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

p-adic 群的亚瑟包的微局域视角的分类结果

• 批准号: RGPIN-2020-05220
• 负责人: Cunningham, Clifton
• 金额: $ 1.75万
• 依托单位: University of Calgary
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740461
• 关键词: Categorical; consequences; microlocal; perspective; Arthur

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.

这项NSERC DG申请涉及朗兰兹计划中关于p-ady群的Arthur包的研究。本研究的最终目的是为当地朗兰兹函电的一种范畴形式探索一种具体的思想。20世纪90年代初，David Vogan提出了利用朗兰兹参数模空间上的微局域分析来研究p-add群的Arthur包的思想。这一思想与Jeffrey Adams和Dan Barbasch在他们1992年的著作《实群的朗兰兹分类和不可约特征标》中共同提出的实群的构造是平行的。沃根在1993年的论文《局部朗兰兹通信》中提出了p-进群的概念。然而，直到詹姆斯·阿瑟在他2013年的著作《内窥镜表示分类：正交和辛群》中为p-Add群产生了对Arthur包的纯局部描述之前，很难检验沃根的想法。到目前为止，它们仍然是悬而未决的问题。在由Clifton Cunningham、Andrew Fiori、Ahmed Moussaoui、James MRacek和Bin Xu所著的《美国数学学会回忆录》(出版中)的例子中，我和我的合作者们将Adams、Barbasch和Vogan的书的许多结果从Real群改编到了p-add群。我们将回忆录中所描述的p-进群的可容许表示的包称为ABV-包。这一建议的主要目的是证明经典p-add群的纯内形式的Arthur包是ABV-包，并证明这一猜想在我们的书中出现的关于稳定分布及其内窥镜变换的精炼。我以例子的形式对这个猜想有相当多的证据。与我的研究小组一起，我也有一个策略来证明这些关于准分裂经典群的纯内形式的猜想，但因为我们需要与Arthur的工作进行仔细的比较，所以我们的论证是逐个类型进行的；有了GL(N)，我们将以SO(2n+1)及其纯内形式为起点来研究经典群。这项研究的核心是两种类型的比较：固定尖点支撑的光滑表示法和某些朗兰兹参数的模空间上的等变反叶轮集。当正确校准时，这两个类别具有相同的Grothendieck组。然而，我们知道，总的来说，这些类别并不等同。我最近的工作建议对这些类别中的每一个进行修改，以确定它们是Koszul对偶。这个想法是出现在本研究提案标题中的关于Arthur Packets的微局部视角的绝对结果，也是我的最终目标：利用我们对Arthur Packets的工作，我希望阐明p-Add群的一个范畴局部Langland对应。