Geometrization of Admissible Distributions and the Local Langlands Conjecture

我的研究计划的长期目标是证明非阿基米德局域场的朗兰兹猜想，这是两类倒立叶之间的对偶的结果。局部朗兰兹猜想保证了局部域的伽罗瓦表示和局部域上的约化群的可容许表示之间的精确关系，因此，这是一个在数论中具有重要意义的中心问题。许多局部朗兰兹猜想的例子现在是已知的，最近由于Arthur的惊人的新结果，但一般情况仍然是开放的，似乎这个主题可能会从新的视角中受益。*在利用最近的进展的同时，我为非阿基米德局部场证明朗兰兹猜想的方法最终依赖于一种关于可容许分布的新的几何和分类观点，该观点改编自George Lusztig关于字符片断的工作。这种适应依赖于算术几何的大量技术，包括非阿基米德局部场上某些代数变体的光滑积分模型、它们的Greenberg变换和Serre-Hazewinkel类场论的新改进。*这个研究项目始于2011年，当时我与合作者Pramod Achar、Masoud Kamgarour和Hadi Salmasian在数学研究所Oberwolfach工作。在这里，我们了解了如何使用从G的朗兰兹群建立的内簇上的等变倒置轨道来几何化和分类所有准分裂群p-add G的完全纯朗兰兹参数。被这张图片的优雅所淹没，我们开始思考由G本身建立的亲族上的等变反轨道倒转轨道弯曲轨道的“对偶”类别，并敢于梦想使用类似于这些类别之间的傅里叶-穆凯变换的东西来理解朗兰兹对应。这个想法，我们称之为斯特拉斯堡梦，导致David Roe和我提出了非阿基米德局部场上Tori的准特征层的概念，并证明了Tori的准特征层和朗兰兹参数都是以准特征层编码的。即使是这种情况也是非常丰富和令人惊讶的，它包括(它必须包括)几何类域理论和环面的纯有理形式。*本研究计划的下一步是通过将Lusztig的特征线构造改写为Takashi Suzuki最近开发的一个范畴，找到准分裂群的准特征线线组的正确概念。准字符滑轮将是简单的倒置滑轮，带有局域场的Weil群的作用(与附近的周期相比较)。其核心思想是，朗兰兹参数和允许分布都将被编码在准特征线束中，并利用这种编码来建立非阿基米德局部域上准分裂群的朗兰兹猜想。

The long-term objective of my research program is to prove the Langlands Conjecture for non-Archimedean local fields as a consequence of a duality between two categories of perverse sheaves. The local Langlands Conjecture promises a precise relation between Galois representations of local fields and admissible representations of reductive groups over local fields and, as such, is a central problem with important implications in number theory. Many instances of the local Langlands Conjecture are now known, most recently due to spectacular new results by Arthur, but the general case remains open and it seems that the subject may benefit from new perspectives. *** While capitalizing on recent progress, my approach to the Langlands Conjecture for non-Archimedean local fields ultimately relies on a novel geometric and categorical perspective on admissible distributions that is adapted from George Lusztig's work on character sheaves. This adaptation relies on an arsenal of techniques from arithmetic geometry, including smooth integral models for certain algebraic varieties over non-Archimedean local fields, their Greenberg transforms and new refinements of Serre-Hazewinkel class field theory.*** This research program began in 2011 when I was working at the Mathematisches Forschungsinstitut Oberwolfach with collaborators Pramod Achar, Masoud Kamgarpour and Hadi Salmasian. There we understood how to geometrize and categorify complete pure Langlands parameters for all quasisplit groups p-adic G using equivariant perverse sheaves on an ind-variety built from the Langlands group of G. Overwhelmed by the elegance of this picture, we started to think about a `dual' category of equivariant anti-orbital perverse sheaves on a pro-variety built from G itself, and dared to dream that the Langlands Correspondence might be understood using something like the Fourier-Mukai transform between these categories. This idea, which we call the Strasbourg Dream, led David Roe and me to the notion of quasicharacter sheaves for tori over non-Archimedean local fields and the proof that both quasicharacters and Langlands parameters for tori are encoded in quasicharacter sheaves. Even this case is very rich and surprising, encompassing, as it must, both geometric class field theory and pure rational forms of tori.*** The next step in this research program is to find the correct notion of quasicharacter sheaves for quasisplit groups by adapting Lusztig's construction of character sheaves to a category developed recently by Takashi Suzuki. Quasicharacter sheaves will be simple perverse sheaves equipped with an action of the Weil group of the local field (compare with nearby cycles). The key idea is that both Langlands parameters and admissible distributions will be encoded in quasicharacter sheaves, and to use this encoding to establish the Langlands Conjecture for quasisplit groups over non-Archimedean local fields.**