Differential-operator representations of Weyl group and singular vectors

TL;DR

This paper establishes a differential-operator representation of the Weyl group on truncated formal power series and uses PDEs to characterize singular vectors in Verma modules, providing a new proof of the BGG-Verma Theorem.

Contribution

It introduces a novel PDE-based approach to study singular vectors and demonstrates a differential-operator representation of the Weyl group on polynomial and power series spaces.

Findings

The space of truncated formal power series forms a differential-operator representation of the Weyl group.

Solutions to a specific PDE system span the Weyl group orbit of 1, identifying singular vectors.

The approach offers a new proof of the BGG-Verma Theorem.

Abstract

Given a suitable ordering of the positive root system associated with a semisimple Lie algebra, there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series is a differential-operator representation of the Weyl group $W$. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of $\{w(1)\ |\ w\in W\}$. Those $w(1)$ that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma Theorem.