The aim of this article is to construct specific Poisson transforms mapping differential forms on the sphere $S^{2n+1}$ endowed with its natural CR structure to forms on complex hyperbolic space. These transforms have coclosed harmonic values, descend to the BGG (Rumin) complex, and intertwine the differential operators in that complex with the exterior derivative.

Passing to the Poincaré ball model, we analyze boundary asymptotics, proving that the values of our transforms admit a continuous extension to the boundary in degrees $\leq n$. Finally, we show that, composing the exterior derivative with the transform in degree n, one obtains an isomorphism between the kernel of the Rumin operator in degree n and a dense subspace of the $L^2$-harmonic forms on complex hyperbolic space. This provides a realization of the discrete series representations of $SU(n+1,1)$ with trivial infinitesimal character in spaces of differential forms on the compact manifold $S^{2n+1}$.

These developments are motivated by a programme of the third author to prove some instances of the Baum-Connes conjecture. The first part of the article is valid in a much more general setting; in particular, it is relevant for cases in which the conjecture is still open.

We obtain intertwining dilation theorems for non-commutative regular domains 𝒟f and non-commutative varieties 𝒱J in B(𝓗)n, which generalize Sarason and Szőkefalvi-Nagy and Foiaş's commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain 𝒟f (respectively, variety 𝒱J ) as well as a Schur-type representation for the unit ball of the Hardy algebra associated with the variety 𝒱J. We provide Andô-type dilations and inequalities for bi-domains 𝒟f ×c 𝒟g consisting of all pairs (X,Y ) of tuples X := (X1,…, Xn1) ∊ 𝒟f and Y := (Y1,…, Yn2) ∊ 𝒟g that commute, i.e. each entry of X commutes with each entry of Y . The results are new, even when n1 = n2 = 1. In this particular case, we obtain extensions of Andô's results and Agler and McCarthy's inequality for commuting contractions to larger classes of commuting operators. All the results are extended to bi-varieties 𝒱J1×c 𝒱J2, where 𝒱J1 and 𝒱J2 are non-commutative varieties generated by weak-operator-topology-closed two-sided ideals in non-commutative Hardy algebras. The commutative case and the matrix case when n1 = n2 = 1 are also discussed.