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

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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.