Geometric quantization on CR manifolds

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