Die metaplektische Darstellung: Holomorphe Fortsetzung und Jordan-theoretische Realisierung

The present thesis is a contribution to "geometric quantization". It is structured in three parts: the first is part of the general Gelfand-Gindikin-programm, and shows that the metaplectic representation can be seen as an extension of a representation of a subsemigroup in the complexification of the real symplectic group. The second part is concerned with a representation of the real symplectic group in terms of Jordan algebras, and in the third, based on the results of part two and the new state space, a projectively flat Hilbertspace bundle is given. A concrete realization of the Shilov boundary of certain complex structures leads, as application, to a concrete description of the fibers over boundary points in the metaplectic corrected bundle extended to this points.