Publication date:
20 August 2017
Source:Advances in Mathematics, Volume 316
Author(s): Charles H. Conley, Dimitar Grantcharov
The Lie algebra of vector fields on ${\mathbb{R}}^m$ acts naturally on the spaces of differential operators between tensor field modules. Its projective subalgebra is isomorphic to ${\mathfrak{sl}}_{m+1}$, and its affine subalgebra is a maximal parabolic subalgebra of the projective subalgebra with Levi factor ${\mathfrak{gl}}_m$. We prove two results. First, we realize explicitly all injective objects of the parabolic category ${\mathcal{O}}^{\mathfrak{g}{\mathfrak{l}}_m}({\mathfrak{sl}}_{m+1})$ of ${\mathfrak{gl}}_m$-finite ${\mathfrak{sl}}_{m+1}$-modules, as submodules of differential operator modules. Second, we study projective quantizations of differential operator modules, i.e., ${\mathfrak{sl}}_{m+1}$-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.