Let $\mathfrak{g}$ be a finite dimensional semi-simple complex Lie algebra. Then, BGG category $\mathcal{O}$ is defined to be the full subcategory of finitely generated $U(\mathfrak{g})$-modules of those modules which are weight modules and locally $U(\mathfrak{n})$-finite. It is known that $\mathcal{O}$ is not extension-closed in the category of (finitely generated) $U(\mathfrak{g})$-modules, see e.g. this math.stackexchange question. In particular (since $\mathcal{O}$ is closed under factor modules), it can't be true that $\mathcal{O}$ contains all finitely generated projective $U(\mathfrak{g})$-modules. I was wondering about the following:
Is there any projective object $P(\lambda)$ in $\mathcal{O}$ which is projective as a finitely generated $U(\mathfrak{g})$-module. Or put another way: Is any finitely generated projective $U(\mathfrak{g})$-module contained in $\mathcal{O}$?
The restriction of a projective $U(\mathfrak{g})$-module to $U(\mathfrak{h})$ is projective, but the restriction of an object of category $\mathcal{O}$ is a direct sum of one-dimensional modules.