Period integrals

Columbia University, New York NY

Investigators

Abstract

Abstract for Jacquet DMS-0245310 The PI proposes to study certain period integrals defined on spaces of automorphic representations. The two main questions are to determine for which representations the periods are non-zero, and to express if possible the periods (which are globally defined) in terms of local data. The main tool of this investigation is the relative trace formula, the study of which was initiated by the PI. One of the main difficulties is a conjectural combinatorial identity (the fundamental lemma). Recently, the PI has obtained the identity in question is an interesting case. The method may apply to other cases as well. From a broader perspective, Langlands program deals with the harmonic analysis of certain linear operators on the spaces of automorphic functions defined on reductive groups. The eigenvalues are related to L-functions with striking analytic properties. On the other hand, the eigenvalues associated with different groups are related in a simple way. In certain specific cases, the two types of relations can be made more precise and more geometric. The goal of the proposal is to study these specific cases.