  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.
