On automorphic descent from $G{L}_7$ to $G_2$

Abstract

In this paper, we study the functorial descent from self-contragredient cuspidal automorphic representations $\pi$ of $G{L}_7(\mathbb{A})$ with $L^S(s,\pi ,{\wedge }^3)$ having a pole at $s=1$ to the split exceptional group $G_2(\mathbb{A})$, using Fourier coefficients associated to two nilpotent orbits of $E_7$. We show that one descent module is generic, and under suitable local conditions, it is cuspidal and $\pi$ is a weak functorial lift of each of its irreducible summands. This establishes the first functorial descent involving the exotic exterior cube $L$-function. However, we show that the other descent module supports not only the nondegenerate Whittaker–Fourier integral on $G_2(\mathbb{A})$ but also every degenerate Whittaker–Fourier integral. Thus it is generic, but not cuspidal.