A Poisson transform adapted to the Rumin complex

Let $G$ be a semisimple Lie group with finite center, $K\subset G$ a maximal compact subgroup, and $P\subset G$ a parabolic subgroup. Following ideas of P.Y.\ Gaillard, one may use $G$-invariant differential forms on $G/K\times G/P$ to construct $G$-equivariant Poisson transforms mapping differential forms on $G/P$ to differential forms on $G/K$. 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 $G/P$ to the associated Bernstein-Gelfand-Gelfand (or BGG) complex in a well defined sense. The main part of the article is devoted to an explicit construction of such transforms with additional favorable properties in the case that $G=SU(n+1,1)$. Thus $G/P$ is $S^{2n+1}$ with its natural CR structure and the relevant BGG complex is the Rumin complex, while $G/K$ is complex hyperbolic space of complex dimension $n+1$. 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.