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 $\lambda$ 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 $\{\sigma(1)\ |\ \sigma\in S_n\}$. We present an explicit formula of the solution $s_\alpha(1)$ for every positive root $\alpha$ and showed directly that $s_\alpha(1)$ is a polynomial if and only if $\langle\lambda+\rho,\alpha\rangle$ is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al.