Dichotomy for generic supercuspidal representations of G₂

Original · EN

The local Langlands conjectures imply that to every generic supercuspidal irreducible representation of G₂ over a p-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp₆ orPGL₃. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of G₂ and other representations of PGSp₆ and PGL₃. This correspondence arises from theta correspondences in E₆ and E₇, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of generic supercuspidal irreducible representations of G₂ to a single conjecture about the parameterization for PGSp₆.