Quantization and reduction for torsion free CR manifolds

Consider a compact torsion free CR manifold X and assume that X admits a compact CR Lie group action G. Let L be a G-equivariant rigid CR line bundle over X. It seems natural to consider the space of G-invariant CR sections in the high tensor powers as quantization space, on which a certain weighted G-invariant Fourier–Szegő operator projects. Under certain natural assumptions, we show that the group invariant Fourier–Szegő projector admits a full asymptotic expansion. As an application, if the tensor power of the line bundle is large enough, we prove that quantization commutes with reduction.