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; ^ ³ / having a pole at s=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. [ABSTRACT FROM AUTHOR]