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.

该提议位于表示论和数论的交叉点，涉及一类称为 Kac-Moody 群的无限维群。我们既专注于开发数论中众所周知的问题的具体应用所必需的工具，也专注于引入无限维表示论的新结构。数论的应用 环群（最简单的无限维 Kac-Moody 群）的爱森斯坦级数理论有两种相关的风格：数域和函数域。在功能域的设置上，与H. Garland和S.D.共同提出了精确的建议。 Miller（以 A. Braverman 和 D. Kazhdan 的工作为基础）将自同构 L 函数从有限维关联到无限维。为了使这个项目取得成果，我们需要分析地继续出现一个新的对象——所谓的负尖角爱森斯坦级数。由于缺乏实环群的球函数理论，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，我们建议研究这些群的高斯测度，并从中提取球函数及其极限的具体公式。在最近与 A. Puskas 的合作中，我们引入了任何单连通 Kac-Moody 群的元波覆盖，并研究了其局部 Whittaker 函数。长期以来，人们一直期望这种封面上的爱森斯坦级数的全局傅立叶系数应该与有关狄利克雷 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初步联系后，我们建议继续这些研究。概括我的博士论文（在函数域设置中）的工作，我们提出了数字域上循环群算术商的几何解释。我们期望这与最近的形式表面 Arakelov 理论（由 J.B.-Bost 提出）相关，并且我们计划应用我们的解释来估计算术表面上丛的某些模空间的有理点的数量。 Kac-Moody 群表示理论的新技术基于我们之前与 A. Braverman 和 D. Kazhdan 在 Hecke 代数方面的合作，我们现在建议研究尖表示的概念（所有表示的构建块），并研究它们的 L 函数。我们还建议与 D. Muthiah 一起研究双仿射 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的“最小”自守表示。他们还基于显着的计算假设存在无限维群的这种表示。我们建议发展无限维基里洛夫-迪弗洛轨道理论的各个方面，以数学方式构建这些理论。