Quantization Commutes with Reduction for Embeddable CR Reductions

Abstract

We prove a canonical quantization-commutes-with-reduction theorem for CR functions. Under the geometric hypotheses of Hsiao--Ma--Marinescu, if the zero moment level is nonempty and the strictly pseudoconvex reduction is globally CR embeddable of real dimension at least three, the CR Guillemin--Sternberg map is a bounded isomorphism at every real Sobolev order. The central new step is an algebraic and complex-analytic elimination of its finite-dimensional Fredholm defect: multiplicativity removes the kernel, while the conductor of the smooth range, integral dependence, and holomorphic removability produce global lifts. Conductor nonvanishing follows from the reduced Szegő singularity; we also prove it using global CR peak functions. The map is an isomorphism of Fréchet algebras, the reduction is connected, and the corresponding maps for holomorphic line bundles positive near the zero level are isomorphisms in every nonnegative tensor degree. Invariant holomorphic sections vanish in every negative degree; when the reduced base has positive complex dimension, the section maps are isomorphisms in every integer degree.