Asymptotics of G-equivariant Szeg\H{o} kernels

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 $G$ action. Under certain natural assumptions about the group $G$ action, we define $G$-equivariant Szeg\H{o} kernels and establish the associated Boutet de Monvel-Sj"ostrand type theorems. When $X$ admits also a transversal CR $S^1$ action, we study the asymptotics of Fourier components of $G$-equivariant Szeg\H{o} kernels with respect to the $S^1$ action.