Dichotomy for generic supercuspidal representations of G_2

摘要

The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of G_2 over a p-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp6 or PGL_3. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of G_2 and other representations of PGSp6 and PGL_3. This correspondence arises from theta correspondences in E_6 and E_7, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of generic supercuspidal irreducible representations of G_2 to a single conjecture about the parameterization for PGSp_6.