POISSON TRANSFORMS, THE BGG COMPLEX AND DISCRETE SERIES REPRESENTATIONS OF S U ( n + 1 , 1 ) $SU(n+1,1)$ upper S upper U left parenthesis n plus 1 comma 1 right parenthesis

Abstract The aim of this article is to construct specific Poisson transforms mapping differential forms on the sphere S 2 n + 1 $S^{2n+1}$ upper S Superscript 2 n plus 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 ≤ n $\leq n$ less than or equals 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 $L^2$ upper L squared -harmonic forms on complex hyperbolic space. This provides a realization of the discrete series representations of S U ( n + 1 , 1 ) $SU(n+1,1)$ upper S upper U left parenthesis n plus 1 comma 1 right parenthesis with trivial infinitesimal character in spaces of differential forms on the compact manifold S 2 n + 1 $S^{2n+1}$ upper S Superscript 2 n plus 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.