On minimal singular vectors of $V^κ(\mathfrak{g})$ for types $B$ and $C$

Abstract

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra of type $B$ or $C$ over ${\mathbb C}$. Given $κ\in{\mathbb C}$, let $V^κ(\mathfrak{g})$ be the universal affine vertex algebra associated with $\mathfrak{g}$ at level $κ$. We characterize the weights of minimal singular vectors of $V^κ(\mathfrak{g})$ with respect to the generalized Cartan matrix, when $V^κ(\mathfrak{g})$ is not simple and $κ$ is not critical. Furthermore, we determine the corresponding longest Weyl elements in the Kashiwara-Tanisaki character theorem of the singular vectors.