$G$-invariant Szeg\"o kernel asymptotics and CR reduction

Let $(X, T^{1,0}X)$ be a compact connected orientable CR manifold of dimension $2n+1$ with non-degenerate Levi curvature. Assume that $X$ admits a connected compact Lie group action $G$. Under certain natural assumptions about the group action $G$, we show that the $G$-invariant Szeg\"o kernel for $(0,q)$ forms is a complex Fourier integral operator, smoothing away $\mu^{-1}(0)$ and there is a precise description of the singularity near $\mu^{-1}(0)$, where $\mu$ denotes the CR moment map. We apply our result to the case when $X$ admits a transversal CR $S^1$ action and deduce an asymptotic expansion for the $m$-th Fourier component of the $G$-invariant Szeg\"o kernel for $(0,q)$ forms as $m \to+\infty$. As an application, we show that if $m$ large enough, quantization commutes with reduction.