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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758360
• 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 appears-the 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.

该提议在于表示理论和数字理论的交集，并涉及一类称为kac-moody群体的无限二维群体。我们既专注于开发具体应用必要的工具，以在数量理论中为众所周知的问题，以及在无限维代表理论中引入新的结构。在数字理论上的应用艾森斯坦在循环组上的序列（无限尺寸kac-moody群体中的最简单）属于两个相关的口味 - 数字字段和功能字段。在功能领域设置中，与H. Garland and S.D.共同合作提出了一项精确的建议。米勒（基于A. Braverman和D. Kazhdan的作品建设），将有限尺寸的自动型L功能与无限维度有关。为了实现这个项目，需要在分析上继续一个新对象，该对象似乎是cuspidal Eisenstein系列。由于缺乏实际循环组的球形函数理论，我们在数字场设置滞后的理解滞后。受概率理论和量子群的技术的启发，我们建议研究对这些组的高斯措施，并从中提取具体公式，以实现球形函数及其限制。在与A. Puskas的最新工作中，我们引入了任何简单相关的Kac-Moody组的Metapcip封面，并研究了当地的Whittaker功能。长期以来，人们一直期望艾森斯坦（Eisenstein）系列的全球傅立叶系数在这种覆盖中应与关于迪里奇（Dirichlet L）函数时刻的非常具体的问题有关。在朝这个方向建立了第一个链接后，我们建议继续这些研究。概括了我的博士学位论文的工作（在函数字段设置中），我们提出了对循环组的算术商在数字字段上的几何解释。我们希望这将与最近的Arakelov理论相关联（由于J.B. BOST），我们计划将解释应用于算术表面上某些捆绑包的某些模量空间的理性点的数量。 KAC-MOODY群体代表理论的新技术基于我们以前与A. Braverman和D. Kazhdan在Hecke代数上的联合合作，我们现在建议研究Cuspidal表示的概念，所有表示的构件，所有表示的基础，并研究其L-功能。我们还与D. muthiah一起研究了双仿射Kazhdan-Lusztig函数的概念。对某些字符串理论的低能限制感兴趣的物理学家导致研究某些特殊的谎言组的“最小”自身形态表示。他们还基于显着的计算，假定了无限维组的这种表示形式的存在。我们建议在无限维度中开发基里洛夫·杜夫洛轨道理论的各个方面，以数学上构建这些方面。