Abstract: Let $G$ be a semisimple Lie group of Hermitian type, $K \subset G$ a maximal compact subgroup, and $P \subset G$ a minimal parabolic subgroup associated to $K$. A finite-dimensional representation of $K$ in a complex vector space determines the associated homogeneous vector bundles on the homogeneous manifolds $G/P$ and $G/K$. The Poisson transform associates to each section of the bundle over $G/P$ a section of the bundle over $G/K$, and it generalizes the classical Poisson integral. Given a discrete subgroup $\Gamma$ of $G$, we prove that the image of a $\G$-invariant section of the bundle over $G/P$ under the Poisson transform is a holomorphic automorphic form on $G/K$ for $\G$. We also discuss the special case of symplectic groups in connection with holomorphic forms on families of abelian varieties.
Full text of the article: