Induced Representations , Gauge Fields , andQuantization on Homogeneous

We study representations of the enveloping algebra of a Lie group G which are induced by representations of a Lie subgroup H, assuming that G=H is reductive. Such representations describe the superselection sectors of a quantum particle moving on G=H. It is found that the representatives of both the generators and the quadratic Casimir operators of G have a natural geometric realization in terms of the canonical connection on the principal H-bundle G. The explicit expression for the generators can be understood from the point of view of conservation laws and moment maps in classical eld theory and classical particle mechanics on G=H. The emergence of classical geometric structures in the quantum-mechanical situation is explained by a detailed study of the domain and possible self-adjointness properties of the relevant operators. A new and practical criterion for essential self-adjointness in general unitary representations is given.