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

绿色用组合方法完整地描述了有限域上一般线性群的复特征标理论[1]。Deligne和Lusztig [2]对有限域上的约化代数群的工作将其推广到几何环境中。在这一点上，群作用于所谓的Deligne-Lusztig簇，并且通过检查etale上同调的诱导作用，可以恢复所有不可约表示。这种方法是上级和更有启发性的，因为它不仅构造字符，但进一步表示，在一个自然的方式。这个项目的总目标是应用几何和上同调方法，如上述代数问题的表示论。一个可能的方向是使用类似的上同调方法（所谓的Deligne-Lusztig构造）研究GL_2（F）（或更一般地GL_n（F））及其对非阿基米德局部场F的表示，也许可以利用有限表示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]的工作中，p-adic Deligne-Lutztig理论与GL_2（F）的局部Langlands和Jacquet Langlands对应都相关，并且在Ivanov [4]（2020）中，提出并研究了p-adic Deligne-Lusztig空间的新定义。群GL_2（F）是一个重要的研究对象，是朗兰兹局部对应的中心对象。此外，作为p-adic群的一个典型例子，GL_2（F）的构造除了本身是有趣的之外，还可以很好地推广到所有或许多p-adic群。通过检索mathscinet上近10年来的文献，可以看出人们对这一问题的兴趣：以“Deligne-Lusztig”为标签检索到177条结果，以“GL（2）”为标签检索到4500多条结果，这表明这是一个研究GL_2（F）表示理论的活跃但尚未得到充分研究的方法。这个项目福尔斯属于EPSRC代数研究领域，并应用于数论。

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.