Representations of p-adic groups and motivic integration

拟议活动的目标是了解在何种程度上表示理论的p-adic组可以做的方式是独立的p.主要工具是动机整合，这是在交叉的代数几何和逻辑的理论。这种方法是T.C. 2001年的黑尔斯。 p-adic群的调和分析和这类群的表示论在它们的基础上使用p-adic积分。Motivic表示理论旨在通过用M引入的某种符号积分（motivic积分）代替普通的p-adic积分，来捕捉p-adic群上调和分析的各种构造的“与p无关”本质。Kontsevich于1995年提出，并由J. Denef，F. Loeser和R.咯咯叫。 这种方法，如果它是成功的，将产生的可能性，转移证明的结果超过功能领域的p-adic领域的特征为0，但许多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，并将澄清表示论中出现的许多构造对p的依赖性。最终，这种方法预计将导致算法的计算量，逃避计算到目前为止，如值的哈里什-钱德拉字符。 动机整合也可以是一个强大的来源的类比允许调查集团在非局部紧值fields.One的第一个主要障碍时，遇到的试图研究这样的群体，是没有哈尔措施。动机整合可以直接使用，或者只是作为一种类比，来提供一种绕过这一障碍的方法。这些结果可以在数论中用来研究自守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.