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We define conic reduction $X^{\mathrm{red}}_{\nu}$ for torus actions on theboundary $X$ of a strictly pseudo-convex domain and for a given weight $\nu$labeling a unitary irreducible representation. There is a natural residualcircle action on $X^{\mathrm{red}}_{\nu}$. We have two natural decompositionsof the corresponding Hardy spaces $H(X)$ and $H(X^{\mathrm{red}}_{\nu})$. Thefirst 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$H(X^{\mathrm{red}}_{\nu})_k$ induced by the residual circle action. The aim ofthis paper is to prove that they are isomorphic for $k$ sufficiently large. Theresult is given for spaces of $(0,q)$-forms with $L^2$-coefficient when $X$ isa CR manifold with non-degenerate Levi-curvature.

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