Cohomology of the de Rham complex twisted by the oscillatory representation

TL;DR

This paper develops a new framework for analyzing the cohomology of the de Rham complex twisted by oscillatory representations using Hilbert $A$-modules and $A$-elliptic complexes on symplectic manifolds.

Contribution

It introduces a Hilbert $A$-module structure on the higher oscillatory module and constructs an $A$-elliptic complex for symplectic manifolds with a symplectic connection.

Findings

Established a Hilbert $A$-module structure on oscillatory modules

Defined an exterior covariant derivative in $A$-Hilbert bundles

Constructed an $A$-elliptic complex for certain symplectic manifolds

Abstract

We introduce a Hilbert $A$-module structure on the higher oscillatory module, where $A$ denotes the $C^*$-algebra of bounded 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 symplectic connection.