Representations of Kac-Moody Groups and Applications to Automorphic Forms

Kac-Moody 群的表示及其在自守形式中的应用

• 批准号: RGPIN-2019-06112
• 负责人: Patnaik, Manish
• 金额: $ 1.38万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=677027
• 关键词: Representations; Kac; Moody; Groups; Applications

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. **

这个建议是在表示理论和数论的交叉点上提出的，涉及到一类被称为Kac-Moody群的无限维群。我们既专注于开发必要的工具来具体应用数论中众所周知的问题，也在无限维表示理论中引入新的结构。******数论的应用******在循环群（无限维Kac-Moody群中最简单的）上的爱森斯坦级数的理论有两种相关的风格——数域和函数域。在函数场设置中，与H. Garland和S.D. Miller（在a . Braverman和D. Kazhdan的工作基础上）共同提出了从有限维到无限维的自同构l函数的精确建议。为了实现这一计划，人们需要分析地继续一个新的对象，即所谓的负倒转爱森斯坦系列。由于实环群的球函数理论的缺失，我们对数域设置的理解落后于函数域设置。受概率论和量子群技术的启发，我们提出在这些群上研究高斯测度，并从中提取球面函数及其极限的具体公式。******在最近与a . Puskas的合作中，我们引入了任何单连通Kac-Moody群的元复盖，并研究了其上的局部Whittaker函数。长期以来，人们一直期望在这样一个盖上，爱森斯坦级数的全局傅立叶系数应该与狄利克雷l函数矩的非常具体的问题联系起来。在这个方向上建立了第一个联系之后，我们建议继续这些研究。******推广我的博士论文（这是在函数域设置）的工作，我们提出了环路群在数域上的算术商的几何解释。我们期望这与最近关于形式曲面的Arakelov理论（由于j.b.b - bost）相联系，并且我们计划将我们的解释应用于估计算术曲面上束的某些模空间的有理点的数量。****** Kac-Moody群表示理论中的新技术******基于我们之前与A. Braverman和D. Kazhdan在Hecke代数上的合作工作，我们现在提议研究cuspidal表示的概念，所有表示的构建块，并研究它们的l -函数。我们还与D. Muthiah一起提出研究双仿射Kazhdan-Lusztig函数的概念。******对某些弦理论的低能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。他们还假设存在，基于非凡的计算，无限维群的这种表示。我们建议发展无限维Kirillov-Duflo轨道理论的各个方面，以数学方式构建这些。**