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

We define conic reductions \(X^{\textrm{red}}_{\nu }\) for torus actions on the boundary X of a strictly pseudo-convex domain and for a given weight \(\nu \) labeling a unitary irreducible representation. There is a natural residual circle action on \(X^{\textrm{red}}_{\nu }\). We have two natural decompositions of the corresponding Hardy spaces H(X) and \(H(X^{\textrm{red}}_{\nu })\). The first one is given by the ladder of isotypes \(H(X)_{k\nu }\), \(k\in {\mathbb {Z}}\); the second one is given by the k-th Fourier components \(H(X^{\textrm{red}}_{\nu })_k\) 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 \(L^2\)-coefficient when X is a CR manifold with non-degenerate Levi form.