Differential Equations and Singular Vectors in Verma Modules over sl(n,C)

Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weightλover sl（n,C）.He gave a differential-operator representation of the symmetric group S_n on the corresponding space of truncated power series and proved that the solution space of the system is spanned by{σ（1）|σ∈S_n}.It is known that S_n is also the Weyl group of sl（n,C）and generated by all reflections sαwith positive rootsα.We present an explicit formula of the solution sα（1）for every positive rootαand show directly that sα（1）is a polynomial if and only if〈λ+ρ,α〉is a nonnegative integer.From this,we can recover a formula of singular vectors given by Malikov et al..