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

Abstract

We compute the spectra of the Tanaka type Laplacians □ = ∂ ¯ Q ∗ ∂ ¯ Q + ∂ ¯ Q ∂ ¯ Q ∗ and Δ = ∂ ¯ Q ∗ ∂ ¯ Q + ∂ Q ∂ Q ∗ on the Rumin complex Q, a quotient of the tangential Cauchy-Riemann complex on the unit sphere S 2 n−1 in ℂ 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 L 2-harmonic ( p, q)-forms on the unit ball. Our results generalize earlier result of Folland.