Laplacians on quotients of Cauchy-Riemann complexes and Szegö map for L-harmonic forms

We compute the spectra of the Tanaka type Laplacians on the Rumin complex Q, a quotient of the tangential Cauchy-Riemann complex on the unit sphere S²ⁿ⁻¹ in ⁿ. We prove that Szegö map is a unitary operator from a subspace of (p, q −1)-forms on the sphere defined by the operators and the normal vector field onto the space of L²-harmonic (p, q)-forms on the unit ball. Our results generalize earlier result of Folland.