Geometric Methods in the Local Langlands Correspondance for p-adic Groups.

p-adic 群的局部 Langlands 对应中的几何方法。

• 批准号: RGPIN-2020-05316
• 负责人: Fiori, Andrew
• 金额: $ 1.68万
• 依托单位: University of Lethbridge
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740505
• 关键词: Geometric; Methods; Local; Langlands; Correspondance

This research program is part of the local Langlands program for p-adic groups. The ambitious long term aim of this research program is the development of a categorical local Langlands correspondence for p-adic groups. The Langlands program is one of the major themes of modern mathematics and consists of a series of conjectures spanning number theory, representation theory, and the theory of automorphic forms. The Langlands program conjectures a correspondence between automorphic representations and representations of Galois groups and includes studying both the existence and functorialities of this correspondence. The local Langlands correspondence for p-adic groups is the part of this program related to algebraic and Galois groups over local fields. Though this correspondence is known in many cases, a conjectural interpretation of the correspondence as a series of functors has not been made. Even a conjectural description of the correspondence in terms of functors would be significant as this would allow for more systematic treatments in those already established cases and allow for proofs which will translate to the remaining unresolved cases. Our approach builds off of ideas of David Vogan, who introduced into our context the equivariant derived category of sheaves with constructable cohomology on the moduli space of Langlands parameters.  This geometric category is used to form a bridge between modules for Hecke algebras (attached to automorphic representations) and Langlands parameters (which are in turn associated to Galois representations). The precise aim of our program is to work towards understanding this bridge as a series of functors. Though the ultimate objectives are ambitious they lead us naturally towards two major research directions. The first research direction concerns the development of an explicit and workable description of the aforementioned geometric category. Within this theme, we propose to develop effective computational techniques to work explicitly with these objects. A second research direction is to reformulate the many functorialities of the Langlands correspondence in terms of functors on these geometric categories. There are reasons to believe that in many cases what one finds in the geometric context is in fact easier to describe than our current descriptions of these functorialities. Aside from the contribution these two tasks would make to our theoretical understanding, this work has the added benefit of providing an alternative approach to performing actual computations in the Langlands program. The development of explicit tools for working with these conjectured functorialities then provides an important, and currently often lacking, ability to test, explore, refine and even correct our existing conjectures.

该研究计划是 p-adic 群体的本地 Langlands 计划的一部分。该研究计划的长期目标是为 p-adic 群体开发明确的本地 Langlands 对应关系。 Langlands 计划是该计划的主要主题之一。现代数学，由一系列涵盖数论、表示论和自守形式理论的猜想组成，朗兰兹纲领猜想了自守表示和伽罗瓦群表示之间的对应关系，并包括研究存在性和自守形式。 p 进数群的局部朗兰兹对应关系是与局部域上的代数群和伽罗瓦群相关的程序的一部分，尽管这种对应关系在许多情况下是已知的，但该对应关系的推测解释是一系列函子。即使是函子方面的对应关系的推测性描述也很重要，因为这将允许对那些已经建立的案例进行更系统的处理，并允许将证据转化为我们剩余的未解决的案例。该方法建立在 David Vogan 的思想之上，他在朗兰兹参数的模空间上引入了具有可构造上同调的滑轮的等变派生范畴​此几何范畴用于在 Hecke 代数模块之间形成桥梁（附加到自守）。表示）和朗兰兹参数（又与伽罗瓦表示相关）。我们程序的确切目标是努力将这座桥理解为一系列函子，尽管最终目标是雄心勃勃的。我们自然会走向两个主要研究方向。第一个研究方向涉及对上述几何类别进行明确且可行的描述，我们建议开发有效的计算技术来明确地处理这些对象。用这些几何范畴上的函子重新表述朗兰兹对应关系的许多函子性 有理由相信，在许多情况下，在几何背景下发现的内容实际上比我们目前对这些函子性的描述更容易描述。这两个人的贡献任务将有助于我们的理论理解，这项工作的另一个好处是提供了一种在朗兰兹程序中执行实际计算的替代方法，然后开发用于处理这些推测的函数的显式工具提供了一种重要的但目前经常缺乏的能力。去测试、探索、完善甚至纠正我们现有的猜想。