Induced $C^{\ast }$-complexes in metaplectic geometry

For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of the Kostant’s symplectic spinor bundle. Defining a Hilbert $C^{\ast }$-structure on this bundle for a suitable $C^{\ast }$-algebra, we obtain an elliptic $C^{\ast }$-complex in the sense of Mishchenko–Fomenko. Its cohomology groups appear to be finitely generated projective Hilbert $C^{\ast }$-modules. The paper can serve as a guide for handling of differential complexes and PDEs on Hilbert bundles