$\Uh$ invariant Quantization of Coadjoint Orbits and Vector Bundles over them

Abstract

Let M be a coadjoint semisimple orbit of a simple Lie group G. Let $U_h(\g)$ be a quantum group corresponding to G. We construct a universal family of $U_h(\g)$ invariant quantizations of the sheaf of functions on M and describe all such quantizations. We also describe all two parameter $U_h(\g)$ invariant quantizations on M, which can be considered as $U_h(\g)$ invariant quantizations of the Kirillov-Kostant-Souriau (KKS) Poisson bracket on M. We also consider how those quantizations relate to the natural polarizations of M with respect to the KKS bracket. Using polarizations, we quantize the sheaves of sections of vector bundles on M as one- and two-sided $U_h(\g)$ invariant modules over a quantized function sheaf.