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

这项研究计划是当地针对p-add族的朗兰兹计划的一部分。这一研究计划雄心勃勃的长期目标是为p-Add群发展一种范畴局部朗兰兹对应。朗兰兹程序是现代数学的主要主题之一，由一系列横跨数论、表示论和自同构形理论的猜想组成。朗兰兹程序猜想了伽罗华群的自同构表示和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函数性。本程序的主要内容是关于局部域上的代数群和伽罗华群的局部Langland对应。虽然这种对应关系在许多情况下是已知的，但还没有人把这种对应关系解释为一系列函子。即使是用函数式的猜测性描述来描述对应关系，也将具有重要意义，因为这将允许对那些已经确定的案件进行更系统的处理，并允许将证据转化为剩余的未解决案件。我们的方法建立在David Vogan的思想基础上，他在我们的上下文中引入了在Langland参数的模空间上具有可构造上同调的等变上同调的层的派生范畴。这个几何范畴被用来在Hecke代数(附加到自同构表示)和Langland参数(反过来又关联到Galois表示)的模之间形成一座桥梁。我们计划的确切目标是努力将这座桥理解为一系列的工作人员。虽然最终目标是雄心勃勃的，但它们自然会引导我们走向两个主要的研究方向。第一个研究方向是发展对上述几何范畴的明确和可行的描述。在这个主题下，我们建议开发有效的计算技术来显式地处理这些对象。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数。有理由相信，在许多情况下，人们在几何环境中发现的东西实际上比我们目前对这些功能的描述更容易描述。除了这两项任务将对我们的理论理解做出贡献外，这项工作还有一个额外的好处，即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，开发用于处理这些猜想功能的显式工具提供了一种重要的、目前通常缺乏的测试、探索、提炼甚至纠正我们现有猜想的能力。

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.