Quantizations of modules of differential operators

Fix a manifold M , and let V be an infinite dimensional simple Lie subalgebra of the Lie algebra VecM of vector fields on M . Assume that V contains a finite dimensional simple maximal subalgebra a(V). We define an a(V)-quantization of a V-module of differential operators on M to be a decomposition of the module into irreducible a(V)-modules. In this article we survey some recent results and open problems involving this type of quantization and its applications to cohomology, indecomposable modules, and geometric equivalences and symmetries of differential operator modules. There are several mathematical theories of quantization. Two of the most important are geometric quantization, which hinges on polarization and is linked to the orbit method in the representation theory of Lie groups, and deformation quantization, in which the classical Poisson algebra structure becomes the first order approximation of an associative star product. In its original physical sense, to quantize a system meant to replace the commutative Poisson algebra of functions on the phase space, the classical observables, with a noncommutative algebra of operators on a Hilbert space, the quantum mechanical observables. In the theory of quantization under consideration here, the role of the noncommutative algebra is played by the differential operators and that of the commutative algebra is played by their symbols. We will consider two cases: the case that V is all of VecM , and the case that M is a contact manifold and V is the Lie algebra ConM of contact vector fields on M . Our approach is algebraic: we assume that M is a Euclidean manifold R and we consider only polynomial vector fields. Thus, writing Di for ∂/∂xi and using the multi-index notation x = x1 1 · · ·xJm m , V ⊆ VecR := SpanC { xDi : 1 ≤ i ≤ m,J ∈ N }