The Fourier-Jacobi map and small representations

Abstract

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.

Eisenstein Series Arising from Jordan Algebras

“…Proof. This is proved by Weissman in [11] if G is split and simply laced. The proof given there extends easily to the more general class of groups considered here.…”

“…In this section k is a p-adic field. The goal of this section is to extend the results of Weissman in [11] to non-split groups.…”

“…For real groups, in the setting of this paper, this was accomplished by Sahi in a couple of papers, [9] and [10]. On the other hand, for p-adic groups, Weissman [11] analyzes the structure of the degenerate principal series representations using a Fourier-Jacobi functor. In a nutshell, this method is a local analogue of Ikeda's method.…”

A generalized Theta lifting, CAP representations, and Arthur parameters

“…Our main technique for proving the vanishing of Fourier coefficients associated to certain small nilpotent orbits is to study associated local Fourier-Jacobi models. These are local analogues of the Fourier-Jacobi coefficients studied in [30], and the functor we study is intimately related to the Fourier-Jacobi map studied by Weissman [41] as well as the generalized Whittaker models of [31], though this is the first occurrence of this technique in the context of higher-degree covering groups.…”

Raising nilpotent orbits in wave-front sets

“…Proof. The first part, the isomorphism given by A ⊗ v ↦ A(v), is in [Wei03]. If ℓ is an arbitrary functional on Hom H (ρ ψ , π) and ℓ y the functional on S(Y ) given by evaluating functions f ∈ S(Y ) at y, then ℓ ⊗ ℓ y transforms under the action of X as ψ y .…”