Differential Equations for Singular Vectors of sp(2n)

Abstract

ABSTRACT Given a weight λ of sp(2n), we derive a system of variable-coefficient second-order linear partial differential equations that determines the singular vectors in the corresponding Verma module. Moreover, we find a family of exact solutions of the system in a certain space of power series. The polynomial solutions correspond to the singular vectors in the Verma module. In particular, we find the explicit expression of a singular vector corresponding to the single condition that ⟨ λ,α⟩ + ht α is a nonnegative integer for some positive root α, whose existence was proven by Jantzen. In the case n = 2, we completely solved the system in a certain space of power series.