Representations of p-adic groups and dg-algebras

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The representation theory of p-adic groups has been a highly active area of research over the last 50 years, since the conjectures of Langlands proposed very deep connections to Number Theory. For example, there have been over 1500 publications which list the corresponding Mathematics Subject Classification (22E50) since 2000, and there are many more in neighbouring areas which overlap.Initially, only complex representations of p-adic groups were studied but more recently representations over other fields and rings have been considered. Of particular interest to number theorists is the case where the coefficient field in some way matches the p-adic field arising in the group - but this is also much more difficult: there are many techniques from the complex world which can no longer be used and, more significantly, many things are genuinely different. A particular example of this concerns the translation of questions about representation of p-adic groups into questions about modules over so-called Hecke algebras: for complex representations, these are somehow the "same thing" but for representations over other fields, they generally are not - somehow, the algebra only sees part of the picture. Fortunately, there is a way to extend this partial picture to reveal the missing parts, at least in principle - and the project is about putting this into practice. The Hecke algebras are the "symmetries" of certain objects, but these object can be enhanced into something called their "injective resolution" - looking at the symmetries of these injective resolutions (called a dg Hecke algebra) should then give a fuller picture of the representations. Ultimately, a complete understanding of all of these, together with the relationship between the dg Hecke algebras for the group and certain subgroups, should be a shell of an overarching picture tying them all together: a 2-category whose 2-endomorphisms are given by the above dg Hecke algebras.The project would begin by looking at complex representations of small groups (GL(1), GL(2), GL(3)) and try to compute these injective resolutions in the simplest case, and the corresponding dg Hecke algebras - even though the representation theory is already understood in these cases, these have not been computed - and look at the relationship between them (for example, for GL(1)GL(1) inside GL(2)). As well as producing new results, the main purpose of this is to be a testing ground to help inform the next stage: to repeat the same computations for p-modular representations. We have clear path to this and yet, even for GL(2), it would be a major achievement in terms of interest in the community. Thus, the major (technical) aims are:1) to compute the injective resolutions of the trivial complex representation of the Iwahori subgroup for GL(1), GL(2) and GL(3), and the corresponding dg Hecke algebra;2) to compute the dg bimodules giving parabolic induction and restriction for these dg Hecke algebras and understand their properties;3) to repeat these steps for the trivial p-modular representation of the pro-p-Iwahori subgroup for GL(1), GL(2) and GL(3);4) to interpret this in terms of the action of a 2-category.