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

TL;DR

This paper investigates the Poisson transform and Hua operators on line bundles over bounded symmetric domains, characterizing their images and identifying elements in the relative discrete series for specific parameters.

Contribution

It provides a characterization of the Poisson transform's image using twisted Hua operators and computes elements in the relative discrete series for special parameters.

Findings

Characterization of the Poisson transform's image via twisted Hua operators

Identification of elements in the relative discrete series for certain parameters

Explicit computation of discrete series elements

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.