On singular vectors of simply-laced universal affine vertex operator algebras

Abstract

Given a finite-dimensional complex simple Lie algebra $\mathfrak{g}$ and a complex number $\kappa$, let $V^{\kappa}(\mathfrak{g})$ be the associated universal affine vertex algebra. Gorelik and Kac [GK07] gave a sufficient and necessary condition for $V^{\kappa}(\mathfrak{g})$ to be simple. In this paper, for simply-laced $\mathfrak{g}$ and non-critical $\kappa$, we determine the weights of singular vectors of $V^{\kappa}(\mathfrak{g})$ with minimal conformal weights, when $V^{\kappa}(\mathfrak{g})$ is not simple. We further determine all the longest Weyl elements in the Kashiwara-Tanisaki character theorem [KT00] which correspond to the weights of the singular vectors.