Representations of p-adic groups and motivic integration

点击翻译按钮获取中文摘要

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.