Local Langlands correspondence for reductive p-adic groups

还原 p-adic 群的局部 Langlands 对应

• 批准号: 1303418
• 负责人: Mark Reeder
• 金额: $ 16.9万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303418&HistoricalAwards=false
• 关键词: Local; Langlands; correspondence; reductive; adic

This proposal is about interactions between Representation Theory and Number Theory, as predicted by the conjectural Local Langlands Correspondence (LLC) for general reductive groups. The LLC is a body of predicted relations between the representation theory of local Galois groups and the representation theory of reductive groups over a p-adic field. The goals of the work in this proposal are aimed toward an explicit understanding of the LLC. There are four topics: i) Extending the proposer's earlier work introducing Geometric Invariant Theory to study representations of p-adic groups, in particular the recently-constructed 'epipelagic' representations. ii) Study of discrete Langlands parameters, including proof of a conjectured inequality for adjoint Swan conductors, atypical behavior at small primes, parameters for Yu's representations, classification of depths and mass formulas. iii) Uniqueness results for the LLC. iv) Jordan decomposition of depth zero representations and LLC.The mathematics in this proposal is rooted in two ancient topics of mathematics: Representation Theory (the study of symmetry) and Number Theory (numerical solutions of equations). Though these appear to be two quite different areas of mathematics, the Local Langlands Correspondence predicts surprising relations between them. Roughly speaking, it says that certain kinds of infinite dimensional symmetries should correspond to certain equations whose solutions have a related collection of finite dimensional symmetries. The aims of the proposal are first, to discover and explicitly verify new and interesting examples of the LLC, and second, to use the predictions of the LLC to make new discoveries in Number Theory and Representation Theory.

这一建议是关于表示论和数论之间的相互作用，正如一般约化群的猜想局部朗兰兹对应(LLC)所预测的那样。LLC是p-ady域上局部Galois群的表示理论和约化群的表示理论之间的一系列预测关系。本提案中工作的目标旨在明确了解有限责任公司。本文共分四个主题：i)推广了作者早期的工作，引入几何不变理论来研究p-Add群的表示，特别是最近构造的‘上域’表示。Ii)离散朗兰兹参数的研究，包括对伴随天鹅导体的一个猜想不等式的证明，小素数的非典型行为，Yu表示的参数，深度的分类和质量公式。Iii)有限责任公司的唯一性结果。IV)深度零表示和LLC的Jordan分解。这项建议中的数学植根于两个古老的数学主题：表示论(对称性研究)和数论(方程的数值解)。虽然这似乎是两个完全不同的数学领域，但当地朗兰兹的通信预测了它们之间令人惊讶的关系。粗略地说，它说某些类型的无限维对称应该对应于某些方程，这些方程的解具有相关的有限维对称集合。该提案的目的首先是发现并明确验证有限责任公司的新的有趣的例子，其次是利用有限责任公司的预测在数论和表示理论中做出新的发现。