  Let $X$ be a compact connected orientable CR manifold with the action of a
connected compact Lie group $G$. Under natural pseudoconvexity assumptions we
show that the CR Guillemin-Strernberg map is Fredholm at the level of Sobolev
spaces of CR functions. 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.
