Differential-Operator Representations of Sn and Singular Vectors in Verma Modules

Given a weight of sl(n, C), we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module, and a differential-operator representation of the symmetric group Sn on the related space of truncated power series. We prove that the solution space of the system of partial differential equations is exactly spanned by {sigma(1) | sigma is an element of S-n}. Moreover, the singular vectors of sl(n, C) in the Verma module are given by those sigma(1) that are polynomials. The well-known results of Verma, Bernstein-Gel'fand-Gel'fand and Jantzen for the case of sl(n, C) are naturally included in our almost elementary approach of partial differential equations.