FRG: Collaborative Research: Characters, Liftings, and Types: Investigations in p-adic Representation Theory

FRG：协作研究：特征、提升和类型：p-adic 表示理论的调查

• 批准号: 0854909
• 负责人: Mark Reeder
• 金额: $ 4.7万
• 依托单位: Boston College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0854909&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Characters; Liftings

The local Langlands conjectures can be viewed as offering two important kinds of connections---first, between the same matrix group, taken with coefficients in two different fields; and second, between two different matrix groups, taken with coefficients in the same field. The conjectures offer an explanation for why the representation theories of real and p-adic, and even of adelic, groups are so similar. The price of this uniformity is that, often, one may no longer speak of individual representations, but rather of finite collections of them (called L-packets). These are expected to encode both number-theoretic (via Galois groups) and algebro-geometric (via many avenues---for example, the theory of stable distributions) information. For some families of representations, the investigators already have clear expectations for what the L-packets should be, but they cannot yet prove that their expectations are correct. For other families, there is not even a reasonable conjecture for what the answer should be. The theory of real groups suggests yet another rewarding perspective, from the point of view of symmetric spaces. For p-adic groups, the serious study of harmonic analysis on such spaces is just beginning to be developed, in large part by the investigators, and the analogues in this setting of the local Langlands conjectures are far from clear. The part of the local Langlands conjectures dealing with functoriality also suggests that the investigators should be able to transfer representation-theoretic information between different matrix groups. A classical realization of this is the theory of lifting, where representations of matrix groups over a large field are related to those of the same group, but with coefficients taken in a smaller field. Most progress in this area has been via somewhat ad hoc methods, but the answers have invariably turned out to be related to natural constructions arising in the symmetric-space setting.Representation theory, broadly understood, has its origins in two classical problems. The first, investigated by Fourier in the 19th century, was an attempt to understand complicated physical processes, such as heat diffusion, by representing them as combinations of simpler processes. The second, initially studied by Frobenius, Schur, and others, was an attempt to understand the structure of a finite collection of symmetries via an associated polynomial known as its group determinant. The surprising fact that the solutions to these two problems are related has turned out to be just the earliest instance of a family of deep and far-reaching connections that have been formalized in a collection of conjectures known collectively as the (local) Langlands conjectures. The depth and broad reach of these conjectures---for example, they encompass a large part of the celebrated recent proof of the centuries-old Fermat's Last Theorem---has meant that progress has so far been relatively slow. This project brings together a group of mathematicians from a broad variety of related backgrounds, whose combined expertise can be expected to allow significant progress both on these conjectures and on related results in representation theory and harmonic analysis.

局部朗兰兹猜想可以看作是提供了两种重要的联系——第一，在相同的矩阵群之间，在两个不同的域中取系数；第二，在两个不同的矩阵群之间，取同一域中的系数。这些猜想提供了一种解释，解释了为什么实群和虚群，甚至是幻群的表征理论如此相似。这种一致性的代价是，人们常常不再谈论单个表示，而是谈论它们的有限集合（称为l包）。它们被期望编码数论（通过伽罗瓦群）和代数几何（通过许多途径——例如，稳定分布理论）信息。对于一些表示族，研究者已经对l包应该是什么有了明确的期望，但是他们还不能证明他们的期望是正确的。对其他家庭来说，答案甚至连一个合理的猜测都没有。实群理论从对称空间的角度提出了另一个有益的观点。对于p进群，在这样的空间上进行谐波分析的严肃研究才刚刚开始，在很大程度上是由研究人员进行的，而在这种局部朗兰兹猜想的背景下的类似物还远远不够清楚。局部朗兰兹猜想处理功能的部分也表明研究者应该能够在不同的矩阵群之间传递表征论信息。一个经典的实现是提升理论，其中大域上矩阵群的表示与相同群的表示相关，但系数取在较小的域上。这一领域的大多数进展都是通过一些特别的方法取得的，但答案总是与对称空间环境中产生的自然结构有关。广义的表征理论起源于两个经典问题。第一个是傅立叶在19世纪研究的，他试图通过将复杂的物理过程（如热扩散）表示为简单过程的组合来理解它们。第二个，最初由Frobenius， Schur和其他人研究，是试图通过一个被称为群行列式的相关多项式来理解有限对称集合的结构。这两个问题的答案是相互关联的，这一令人惊讶的事实证明，这只是一系列深刻而深远的联系的最早例子，这些联系已被形式化地形成了一系列猜想，统称为（当地的）朗兰兹猜想。这些猜想的深度和广度——例如，它们包含了最近著名的几个世纪前的费马大定理的大部分证明——意味着迄今为止进展相对缓慢。该项目汇集了一组来自各种相关背景的数学家，他们的综合专业知识有望在这些猜想以及表示理论和谐波分析的相关结果上取得重大进展。