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

Abstract

We define conic reduction $X^{\mathrm{red}}_{\nu}$ 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 $X^{\mathrm{red}}_{\nu}$. We have two natural decompositions of the corresponding Hardy spaces $H(X)$ and $H(X^{\mathrm{red}}_{\nu})$. 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 $H(X^{\mathrm{red}}_{\nu})_k$ 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-curvature.