Representations of p-adic groups and dg-algebras

自从朗兰兹的猜想与数论有了很深的联系以来，p-add群的表示理论在过去的50年里一直是一个非常活跃的研究领域。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学科分类(22E50)，而在邻近地区有更多的出版物是重叠的。最初，只研究了p-进群的复杂表示，但最近考虑了其他域和环上的表示。数学家特别感兴趣的是，系数场在某种程度上与群中出现的p-ady场相匹配--但这也要困难得多：有许多来自复杂世界的技术不再能使用，更重要的是，许多事情确实不同。这方面的一个具体例子涉及将关于p-进群的表示的问题转化为关于所谓的Hecke代数上的模的问题：对于复杂的表示，这些在某种程度上是相同的，但对于其他域的表示，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，有一种方法可以扩展这一部分图像，以揭示缺失的部分，至少在原则上--该项目是关于将其付诸实践的。Hecke代数是某些对象的“对称性”，但这些对象可以被增强为所谓的“内射解析”--观察这些内射解析的对称性(称为dg Hecke代数)，应该会给出表示的更全面的图景。最终，对所有这些以及群的dg-Hecke代数和某些子群的dg-Hecke代数之间的关系的完整理解，应该是将它们全部联系在一起的一个总体图画的外壳：2-范畴，其2-自同态是由上面的dg-Hecke代数给出的。该项目将从查看小群(GL(1)、GL(2)、GL(3))的复杂表示开始，并试图在最简单的情况下计算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理论还没有被计算--并查看它们之间的关系(例如，适用于GL(1)GL(1)在GL(2)内)。除了产生新的结果，它的主要目的是作为一个试验场，帮助告知下一阶段：为p-模表示重复相同的计算。我们已有明确的路向，但即使对GL(2)来说，就社会利益而言，这也是一项重大成就。因此，主要的(技术)目标是：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；2)计算给出这些DG Hecke代数的抛物线归纳和限制的DG双模，并了解它们的性质；3)对GL(1)、GL(2)和GL(3)的Prop-Iwahori子群的平凡p-模表示重复这些步骤；4)用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.