Geometrization of Admissible Distributions and the Local Langlands Conjecture

我的研究计划的长期目标是证明朗兰兹猜想的非阿基米德局部领域之间的对偶性两类反常层的后果。局部朗兰兹猜想保证了局部域的伽罗瓦表示与局部域上约化群的可容许表示之间的精确关系，因此是数论中具有重要意义的中心问题。现在人们已经知道了许多局部朗兰兹猜想的例子，最近的一次是由于亚瑟的惊人的新结果，但是一般的情况仍然是开放的，似乎这个主题可能会受益于新的观点。* 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一个新的几何和分类观点，这个观点是从乔治卢斯蒂格关于特征层的工作中改编而来的。这种适应依赖于算术几何的一系列技术，包括非阿基米德局部域上某些代数簇的光滑积分模型，它们的格林伯格变换和Serre-Hazewinkel类场论的新改进。这项研究计划始于2011年，当时我在Oberwolfach数学研究所与合作者Pramod Achar，Masoud Kamgarpour和Hadi Salmasian合作。在那里，我们理解了如何几何化和分类所有拟分裂群p进G的完全纯朗兰兹参数，使用从G的朗兰兹群构建的ind簇上的等变逆层。我们被这幅图画的优美所震撼，开始思考在由G本身构建的亲簇上的等变反轨道反常层的“对偶”范畴，并且敢于梦想朗兰兹对应可以用类似于这些范畴之间的傅里叶-向井变换的东西来理解。这个想法，我们称之为斯特拉斯堡梦，引导大卫罗伊和我的概念，拟字符层的环面超过非阿基米德的地方领域和证明，拟字符和朗兰兹参数环面编码在拟字符层。甚至这个例子也是非常丰富和令人惊讶的，它必然包含几何类场论和环面的纯理性形式。在这项研究计划的下一步是找到正确的概念quasicharacter层的quasissplit组适应Lusztig的建设性质层的一个类别最近开发的铃木隆。拟特征层将是简单的反常层，配备有局部场的Weil群的作用（与附近的圈相比）。关键思想是朗兰兹参数和容许分布都将被编码在拟特征层中，并使用这种编码建立非阿基米德局部域上拟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.**