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

Abstract

Xu introduced a system of partial differential equations to investigate singular vectors in the Verma module of highest weight λ over 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 s _α with positive roots α. We present an explicit formula of the solution s _α(1) for every positive root α and show directly that s _α(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..