A Family of New-Way Integrals for the Standard ℒ-function of Cuspidal Representations of the Exceptional Group of TypeG2

Abstract

Let |$\mathcal{L}^{S}\!\left({s,\pi,\chi,\operatorname{\mathfrak{st}}}\right)\!$| be a standard twisted partial |$\mathcal{L}$|-function of degree |$7$| of the cuspidal automorphic representation |$\pi$| of the exceptional group of type |$G_2$|⁠. In this paper, we consider a family of Rankin–Selberg integrals and prove that it represents this |$\mathcal{L}$|-function. As an application, we prove that the representations attaining certain prescribed poles are exactly the representations attained by |$\theta$|-lift from a group of\break finite type.