Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds

Given a CR manifold with non-degenerate Levi form, we show that the operators of the functional calculus for Toeplitz operators are complex Fourier integral operators of Szegő type. As an application, we establish semi-classical asymptotics for the dimension of the spectral spaces of Toeplitz operators. We then consider a CR manifold with a compact Lie group action G and we establish quantization commutes with reduction for Toeplitz operators. Moreover, we also compute semi-classical asymptotics for the dimension of the spectral spaces of G-invariant Toeplitz operators.