Cotangent Bundles as Coadjoint Orbits and Asymptotic Character Formulas

Abstract

We present geometric quantization on cotangent bundles as a special instance of Kirillov's orbit method. To this end, the cotangent bundle is realized as a coadjoint orbit of an infinite-dimensional Lie group for a variety of topologies on the dual of its Lie algebra. The group, central to quantum mechanics on manifolds, is the semidirect product constructed from the diffeomorphism group acting on the abelian group of smooth functions on the manifold. We then develop asymptotic character formulas by employing the tangent groupoid of the manifold and investigating the interplay between its groupoid C${}^{\ast }$-algebra and the infinite-dimensional Lie group.

arXiv commentRevised and expanded version. Includes a new exact trace formula (Corollary 3.16), an appendix constructing quantization schemes on the tangent groupoid (Appendix C) and a thorough discussion of future research directions (Conclusion and Outlook). To appear in Annales Henri Poincaré. 49 pages, 1 figure