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

Abstract

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

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