Representations of p-adic groups and motivic integration

所提议的活动的目标是了解p进群的表示理论在多大程度上可以独立于p。主要工具是动机整合理论，它位于代数几何和逻辑的交叉点。这种方法是T.C.黑尔斯在2001年宣布的一项发展动机表征理论的长期计划的一部分。p进群的调和分析及其群的表示理论以p进积分为基础。动机表征理论旨在通过用M. Kontsevich在1995年引入的某种符号积分（动机积分）取代普通的p进积分，并由J. Denef、F. Loeser和R. Cluckers发展，捕捉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，将有可能将在函数域上证明的结果转移到特征为0的p进域，除了有限多个p，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，并将澄清在表示理论中出现的许多结构对p的依赖。最终，这种方法有望导致计算迄今为止无法计算的数量的算法，例如哈里什-钱德拉字符的值。动机整合也可以是一个强大的类比来源，允许研究非局部紧致值域上的群。在试图研究这类群体时，人们遇到的第一个主要障碍是哈尔测度的缺失。动机整合可以直接使用，或者只是作为一个类比，来提供绕过这个障碍的方法。结果可以潜在地用于数论研究自同构l -函数。

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.