Poles of triple product $L$-functions involving monomial representations

Abstract

In this paper, we study the order of the pole of the triple tensor product $L$-functions $L(s,{\pi }_1\times {\pi }_2\times {\pi }_3,{\otimes }^3)$ for cuspidal automorphic representations ${\pi }_i$ of ${\mathrm{GL}}_{n_i}({\mathbb{A}}_F)$ in the setting where one of the ${\pi }_i$ is a monomial representation. In the view of Brauer theory, this is a natural setting to consider. The results provided in this paper give crucial examples that can be used as a point of reference for Langlands' beyond endoscopy proposal.