Geometric quantization on CR manifolds

Let X be a compact connected orientable Cauchy-Riemann (CR) manifold with the action of a compact Lie group G. Under natural pseudoconvexity assumptions we show that the CR Guillemin-Sternberg map is an isomorphism at the level of Sobolev spaces of CR functions, modulo a finite-dimensional subspace. As application we study this map for holomorphic line bundles which are positive near the inverse image of 0 by the momentum map. We also show that "quantization commutes with reduction" for Sasakian manifolds.