Intertwining periods, L-functions and local-global principles for distinction of automorphic representations

TL;DR

This paper establishes a local-global criterion for the non-vanishing of period integrals on automorphic representations of general linear groups, connecting local distinction, L-functions, and global periods, and generalizing previous results.

Contribution

It introduces a new local-global principle for period integrals involving division algebras, extending known results and proving a key part of the Guo-Jacquet conjecture.

Findings

Established a criterion linking local distinction and global L-functions.

Generalized results to inner forms of GL(n).

Provided new results for twisted linear periods in split cases.

Abstract

We provide a criterion for non-vanishing of period integrals on automorphic representations of a general linear group over a division algebra. We consider three different periods: linear periods, twisted-linear periods and Galois periods. Our criterion is a local-global principle, which is stated in terms of local distinction, a further local obstruction, and poles of certain global L-functions associated to the underlying involution via the Jacquet-Langlands correspondence. Our local-global principle follows from a new method, relying on the Maass-Selberg relations and a careful analysis of singularities of local and global intertwining periods. Our results generalize to inner forms, known results for split general linear groups. Moreover, our result for twisted linear periods is new even in the split situation. As a consequence of our local-global principle, we complete the proof of one direction of the Guo-Jacquet conjecture.