Classification of simple strong Harish-Chandra modules over the Lie superalgebra of vector fields on $\C^{m|n}$

TL;DR

This paper classifies simple strong Harish-Chandra modules over the Lie superalgebra of vector fields on complex supermanifolds, showing they are submodules of tensor modules constructed from Weyl and general linear superalgebra modules.

Contribution

It provides a complete classification of these modules, linking them to tensor modules built from simpler modules over Weyl and general linear superalgebras.

Findings

Any simple strong Harish-Chandra module is a unique simple submodule of a tensor module.

The tensor modules are constructed from simple weight modules over Weyl superalgebra and general linear superalgebra.

The classification offers a structural understanding of modules over the Lie superalgebra of vector fields.

Abstract

In this paper, we classify simple strong Harish-Chandra modules over the Lie superalgebra $W_{m,n}$ of vector fields on $\C^{m|n}$. Any such module is the unique simple submodule of some tensor module $F(P,V)$ for a simple weight module $P$ over the Weyl superalgebra $\mathcal K_{m,n}$ and a simple weight module $V$ over the general linear superalgebra $\gl_{m,n}$.