Differential Equations for Singular Vectors of sl(n)

Given a weight of sl(n), we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module. Moreover, we completely solve the system in a certain space of power series. The polynomial solutions correspond to the singular vectors in the Verma module. The well-known results of Verma, Bernstein-Gel'fand-Gel'fand and Jantzen for the case of $sl(n)$ are naturally included in our almost elementary approach of partial differential equations.