Differential equations and singular vectors in Verma modules

Xu introduced a system of partial differential equations to investigate singular vectors in the Verma modules of highest weight $λ$ over $\mathfrak{sl}(n,\mathbb{C})$. He proved that the solution space of this system in the space of truncated power series is spanned by $\{σ(1)\ |\ σ\in S_n\}$. We present an explicit formula of the solution $s_α(1)$ for every positive root $α$ and showed directly that $s_α(1)$ is a polynomial if and only if $\langleλ+ρ,α\rangle$ is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al.

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