New families of irreducible weight modules over $\mathfrak{sl}_{3}$

TL;DR

This paper introduces new families of irreducible, infinite-dimensional weight modules over rak{sl}_3, constructed via tensor fields on a 2-dimensional torus, expanding the known classes of such modules.

Contribution

The paper constructs and proves the irreducibility of new rak{sl}_3-modules outside the Gelfand-Tsetlin category, using tensor field modules on a 2-torus.

Findings

Established irreducibility of the modules for generic parameters.

Modules have infinite-dimensional weight spaces.

These modules are new and not Gelfand-Tsetlin modules.

Abstract

Let $n>1$ be an integer, $\alpha\in{\mathbb C}^n$, $b\in{\mathbb C}$, and $V$ a $\mathfrak{gl}_n$-module. We define a class of weight modules $F^\alpha_{b}(V)$ over $\sl_{n+1}$ using the restriction of modules of tensor fields over the Lie algebra of vector fields on $n$-dimensional torus. In this paper we consider the case $n=2$ and prove the irreducibility of such 5-parameter $\mathfrak{sl}_{3}$-modules $F^\alpha_{b}(V)$ generically. All such modules have infinite dimensional weight spaces and lie outside of the category of Gelfand-Tsetlin modules. Hence, this construction yields new families of irreducible $\mathfrak{sl}_{3}$-modules.