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部场的朗兰兹猜想，这是两类反常束之间的对偶性的结果。局部朗兰兹猜想保证了局部域的伽罗瓦表示和局部域上约化群的可容许表示之间的精确关系，因此，它是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已经为人所知，最近由于亚瑟惊人的新结果，但一般情况仍然开放，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，我对非阿基米德局部场的朗兰兹猜想的方法最终依赖于对可容许分布的一种新的几何和分类观点，这种观点改编自George Lusztig关于字符束的工作。这种适应依赖于算术几何中的一系列技术，包括非阿基米德局部域上某些代数变体的光滑积分模型、它们的格林伯格变换和Serre-Hazewinkel类场论的新改进。***这个研究项目开始于2011年，当时我在Oberwolfach数学研究所与合作者Pramod Achar， Masoud Kamgarpour和Hadi Salmasian一起工作。在那里，我们了解了如何对所有准分裂群p进G的完全纯朗兰兹参数进行几何化和分类，使用的是由G的朗兰兹群构成的正变体上的等变反常束。被这幅图的优美所折折感，我们开始考虑在由G本身构成的正变体上的等变反轨道反常束的对偶范畴。并且敢于梦想朗兰兹对应性可以用傅里叶-穆凯变换之类的东西来理解。这个想法，我们称之为斯特拉斯堡梦，引导David Roe和我提出了环面在非阿基米德局部域上的准字符束的概念，并证明了环面的准字符和朗兰兹参数都被编码在准字符束中。甚至这种情况也是非常丰富和令人惊讶的，它包含了几何类场论和环面的纯理性形式。***本研究计划的下一步是通过将Lusztig的字符束构造应用于Takashi Suzuki最近开发的一个范畴，找到准分裂群的准字符束的正确概念。准字符轴将是具有局部场的韦尔群作用的简单逆轴（与附近的周期相比）。关键思想是将Langlands参数和可容许分布都编码在准字符串中，并利用这种编码建立非阿基米德局部域上准分裂群的Langlands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.**