Differential equations and singular vectors in Verma modules over sl(n, .)

Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weight . over sl(n,.). 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 {.(1) | . . S n }. It is known that S n is also the Weyl group of sl(n,.) and generated by all reflections s . with positive roots .. We present an explicit formula of the solution s .(1) for every positive root . and show directly that s .(1) is a polynomial if and only if .. + ., .. is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al..