Quantization and injective submodules of differential operator modules

Abstract

The Lie algebra of vector fields on Rm acts naturally on the spaces of differential operators between tensor field modules. Its projective subalgebra is isomorphic to slm+1, and its affine subalgebra is a maximal parabolic subalgebra of the projective subalgebra with Levi factor glm. We prove two results. First, we realize explicitly all injective objects of the parabolic category Oglm(slm+1) of glm-finite slm+1-modules, as submodules of differential operator modules. Second, we study projective quantizations of differential operator modules, i.e.,slm+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.