Hua operators, Poisson transform and relative discrete series on line bundles over bounded symmetric domains

Abstract

For bounded symmetric domains Ω=G/K of tube type and general domains of type I, we consider the action of G on sections of a homogeneous line bundle over Ω and the corresponding eigenspaces of G-invariant differential operators. The Poisson transform maps hyperfunctions on the Shilov boundary S=K/L to the eigenspaces. We characterize the image in terms of twisted Hua operators. For some special parameters the Poisson transform is of Szegö type whose image is in a relative discrete series; we compute the corresponding elements in the discrete series.