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 $\mathfrak{s}\mathfrak{l}(n,\mathbb{C})$. He proved that the solution space of this system in the space of truncated power series is spanned by ${\sigma(1)\ |\ \sigma\in S_n}$\mathrm{.}Wepresentanexplicitformulaofthesolution$s_\alpha(1)$ for every positive root $\alpha$ and showed directly that $s_{\alpha }(1)$ is a polynomial if and only if $⟨\lambda +\rho ,\alpha ⟩$ is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al.