A Poisson transform adapted to the Rumin complex

Abstract

Let [Formula: see text] be a semisimple Lie group with finite center, [Formula: see text] a maximal compact subgroup, and [Formula: see text] a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use [Formula: see text]-invariant differential forms on [Formula: see text] to construct [Formula: see text]-equivariant Poisson transforms mapping differential forms on [Formula: see text] to differential forms on [Formula: see text]. Such invariant forms can be constructed using finite-dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on [Formula: see text] to the associated Bernstein–Gelfand–Gelfand (or BGG) complex in a well defined sense. The main part of this paper is devoted to an explicit construction of such transforms with additional favorable properties in the case that [Formula: see text]. Thus, [Formula: see text] is [Formula: see text] with its natural CR structure and the relevant BGG complex is the Rumin complex, while [Formula: see text] is complex hyperbolic space of complex dimension [Formula: see text]. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail.