Commutativity of quantization with conic reduction for torus actions on compact CR manifolds

We define conic reductions Xνred for torus actions on the boundary X of a strictly pseudo-convex domain and for a given weight ν labeling a unitary irreducible representation. There is a natural residual circle action on Xνred . We have two natural decompositions of the corresponding Hardy spaces H(X) and H(Xνred) . The first one is given by the ladder of isotypes H(X) kν , k∈ Z ; the second one is given by the k-th Fourier components H(Xνred)k induced by the residual circle action. The aim of this paper is to prove that they are isomorphic for k sufficiently large. The result is given for spaces of (0, q)-forms with L2 -coefficient when X is a CR manifold with non-degenerate Levi form.