne and Lusztig [2] for reductive algebraic groups over a finite field. In this, the group acts on the so called Deligne-Lusztig variety, and by examining the induced action on etale cohomology one can recover all irreducible representations. This method is superior and more enlightening in that it not only constructs the characters but further the representations, in a natural way.The general aim of this project is to apply geometric and cohomological methods such as the above to algebraic problems in representation theory. One possible direction is to study GL_2(F) (or more generally GL_n(F)) and its representations for F a non-Archimedean local field using similar cohomological methods (so called Deligne-Lusztig constructions), perhaps with techniques and methods from finite representation theory used as inspiration. This is already an ongoing field of research, for example in the works of Chen [3], a p-adic Deligne-Lutsztig theory is related to both the local Langlands and Jacquet Langlands correspondence for GL_2(F), and in Ivanov [4] (2020), a new definition of p-adic Deligne-Lusztig spaces is proposed and studied.The group GL_2(F) is an important object of study, being a central object in the local Langlands correspondence. Further, being a prototypical example of p-adic group, constructions for GL_2(F) may well generalise to all or many p-adic groups, besides being interesting in their own right. The active interest in this topic is demonstrated by a search for publications in the last 10 years on mathscinet: the tag "Deligne-Lusztig" returns 177 results and "GL(2)" over 4500, suggesting that this is an active but perhaps understudied approach to the representation theory of GL_2(F).This research complements well with that of Ardakov (supervisor). This project falls within the EPSRC Algebra research area, with applications to Number Theory.

有限域上一般线性群的复特征标理论由Green用组合方法完全描述[1]。通过Deligne和Lusztig[2]关于有限域上的约化代数群的工作，这被推广到几何环境中。在这一点上，群作用于所谓的Deligne-Lusztig簇，并且通过检查对上同调的诱导作用，一个人可以恢复所有不可约表示。这种方法不仅构造了特征标，而且以自然的方式进一步构造了表示。本项目的总体目标是将上述几何和上同调方法应用于表示论中的代数问题。一个可能的方向是利用类似的上同调方法(所谓的Deligne-Lusztig构造)来研究非阿基米德局部域F的GL_2(F)(或者更广泛地说GL_n(F))及其表示，也许可以借鉴有限表示理论的技巧和方法。这已经是一个正在进行的研究领域，例如，在Chen[3]的工作中，p-adic Deligne-Lutsztig理论与GL_2(F)的局部Langland对应和Jacquet-Lutsztig对应有关；在Ivanov[4](2020)中，提出并研究了p-进Deligne-Lusztig空间的新定义。此外，GL_2(F)的构造作为p-进群的一个典型例子，除了本身有趣之外，还可以很好地推广到所有或许多p-进群。人们对这一主题的浓厚兴趣体现在过去10年在Mathscnet上的一次搜索中：标签“Deligne-Lusztig”返回177个结果，“GL(2)”返回4500个结果，这表明这是一个活跃但可能未被充分研究的GL_2(F)表征理论的方法。这项研究与Ardakov(主管)的研究是很好的补充。这个项目属于EPSRC代数研究领域，以及数论的应用。