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

Abstract

Given a weight of sl(n, ${\mathbb C}$ ), 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 S n 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 ∈ S n }. Moreover, the singular vectors of sl(n, ${\mathbb C}$ ) 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, ${\mathbb C}$ ) are naturally included in our almost elementary approach of partial differential equations.