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Given a suitable ordering of the positive root system associated with asemisimple Lie algebra, there exists a natural correspondence between Vermamodules and related polynomial algebras. With this, the Lie algebra action on aVerma module can be interpreted as a differential operator action onpolynomials, and thus on the corresponding truncated formal power series. Weprove that the space of truncated formal power series is adifferential-operator representation of the Weyl group $W$. We also introduce asystem of partial differential equations to investigate singular vectors in theVerma module. It is shown that the solution space of the system in the space oftruncated 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 newproof of the well-known BGG-Verma Theorem.

Differential-operator representations of Weyl group and singular vectors