Tensor Products with Verma Module and Restriction to Parabolic Subalgebra

Using the tensor identity, we obtain decomposition results for the tensor product of a generalized Verma module with a module $M$ in the category $\mathcal{O}^{\mathfrak{p}}$, based on the decomposition of the restriction of $M$ to the parabolic subalgebra $\mathfrak{p}$. We show that this restriction admits an essentially unique decomposition into indecomposable $\mathfrak{p}$-modules and identify two particular types of direct summands: finite-dimensional quotients and tilting submodules. Finally, we give the complete $\mathfrak{b}$-decomposition of all indecomposable $M \in \mathcal{O}$ in $\mathfrak{sl}_2$, and of all Verma modules in the block of a dominant integral weight in $\mathfrak{sl}_3$, from which we derive explicit computations.