Results on symplectic spinors and their higher spin versions, concerning
representation theory and cohomology properties are presented. Exterior forms
with values in the symplectic spinors are decomposed into irreducible modules
including finding the hidden symmetry (Schur--Weyl--Howe type duality) given by
a representation of the Lie superalgebra osp(1∣2) in this case. We
also determine ranges of the induced exterior symplectic spinor derivatives
when they are restricted to bundles induced by the irreducible submodules
mentioned. This duality is used to decompose curvature tensors of covariant
derivatives induced by a Fedosov connection to a symplectic spinor bundle, for
characterizing a subcomplex of the de Rham complex twisted by the spinors and
for proving that the complex is of elliptic type. Part of the dual is used to
characterize Fedosov manifolds admitting symplectic Killing spinors and for
relating the spectra of the symplectic Rarita--Schwinger and the symplectic
Dirac operator of Habermann. Further, we use the Fomenko--Mishchenko
generalization of the Atiyah--Singer index theorem to prove a kind of Hodge
theory is valid for elliptic complexes with differentials on projective and
finitely generated bundles over the algebra of compact operators and certain
further ones.Comment: 41 pages; habilitation thesis (short version) for the process of
association to Charles University in Pragu