Szeg\H{o} kernel expansion and equivariant embedding of CR manifolds with circle action

Let $X$ be a compact strongly pseudoconvex CR manifold with a transversal CR $S^1$-action. In this paper, we establish the asymptotic expansion of Szeg\H{o} kernels of positive Fourier components and by using the asymptotics, we show that $X$ can be equivariant CR embedded into some ${\mathbb{C}}^N$ equipped with a simple $S^1$-action. An equivariant embedding of quasi-regular Sasakian manifold is also derived.