Laplacians on quotients of Cauchy-Riemann complexes and Szeg\"o map for $L^2$-harmonic forms

Abstract

We compute the spectra of the Tanaka type Laplacians on the Rumin complex, a quotient of the tangential Cauchy-Riemann complex on the unit sphere in $C^n$. We prove that Szeg\"o map is a unitary operator from a subspace of $(p, q-1)$-forms on the sphere defined by the Tanaka 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.