On some non-principal locally analytic representations induced by Whittaker modules

TL;DR

This paper constructs and analyzes certain non-principal, locally analytic representations of split semi-simple p-adic groups induced by Whittaker modules, revealing their irreducibility and inadmissibility properties.

Contribution

It introduces a new class of locally analytic representations induced by Whittaker modules and studies their irreducibility, inadmissibility, and Jacquet functor behavior.

Findings

Representations are topologically irreducible when the Whittaker module is simple.

These representations are inadmissible.

The naive Jacquet functor vanishes for all parabolic subgroups.

Abstract

Let G be a connected split adjoint semi simple p-adic Lie group. This paper can be seen as a continuation of [12] and is about the construction of locally analytic G-representations which do not lie in the principal series. Here we consider locally analytic representations which are induced by Whittaker modules of the attached Lie algebra. We prove that they are inadmissible and topologically irreducible in case the Whittaker module is simple. On the other hand, we show that the naive Jacquet functor of these representations vanishes for all parabolic subgroups. However, they do not satisfy the definition of supercuspidality in the sense of Kohlhaase.