Quantization and injective submodules of differential operator modules

Abstract

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 all injective objects of the parabolic category O$^{gl_m}(sl_{m+1})$ 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.