Representations of p-adic groups and motivic integration

p-adic 群的表示和动机整合

• 批准号: 330945-2006
• 负责人: Gordon, Julia
• 金额: $ 2.91万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: University Faculty Award
• 财政年份: 2008
• 资助国家: 加拿大
• 起止时间: 2008-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=398163
• 关键词: Representations; adic; groups; motivic; integration

The goal of the proposed activity is to understand to what extent representation theory of p-adic groups can be done in a way that is independent of p. The main tool is the theory of motivic integration, which lies at the intersection of algebraic geometry and logic. This approach  is part of a long-term program to develop motivic representation theory that was announced by T.C. Hales in 2001.       Harmonic analysis on p-adic groups and representation theory of such groups use p-adic integration at their very basis. Motivic representation theory is intended to capture the ``independent of p'' essence of various constructions of harmonic analysis on p-adic groups by  replacing ordinary p-adic integration with a certain symbolic integration (motivic integration)  introduced by M. Kontsevich in 1995 and developed by J. Denef, F. Loeser, and R. Cluckers.       This approach, if it is successful, will yield the possibility to transfer results proved over function fields to p-adic fields of characteristic 0 for all but finitely many p, and vice versa. It will give us better understanding of the connection between representations of p-adic groups and algebraic geometry, and it will clarify the dependence on p of many constructions appearing in representation theory. Ultimately, this approach is expected to lead to algorithms for calculation of the quantities that have eluded computation so far, such as values of Harish-Chandra characters.          Motivic integration can also be a powerful source of analogies allowing to investigate groups over non-locally compact valued fields.One of the first major obstacles one encounters when trying to study such groups, is the absence of Haar measure. Motivic integration can be used directly, or just as an analogy, to provide a way around this obstacle. The results then can potentially be used in number theory to study automorphic L-functions.

拟议活动的目标是了解 p 进群的表示论可以在多大程度上以独立于 p 的方式完成。主要工具是动机积分理论，它位于代数几何和逻辑的交叉点。这种方法是 T.C. 宣布的开发动机表征理论的长期计划的一部分。 Hales，2001。 p-adic 群的调和分析和此类群的表示论在其基础上使用 p-adic 积分。动机表示理论旨在通过用 M. Kontsevich 于 1995 年引入并由 J. Denef、F. Loeser 和 R. Cluckers 开发的某种符号积分（动机积分）代替普通的 p-adic 积分来捕获 p 进群调和分析的各种结构的“独立于 p”本质。       这种方法如果成功的话，将有可能将函数域上证明的结果转移到特征为 0 的 p 进数域（除了有限多个 p 之外），反之亦然。它将使我们更好地理解 p 进群的表示与代数几何之间的联系，并将阐明表示论中出现的许多构造对 p 的依赖。最终，这种方法预计将导致计算迄今为止无法计算的数量的算法，例如 Harish-Chandra 字符的值。          动机整合也可以是类比的强大来源，允许研究非局部紧凑值域上的群体。在尝试研究此类群体时遇到的第一个主要障碍是缺乏哈尔测度。动机整合可以直接使用，或者只是作为类比，来提供绕过这个障碍的方法。结果可用于数论中研究自同构 L 函数。