G-invariant Szegő kernel asymptotics and CR reduction

Let (X, T^{1 , 0}X) be a compact connected orientable CR manifold of dimension 2 n+ 1 with non-degenerate Levi curvature. Assume that X admits a connected compact Lie group G action. Under certain natural assumptions about the group G action, we show that the G-invariant Szegő kernel for (0, q) forms is a complex Fourier integral operator, smoothing away μ^{- 1}(0) and there is a precise description of the singularity near μ^{- 1}(0) , where μ denotes the CR moment map. We apply our result to the case when X admits a transversal CR S¹ action and deduce an asymptotic expansion for the mth Fourier component of the G-invariant Szegő kernel for (0, q) forms as m→ + ∞ and when q= 0 , we recover Xiaonan Ma and Weiping Zhang’s result about the existence of the G-invariant Bergman kernel for ample line bundles. As an application, we show that if m large enough, quantization commutes with reduction.