Representations of Kac-Moody Groups and Applications to Automorphic Forms

这一建议是表示论和数论的交集，涉及一类被称为Kac-Moody群的无限维群。我们既专注于开发工具来具体应用于数论中的著名问题，也专注于引入无限维表示论中的新结构。*应用于数论*循环群上的Eisenstein级数理论(无限维Kac-Moody群中最简单的)有两个相关的类型--数域和函数域。在函数域的设置上，与H.Garland和S.D.Miller(在A.Braverman和D.Kazhdan工作的基础上)将自同构的L函数从有限维到无限维提出了一个精确的建议。为了使这个项目取得成果，人们需要分析地继续一个新的对象，这个新对象出现在所谓的负尖端爱森斯坦级数。我们对数域设置的理解滞后于函数域设置，这是因为缺乏实循环群的球函数理论。受概率论和量子群技巧的启发，我们建议研究这些群上的一种高斯测度，并从中提取球函数及其极限的具体公式。*在最近与A.Puskas的工作中，我们引入了任何单连通Kac-Moody群的亚辛覆盖，并研究了它上的局部Whittaker函数。长期以来，人们一直期望Eisenstein级数在这种覆盖下的整体傅立叶系数与Dirichlet L函数的矩有关。在建立了这个方向的第一个环节后，我们建议继续这些研究。*推广我的博士论文(在函数域环境中)的工作，我们提出了数域上循环群的算术商数的几何解释。我们期望这与最近关于形式曲面的Arakelov理论有关(由于J.B.-Bost)，并且我们计划应用我们的解释来估计算术曲面上的丛的某些模空间的有理点的数目。*Kac-Moody群表示理论中的新技巧*基于我们以前与A.Braverman和D.Kazhdan在Hecke代数上的合作，我们现在建议研究尖点表示的概念，它是所有表示的基石，并研究它们的L函数。与D.Muthiah一起，我们还建议研究双仿射Kazhdan-Lusztig函数的概念。*对某些弦理论的低能量极限感兴趣的物理学家被引导去研究某些特殊李群上的“最小”自同构表示。基于非凡的计算，他们还假设了无限维群的这种表示的存在。我们建议发展无限维Kirillov-Duflo轨道理论的各个方面来从数学上构造这些轨道。**

This proposal lies at the intersection of representation theory and number theory and concerns a class of infinite-dimensional groups known as Kac-Moody groups. We focus both on developing tools necessary for concrete applications to well-known questions in number theory, and also on introducing new constructions in infinite-dimensional representation theory.******Applications to Number Theory******The theory of Eisenstein series on loop groups (the simplest of the infinite dimensional Kac-Moody groups) comes in two related flavours-- number fields and the function fields. In the function field setting, a precise proposal was put forward in joint work with H. Garland and S.D. Miller (building on work of A. Braverman and D. Kazhdan) relating automorphic L-functions from finite dimensions to infinite-dimensions. To carry this project to fruition, one needs to analytically continue a new object which appearsthe so called negative cuspidal Eisenstein series. Our understanding in the number field setting lags behind the function field setting because of the absence of a theory of spherical functions for real loop groups. Inspired by techniques from probability theory and quantum groups, we propose to study a Gaussian measure on these groups and extract from them concrete formulas for spherical functions and their limits.******In recent work with A. Puskas, we have introduced a metaplectic cover of any simply-connected Kac-Moody group and studied local Whittaker functions on it. It has long been expected that global Fourier coefficients of Eisenstein series on such a cover should be linked to very concrete questions about moments of Dirichlet L-functions. Having made the first links in this direction, we propose to continue these studies.******Generalizing the work of my PhD thesis (which was in the function field setting), we propose a geometric interpretation of arithmetic quotients of loop groups over number fields. We expect this to be connected to a recent Arakelov theory for formal surfaces (due to J.B.-Bost) and we plan to apply our interpretation to estimate for the number of rational points of certain moduli spaces of bundles on an arithmetic surface. ******New Techniques in Representation Theory of Kac-Moody Groups******Based on our previous joint work with A. Braverman and D. Kazhdan on Hecke algebras, we now propose to investigate the notion of cuspidal representations, the building blocks of all representations, and study their L-functions. Together with D. Muthiah, we also propose to study a notion of double affine Kazhdan-Lusztig functions.******Physicists interested in low-energy limits of certain string theories were led to studying “minimal” automorphic representations on certain exceptional Lie groups. They also postulated the existence, based on remarkable calculations, of such representations for the infinite-dimensional groups. We propose to develop aspects of a Kirillov-Duflo orbital theory in infinite-dimensions to mathematically construct these. **