Differential equations and singular vectors in Verma modules

TL;DR

This paper provides explicit formulas for solutions to differential equations related to singular vectors in Verma modules of ng Lie algebras, clarifying when these solutions are polynomials and recovering known singular vector formulas.

Contribution

It explicitly computes solutions for all positive roots and characterizes when they are polynomials, connecting to and recovering Malikov et al.'s singular vector formulas.

Findings

Explicit formulas for solutions s_() for all positive roots.

Characterization of when solutions are polynomials based on + ho pairing.

Recovery of Malikov et al.'s singular vector formulas.

Abstract

Xu introduced a system of partial differential equations to investigate singular vectors in the Verma modules of highest weight $\lambda$ over $\mathfrak{sl}(n,\mathbb{C})$. He proved that the solution space of this system in the space of truncated power series is spanned by $\{\sigma(1)\ |\ \sigma\in S_n\}$. We present an explicit formula of the solution $s_\alpha(1)$ for every positive root $\alpha$ and showed directly that $s_\alpha(1)$ is a polynomial if and only if $\langle\lambda+\rho,\alpha\rangle$ is a nonnegative integer. From this, we can recover a formula of singular vectors given by Malikov et al.