A Family of New-way Integrals for the Standard $\mathcal{L}$-function of Cuspidal Representations of the Exceptional Group of Type $G_2$

TL;DR

This paper constructs a new family of integrals representing the degree 7 standard L-function for cuspidal automorphic representations of the exceptional group G_2, linking poles to theta lifts.

Contribution

It introduces a novel family of Rankin-Selberg integrals for the G_2 L-function, providing a new approach to analyze its poles and automorphic representations.

Findings

Constructed integrals represent the degree 7 L-function of G_2

Identified representations with prescribed poles as theta lifts

Established a connection between poles and automorphic lifts

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 construct a family of Rankin-Selberg integrals representing 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 finite type.