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 this paper 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.

令G为具有有限中心的半单李群，$ķ\subset G$一个最大紧子群，和$磷\subset G$一个抛物线子群。遵循 PY Gaillard 的想法，可以使用G不变的微分形式$G/ķ\times G/磷$构造G等变泊松变换将G/P上的微分形式映射到G/K上的微分形式。这种不变的形式可以使用有限维表示理论来构建。在这个一般设置中，我们首先证明总是产生调和形式的变换正是从G/P上的 de Rham 复合体下降到定义明确的相关 Bernstein-Gelfand-Gelfand（或 BGG）复合体的变换。

本文的主要部分致力于在以下情况下明确构造具有额外有利属性的此类变换：$G=\mathrm{小号}ü(n+1,1)$. 因此，G/P是${\mathrm{小号}}^{2n+1}$其自然的 CR 结构和相关的 BGG 复合体是 Rumin 复合体，而G/K是复维数的复双曲空间$n+1$. 对复微分形式和实微分形式都进行了构造，并详细分析了变换与在其源和目标上可用的自然算子的兼容性。