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

Green用组合均值[1]完整地描述了有限域上一般线性群的复特征理论。通过Deligne和Lusztig[2]对有限域上的约化代数群的研究，将其推广到几何环境中。在这种情况下，群作用于所谓的delign - lusztig变异，并且通过检查诱导作用于etale上同调可以恢复所有不可约表示。这种方法更优越，更具有启发性，因为它不仅以自然的方式构建字符，而且进一步构建表征。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理论中的代数问题。一个可能的方向是研究GL_2(F)（或更一般的GL_n(F)）及其对F的非阿基米德局部域的表示，使用类似的上同调方法（所谓的delignee - lusztig构造），也许可以使用有限表示理论的技术和方法作为灵感。这已经是一个正在进行的研究领域，例如在Chen[3]的作品中，p进delign - lutztig理论与GL_2(F)的局部Langlands和Jacquet Langlands对应相关，而在Ivanov[4]（2020）中，提出并研究了p进delign - lusztig空间的新定义。GL_2(F)群是一个重要的研究对象，是局部朗兰兹对应的中心对象。此外，作为p进群的一个典型例子，GL_2(F)的构造除了本身很有趣外，还可以很好地推广到所有或许多p进群。通过在mathscinet上搜索过去10年的出版物，可以证明对这个主题的积极兴趣：标签“delign - lusztig”返回177个结果，“GL(2)”超过4500个，这表明这是一个活跃但可能未被充分研究的GL_2(F)表示理论方法。本研究与Ardakov（导师）的研究相辅相成。该项目属于EPSRC代数研究领域，应用于数论。

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.