Asymptotics of G-equivariant Szegő kernels

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 define G-equivariant Szegő kernels and establish the associated Boutet de Monvel–Sjöstrand type theorems. When X admits also a transversal CR S¹ action, we study the asymptotics of Fourier components of G-equivariant Szegő kernels with respect to the S¹ action.