Differential - Operator Representations of S n and Singular Vectors in Verma Modules 1 Xiaoping

Given a weight of sl(n,C), we derive a system of variable-coefficient secondorder 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 {σ(1) | σ ∈ Sn}. Moreover, the singular vectors of sl(n,C) in the Verma module are given by those σ(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.