New families of irreducible weight modules over ${\mathfrak{sl}}_3$

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 $F_b^{\alpha }(V)$ 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 $F_b^{\alpha }(V)$ 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.

arXiv comment14 pages