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

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 $\left(\widetilde{S{L}_2},G_2\right)$ in $\widetilde{S{p}_{14}}$. 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.