On Shalika Periods and a Theorem of Jacquet-Martin

TL;DR

This paper provides a complete characterization of when cuspidal automorphic representations of inner forms of GL_2 admit Shalika periods, using theta correspondence, extending known results for GL_4.

Contribution

It offers a full solution to the existence of Shalika periods for inner forms of GL_2 using theta correspondence, complementing prior partial results.

Findings

Complete criteria for Shalika periods on inner forms of GL_2.

Extension of global and local results via theta correspondence.

Resolution of the analogous local problem.

Abstract

Let \pi be a cuspidal automorphic representation of GL_4 with central character \mu^2. It is known that \pi has Shalika period with respect to \mu if and only if the L-function L^S(s, \pi, \bigwedge^2\otimes\mu^{-1}) has a pole at s=1. Recentlt, Jacquet and Martin considered the analogous question for cuspidal representations \pi_D of the inner form GL_2(D)(\A), and obtained a partial result via the relative trace formula. In this paper, we provide a complete solution to this problem via the method of theta correspondence, and give necessary and sufficient conditions for the existence of Shalika period for \pi_D. We also resolve the analogous question in the local setting.