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 sl(m+1), and its affine subalgebra is a maximal parabolic subalgebra of the projective subalgebra with Levi factor gl(m). We prove two results. First, we realize explicitly all injective objects of the parabolic category O-glm (sl(m+1)) of gl(m)-finite sl(m+1)-modules, as submodules of differential operator modules. Second, we study projective quantizations of differential operator modules, i.e., sl(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. (C) 2017 Elsevier Inc. All rights reserved.