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

局部朗兰兹图可以被看作是提供了两种重要的连接-第一，在同一矩阵组之间，在两个不同的领域采取的系数;第二，在两个不同的矩阵组之间，在同一领域采取的系数。这些理论解释了为什么真实的和p-adic，甚至是adelic群的表示理论如此相似。这种一致性的代价是，人们通常不再谈论单个表示，而是谈论它们的有限集合（称为L-包）。这些都有望编码数论（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）的信息。对于某些表示族，研究人员已经对L-包应该是什么有了明确的预期，但他们还不能证明他们的预期是正确的。对于其他家庭来说，甚至没有一个合理的猜测答案应该是什么。从对称空间的观点来看，真实的群理论提出了另一种有益的观点。对于p-adic群，对这类空间的调和分析的认真研究才刚刚开始，主要是由研究者进行的，在这种背景下的局部朗兰兹拓扑的类似物还远不清楚。局部朗兰兹图中处理函性的部分也表明，研究者应该能够在不同的矩阵群之间传递表示论信息。一个经典的实现是提升理论，其中矩阵群在一个大的域上的表示与同一个群的表示相关，但系数在一个较小的域中。这一领域的大多数进展都是通过一些特别的方法，但答案总是与在空间环境中产生的自然结构有关。第一种是傅立叶在世纪研究的，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，最初由弗罗贝纽斯，舒尔和其他人研究，是试图通过一个相关的多项式来理解有限对称集合的结构，称为其群行列式。这两个问题的解决方案是相关的，这一令人惊讶的事实被证明只是一个家庭的最早的例子，这些家庭的深刻而深远的联系已经被正式化，在一系列被统称为（本地）朗兰兹图的集合中。这些理论的深度和广度--例如，它们包含了最近对有几百年历史的费马大定理的著名证明的很大一部分--意味着到目前为止进展相对缓慢。这个项目汇集了一组数学家从各种各样的相关背景，其综合的专业知识，可以预期，使显着的进展都在这些programmures和相关的结果在表示论和谐波分析。

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.