Quantization and injective submodules of differential operator modules

The Lie algebra of vector fields on $R^m$ acts naturally on the spaces of differential operators between tensor field modules. Its projective subalgebra is isomorphic to $s{l}_{m+1}$, and its affine subalgebra is a maximal parabolic subalgebra of the projective subalgebra with Levi factor $g{l}_m$. We prove two results. First, we realize all injective objects of the parabolic category O${}^{g{l}_m}(s{l}_{m+1})$ of $g{l}_m$-finite $s{l}_{m+1}$-modules as submodules of differential operator modules. Second, we study projective quantizations of differential operator modules, i.e., $s{l}_{m+1}$-invariant splittings of their order filtrations. In the case of modules of differential operators from a tensor density module to an arbitrary tensor field module, we determine when there exists a unique projective quantization, when there exists no projective quantization, and when there exist multiple projective quantizations.