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

TL;DR

This paper introduces a Rankin-Selberg integral on the exceptional group G_2 that captures the L-function for generic cuspidal representations of SL_2 tilde  GL_2, with applications to non-vanishing Fourier-Jacobi periods.

Contribution

It constructs a new integral representation for the L-function associated with SL_2 tilde  GL_2 on the exceptional group G_2, linking it to Fourier-Jacobi periods.

Findings

The integral represents the L-function for the specified representations.

Certain Fourier-Jacobi periods on G_2 are shown to be non-vanishing.

The work connects automorphic L-functions with special periods on exceptional groups.

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.