Representations of p-adic groups and dg-algebras

在过去的50年里，p进群的表示理论一直是一个非常活跃的研究领域，因为朗兰兹的著作提出了与数论非常深刻的联系。例如，自2000年以来，已经有超过1500种出版物列出了相应的数学主题分类（22 E50），并且在相邻领域中有更多的重叠。最初，只研究了p进群的复表示，但最近考虑了其他域和环的表示。数论学家特别感兴趣的是系数场在某种程度上与群中产生的p进场相匹配的情况--但这也要困难得多：复杂世界中有许多技术不再能用了，更重要的是，许多东西真的不同了。这方面的一个特别的例子涉及到将关于p-adic群的表示的问题转化为关于所谓的Hecke代数上的模的问题：对于复表示，它们在某种程度上是“同一件事”，但对于其他域上的表示，它们通常不是--不知何故，代数只看到了画面的一部分。幸运的是，至少在原则上，有一种方法可以扩展这一部分，以揭示缺失的部分-该项目就是将其付诸实践。赫克代数是某些对象的“对称性”，但这些对象可以被增强为它们的“内射归结”--研究这些内射归结的对称性（称为dg赫克代数）应该会给出更全面的表示。最终，对所有这些的完整理解，以及群和某些子群的dg Hecke代数之间的关系，应该是将它们联系在一起的总体图景的外壳：一个2-范畴，它的2-自同态由上述dg Hecke代数给出。该项目将开始，首先研究小群的复表示（GL（1），GL（2），GL（3）），并尝试计算最简单情况下的这些内射分解，以及相应的dg Hecke代数-尽管在这些情况下表示论已经被理解，但这些还没有被计算-并查看它们之间的关系（例如，对于GL（1）GL（1）在GL（2）内）。除了产生新的结果外，这样做的主要目的是成为一个测试场，以帮助为下一阶段提供信息：对p-模表示重复相同的计算。我们有明确的道路，但即使是GL（2），这将是一个重大的成就，在社会的利益。因此，少校（技术）目的是：1）计算GL（1），GL（2）和GL（3）的Iwahori子群的平凡复表示的内射分解，以及相应的dg Hecke代数;2）计算这些dg Hecke代数的给出抛物归纳和限制的dg双模，并理解它们的性质;3）对GL（1），GL（2）和GL（3）的pro-p-Iwahori子群的平凡p-模表示重复这些步骤;（4）根据2-范畴的作用来解释这一点。

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.