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

TL;DR

This paper proves that, under certain conditions, the decomposition of Hardy spaces on CR manifolds with torus actions remains consistent when applying conic reduction, demonstrating a form of commutativity between quantization and reduction.

Contribution

It establishes an isomorphism between two natural decompositions of Hardy spaces after conic reduction for torus actions on CR manifolds, extending understanding of quantization-reduction interplay.

Findings

Isomorphism between Hardy space decompositions for large k

Compatibility of quantization with conic reduction

Applicable to CR manifolds with non-degenerate Levi-curvature

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.