We study the “Fourier-Jacobi” functor on smooth representations of split, simple, simply-laced p p -adic groups. This functor has been extensively studied on the symplectic group, where it provides the representation-theoretic analogue of the Fourier-Jacobi expansion of Siegel modular forms. Our applications are different from those studied classically with the symplectic group. In particular, we are able to describe the composition series of certain degenerate principal series. This includes the location of minimal and small (in the sense of the support of the local character expansion) representations as spherical subquotients.

Smooth irreducible representations of tori over local fields have been parameterized by Langlands, using class field theory and Galois cohomology. This paper extends this parameterization to some central extensions of such tori, which arise naturally in the setting of nonlinear covers of reductive groups.

We incorporate covers of quasisplit reductive groups into the Langlands program, defining an L-group associated to such a cover. We work with all covers that arise from extensions of quasisplit reductive groups by K 2the class studied by Brylinski and Deligne. We use this L-group to parameterize genuine irreducible representations in many contexts, including covers of split tori, unramified representations, and discrete series for double covers of semisimple groups over R. An appendix surveys torsors and gerbes on the étale site, as they are used in the construction of the L-group.

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 PGSp 6 or PGL 3 . We prove this conjectural dichotomy, demonstrating a precise correspondence between certain representations of G 2 and other representations of PGSp 6 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 .