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

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. In [GK07], the authors gave a sufficient and necessary condition for $V^{\kappa}(\mathfrak{g})$ to be simple. In this paper, we determine the weights of singular vectors of $V^{\kappa}(\mathfrak{g})$ with minimal conformal weights, when $\mathfrak{g}$ is simply-laced and $V^{\kappa}(\mathfrak{g})$ is not simple.