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

这个研究计划是局部朗兰兹计划的一部分，为p-adic集团。这项研究计划的雄心勃勃的长期目标是为p-adic群发展一个分类的局部朗兰兹对应。朗兰兹纲领是现代数学的主要主题之一，它由一系列跨越数论、表示论和自守形式理论的理论组成。朗兰兹计划描述了自守表示和伽罗瓦群表示之间的对应关系，并包括研究这种对应关系的存在性和功能性。p-adic群的局部朗兰兹对应是本程序中与局部域上的代数群和伽罗瓦群相关的部分。 虽然这种对应关系在许多情况下是已知的，但还没有将这种对应关系解释为一系列函子。即使是用函子来描述对应关系也是很重要的，因为这将允许对那些已经成立的案例进行更系统的处理，并允许证明将转化为剩余的未解决的案例。我们的方法建立在大卫沃根的思想基础上，沃根在我们的上下文中引入了在朗兰兹参数的模空间上具有可构造上同调的层的等变导出范畴。 这个几何范畴被用来在Hecke代数的模（附在自守表示上）和Langlands参数（又与伽罗瓦表示相关联）之间形成桥梁。我们的程序的确切目标是努力将这个桥理解为一系列函子。虽然最终的目标是雄心勃勃的，但它们自然地将我们引向两个主要的研究方向。第一个研究方向涉及上述几何范畴的明确和可行的描述的发展。在这个主题中，我们建议开发有效的计算技术来明确地处理这些对象。第二个研究方向是用这些几何范畴上的函子来重新表述朗兰兹对应的许多函性。有理由相信，在许多情况下，人们在几何背景下发现的东西实际上比我们目前对这些功能性的描述更容易描述。 除了这两项任务将使我们的理论理解的贡献，这项工作有额外的好处，提供了一种替代方法来执行实际计算的朗兰兹程序。开发明确的工具来处理这些复杂的功能，然后提供了一个重要的，目前经常缺乏的能力来测试，探索，改进甚至纠正我们现有的结构。

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.