Quasi-Whittaker modules

TL;DR

This paper introduces quasi-Whittaker modules over nonsemisimple Lie algebras, providing conditions for irreducibility, classifying irreducible modules, and constructing new smooth modules for certain Lie algebras.

Contribution

It generalizes Whittaker modules to a broader class called quasi-Whittaker modules and establishes criteria for their irreducibility and classification.

Findings

Determined necessary and sufficient conditions for irreducibility.

Classified irreducible quasi-Whittaker modules for many Lie algebras.

Constructed new irreducible smooth modules for al W modules of height 2.

Abstract

In this paper, a general setting is proposed to define a class of modules over nonsemisimple Lie algebras $\mathfrak{g}$ induced by a nonperfect ideal $\mathfrak{p}$. This class of Lie algebras includes many well-known Lie algebras, and some of this class of modules are Whittaker modules and others are not. We call these modules quasi-Whittaker modules. By introducing a new concept: the Whittaker annihilator for universal quasi-Whittaker modules, we are able to determine the necessary and sufficient conditions for the irreducibility of the universal quasi-Whittaker modules. In the reducible case, we can obtain some maximal submodules. In particular, we classify the irreducible quasi-Whittaker modules for many Lie algebras, and obtain a lot of irreducible smooth $\mathcal{W}_n^+$-modules of height $2$.