Laplacians on quotients of Cauchy-Riemann complexes and Szegö map for L2-harmonic forms

We compute the spectra of the Tanaka type Laplacians □=∂¯Q∗∂¯Q+∂¯Q∂¯Q∗andΔ=∂¯Q∗∂¯Q+∂Q∂Q∗ on the Rumin complex Q, a quotient of the tangential Cauchy-Riemann complex on the unit sphere S2n−1 in ℂn. 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.