Categorical consequences of the microlocal perspective on Arthur packets for p-adic groups

这NSERC DG应用程序涉及的研究在朗兰兹计划有关亚瑟包的p进群。本研究的最终目的是探索局部朗兰兹对应的范畴形式的一个具体概念。在20世纪90年代早期，大卫沃根（David Vogan）引入了利用微局部分析的思想，在朗兰兹参数的模空间上研究p-adic群的亚瑟包。这个想法与杰弗里·亚当斯（Jeffrey Adams）和丹·巴巴什（Dan Barbasch）在1992年的著作《朗兰兹分类和实数群的不可约特征标》（The Langlands Classification and Irreducible Characters of Real Groups）中提出的真实的群的构造是平行的。沃根的想法p进群提出了他在1993年的论文本地朗兰兹对应。然而，直到詹姆斯·亚瑟在他2013年的著作《表示的内窥镜分类：正交和辛群》中给出了p-adic群的亚瑟包的纯局部描述，才很难测试沃根的想法。迄今为止，这些问题仍然悬而未决。在亚瑟包的p-adic群的方式的微局部消失周期的反常层，与例子克利夫顿坎宁安，安德鲁菲奥里，艾哈迈德Moussaoui，詹姆斯Mracek和徐斌，回忆录的美国数学学会（在出版），我的合作者和我已经适应了许多结果的书由亚当斯，Barbasch和Vogan从真实的群体的p-adic群体。我们指的是我们的回忆录中描述的ABV-包的p-adic群的可接受表示的包。该提案的主要目标是证明经典p-adic群的纯内部形式的亚瑟包是ABV包，并证明我们在有关稳定分布及其内窥镜转移的书中出现的该猜想的细化。我有相当多的证据证明这一猜想的形式的例子。与我的研究小组一起，我也有一个策略来证明准分裂经典群的纯内部形式的这些命题，但是因为我们需要与亚瑟的工作进行仔细的比较，我们的论证逐类进行;有了GL（n），我们将从SO（2n+1）及其纯内部形式开始研究经典群。在这项研究的核心是比较两类：表示的光滑表示固定尖点支持;和等变的反常层的模空间的某些朗兰兹参数。当正确校准时，这两个类别具有相同的Grothendieck群。然而，我们知道，这些类别一般来说并不等同。我最近的工作建议对这些范畴中的每一个进行修改，以确定它们是Koszul对偶的。这个想法是出现在这个研究计划的标题中的亚瑟包的微局部视角的分类结果，也是我的最终目标：利用我们对亚瑟包的研究，我希望阐明p进群的分类局部朗兰兹对应。

This NSERC DG application concerns research in the Langlands programme regarding Arthur packets for p-adic groups. The ultimate objective of this research is to explore a specific idea for a categorical form of the local Langlands correspondence. In the early 1990s, David Vogan introduced the idea of using microlocal analysis on a moduli space of Langlands parameters to study Arthur packets for p-adic groups. This idea is parallel to a construction for Real groups developed in conjunction with Jeffrey Adams and Dan Barbasch in their 1992 book The Langlands Classification and Irreducible Characters of Real Groups. Vogan's idea for p-adic groups was presented in his 1993 paper The Local Langlands Correspondence. However, until James Arthur produced a purely local description of Arthur packets for p-adic groups in his 2013 book The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, it was difficult to test Vogan's ideas. They remain open problems to this date. In Arthur packets for p-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples by Clifton Cunningham, Andrew Fiori, Ahmed Moussaoui, James Mracek and Bin Xu, Memoirs of the American Mathematical Society (in press), my collaborators and I have adapted many of the results of the book by Adams, Barbasch and Vogan from Real groups to p-adic groups. We refer to the packets of admissible representations of p-adic groups described in our memoir as ABV-packets. The main objective of this proposal is to prove that Arthur packets for pure inner forms of classical p-adic groups are ABV-packets and to prove the refinement of this conjecture appearing in our book pertaining to stable distributions and their endoscopic transfers. I have considerable evidence for this conjecture in the form of examples. Together with my research group, I also have a strategy for the proof of these conjectures for pure inner forms of quasi-split classical groups but because we need to make a careful comparison with Arthur's work, our argument proceeds type by type; with GL(n) in hand, we will work our way through the classical groups beginning with SO(2n+1) and its pure inner forms. At the heart of this research is a comparison of two categories: representations of smooth representations with fixed cuspidal support; and equivariant perverse sheaves on the moduli space of certain Langlands parameters. When properly calibrated, these two categories have the same Grothendieck groups. However, we know that these categories are not, in general, equivalent. My recent work suggests a modification to each of these categories that might be used to establish that they are Koszul dual. This idea is the categorical consequence of the microlocal perspective on Arthur packets that appears in the title of this research proposal and is my ultimate objective: using our work on Arthur packets, I expect to shed light on a categorical local Langlands correspondence for p-adic groups.