Let $\mathfrak{g}$ be a finite-dimensional semisimple Lie algebra over $\mathbb{C}$ with a fixed Cartan subalgebra $\mathfrak{h}$ and a fixed system of simple roots. It is stated in Exercise 3.11 of Humphreys' book "Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$" that

*The Verma $U(\mathfrak{g})$-module $M(\lambda)$ of highest weight $\lambda$ is projective only if $\lambda$ is dominant.*

Can anyone give me some hint to prove this proposition? It seems related to the BGG reciprocity.