Whittaker Modules for W type Cartan Lie superalgebras

TL;DR

This paper classifies Whittaker modules for the Lie superalgebra of vector fields on complex supermanifolds, establishing equivalences with modules over certain subsuperalgebras and describing simple modules with finite-dimensional Whittaker vectors.

Contribution

It introduces a new classification framework for Whittaker modules of W type Lie superalgebras, linking them to modules over specific subsuperalgebras and describing their simple modules.

Findings

Equivalence between blocks of Whittaker modules and modules over a subsuperalgebra.

Description of simple Whittaker modules with non-singular parameters.

Application of covering technique to analyze module categories.

Abstract

We consider the category of Whittaker modules for the Lie superalgebra $W_{m,n}$ of vector fields on $\mathbb{C}^{(m|n)}$. For any $\mathbf{a}\in \mathbb{C}^m$ we show the equivalence between the blocks $\Omega_{\mathbf a}^{\widetilde{W}_{m,n}}$ of the category of $(AW)_{m,n}$-Whittaker modules with finite-dimensional Whittaker vector spaces and the category of finite-dimensional modules over certain Lie subsuperalgebra $T_{m,n}$ of $(AW)_{m,n}$ (and also of $\mathfrak{gl}{(m,n)})$. Then we apply the covering technique to study Whittaker $W_{m,n}$-modules and describe simple modules in the category $\Omega_{\mathbf a}^{{W}_{m,n}}$ of such modules with finite-dimensional Whittaker vector spaces and with non-singular ${\mathbf a}$.