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

局部朗兰兹猜想可以被视为提供了两种重要的联系-第一，在同一矩阵群之间，取两个不同域中的系数；第二，两个不同矩阵组之间，取同一域中的系数。这些猜想解释了为什么实群和p-进群，甚至非实群的表示理论如此相似。这种一致性的代价是，通常，人们可能不再谈论个别表示，而是谈论它们的有限集合(称为L包)。它们有望同时编码数论(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。对于一些代表家属来说，调查人员对L的包裹应该是什么已经有了明确的期望，但他们还不能证明他们的期望是正确的。对于其他家庭来说，对于答案应该是什么，甚至没有合理的猜测。实群理论从对称空间的角度提出了另一种有益的观点。对于p-add群，对这类空间上的调和分析的认真研究才刚刚开始，这在很大程度上是由研究人员开展的，而在这种局部朗兰兹猜想的背景下，类似的情况还远远不清楚。局部朗兰兹猜想中涉及函数性的部分也表明，研究者应该能够在不同的矩阵群之间传递表征理论信息。这一点的一个经典实现是提升理论，其中大域上的矩阵群的表示与同一群的矩阵群的表示有关，但系数取在较小的域中。这一领域的大多数进展都是通过某种特殊的方法，但答案总是与对称空间环境中出现的自然结构有关。广义上的表征理论起源于两个经典问题。第一个是由傅立叶在19世纪研究的，它试图通过将复杂的物理过程表示为更简单的过程的组合来理解复杂的物理过程，如热扩散。第二个最初是由Frobenius、Schur和其他人研究的，它试图通过一个称为群行列式的相关多项式来理解对称的有限集合的结构。令人惊讶的事实是，这两个问题的解决方案是相关的，事实证明，这只是一系列深刻而深远的联系的最早例子，这些联系已经在一系列猜想中形式化，统称为(当地)朗兰兹猜想。这些猜想的深度和广度-例如，它们包含了数百年前费马大定理最近著名的证明的很大一部分-意味着到目前为止进展相对缓慢。这个项目汇集了一群来自广泛相关背景的数学家，他们的综合专业知识有望在这些猜想以及表示理论和调和分析的相关结果方面取得重大进展。

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.