Poles of the Standard $\mathcal{L}$-function of $G_2$ and the Rallis-Schiffmann Lift

TL;DR

This paper characterizes certain non-tempered cuspidal representations of the exceptional group G_2 through their standard L-function poles at s=2, linking them to the Rallis-Schiffmann lift via Rankin-Selberg integrals.

Contribution

It provides a new characterization of G_2 representations with poles in their L-function, connecting them to a specific lift and utilizing recent integral constructions.

Findings

Identifies cuspidal G_2 representations with poles at s=2 as Rallis-Schiffmann lifts.

Establishes the non-tempered nature of these representations.

Uses new Rankin-Selberg integrals to analyze the standard L-function.

Abstract

We characterize the cuspidal representations of $G_2$ whose standard $\mathcal{L}$-function admits a pole at $s=2$ as the image of Rallis-Schiffmann lift for the commuting pair $\left(\widetilde{SL_2}, G_2\right)$ in $\widetilde{Sp_{14}}$. The image consists of non-tempered representations. The main tool is the recent construction, by the second author, of a family of Rankin-Selberg integrals representing the standard $\mathcal{L}$-function.