Poisson liftings of holomorphic automorphic forms on semisimple Lie groups

Let G be a semisimple Lie group of Hermitian type, K subset of G a maximal compact subgroup, and P subset of G a minimal parabolic subgroup associated to K. If sigma is a finite-dimensional representation of It in a complex vector space, it 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 Gamma-invariant section of the bundle over G/P under the Poisson transform is a holomorphic automorphic form on G/K for Gamma. We also discuss the special case of symplectic groups in connection with holomorphic forms on families of abelian varieties.