Motivation: Consider $\mathfrak{sl}_2$. Given an integral weight $\lambda\in\mathbf Z$, then the only indecomposable modules contained in $\mathcal O_\lambda$ are the Verma modules and the simple modules with highest weight linked to $\lambda$ as well es their projective covers and injective hulls, cf. for example prop. 3.12 in Humphreys’ category .
Question: Do similar statements hold more generally? At least for $\mathfrak{sl}_3$ or $\mathfrak{sl}_n$? I doubt, but cannot find any clear statements.
The representation types of the blocks of ${\mathscr O}$ have been completely determined here.