Generalized Whittaker models beyond $\mathfrak{sl}_{2}$-triples

TL;DR

This paper broadens the concept of generalized Whittaker models for p-adic groups by incorporating more general representations of unipotent subgroups, extending their applicability and establishing foundational properties.

Contribution

It introduces a generalized framework for Whittaker models beyond characters and Weil representations, utilizing Kirillov's orbit method and proving new multiplicity one results.

Findings

Defined new classes of generalized Whittaker models

Proved basic properties of the associated Jacquet functors

Established new instances of multiplicity one

Abstract

We extend the notion of generalized Whittaker models by allowing them to be built upon smooth irreducible representations of unipotent subgroups of a $p$-adic reductive group that are not necessarily characters, nor induced from Weil representations. This notion generalizes the usual notion of generalized Whittaker models, which include the Bessel and Fourier--Jacobi models. Using Kirillov's orbit method for unipotent groups, we define and prove basic properties of the corresponding Jacquet functors. We also provide new instances of generalized Whittaker models of multiplicity one.