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

Abstract

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