On a Rankin-Selberg integral of the L-function for $\widetilde{SL}_2\times GL_2$

Abstract

We present a Rankin-Selberg integral on the exceptional group $G_2$ which represents the L-function for generic cuspidal representations of $\widetilde{SL}_2\times GL_2$. As an application, we show that certain Fourier-Jacobi type periods on $G_2$ are non-vanishing.