On automorphic descent from GL7 to G2

In this paper, we study the functorial descent from self-contragredient cuspidal automorphic representations π of GL7.A/ with LS .s; π;Λ3/ having a pole at s D 1 to the split exceptional group G2.A/, using Fourier coefficients associated to two nilpotent orbits of E7. We show that one descent module is generic, and under suitable local conditions, it is cuspidal and π 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 G2.A/ but also every degenerate Whittaker- Fourier integral. Thus it is generic, but not cuspidal.