Geometric quantization and the universal enveloping algebra of a nilpotent Lie group

Abstract:

We study geometric quantization in connection with connected nilpotent Lie groups. First it is proved that the quantization map associated with a (real) polarized coadjoint orbit establishes an isomorphism between the space of polynomial quantizable functions and the space of polynomial quantized operators. Our methods allow noninductive proofs of certain basic facts from Kirillov theory. It is then shown how the quantization map connects with the universal enveloping algebra. This is the main result of the paper. Finally we show how one can explicitly compute global canonical coordinates on coadjoint orbits, and that this can be done simultaneously on all orbits contained in a given stratum of what we call "the fine $\mathcal {F}$-stratification of the dual of the Lie algebra". This is a generalization of a result of M. Vergne about simultaneous canonical coodinates for orbits in general position.