Cohomology of the de Rham complex twisted by the oscillatory representation

We introduce a Hilbert A-module structure on the higher oscillatory module, where A   denotes the C⁎C⁎-algebra of continuous endomorphisms of the basic oscillatory module. We also define the notion of an exterior covariant derivative in an A-Hilbert bundle and use it for a construction of an A-elliptic complex of differential operators for certain symplectic manifolds equipped with a flat symplectic connection.