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

Abstract

Let $\Omega=G/K$ be a bounded symmetric domain and $S=K/L$ its Shilov boundary. We consider the action of $G$ on sections of a homogeneous line bundle over $\Omega$ and the corresponding eigenspaces of $G$-invariant differential operators. The Poisson transform maps hyperfunctions on the $S$ to the eigenspaces. We characterize the image in terms of twisted Hua operators. For some special parameters the Poisson transform is of Szeg\"o type mapping into the relative discrete series; we compute the corresponding elements in the discrete series.