Universal property of a Verma module
= Universal property of a Verma module
If a $\mathfrak g$-module contains a vector $v$ of weight $\lambda$ annihilated by $\mathfrak n^+$, there is a unique homomorphism $M_\lambda\to V$ sending the canonical highest-weight vector to $v$. Every highest-weight module of weight $\lambda$ is therefore a quotient of $M_\lambda$.