Dichotomy for generic supercuspidal representations ofG2

Abstract

AbstractThe local Langlands conjectures imply that to every generic supercuspidal irreducible representation ofG2over ap-adic field, one can associate a generic supercuspidal irreducible representation of either PGSp6or PGL3. We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations ofG2and other representations of PGSp6and PGL3. This correspondence arises from theta correspondences inE6andE7, analysis of Shalika functionals, and spin L-functions. Our main result reduces the conjectural Langlands parameterization of generic supercuspidal irreducible representations ofG2to a single conjecture about the parameterization for PGSp6.