Vector-valued holomorphic discrete series and the Laplace transform : an example

TL;DR

This paper explores holomorphic vector-valued representations on Hermitian symmetric tube domains, identifying the Wallach set using Laplace transform techniques to realize these representations as weighted L2-spaces.

Contribution

It provides a detailed analysis of the Wallach set for these representations and introduces a realization via Laplace transform on the cone, offering new insights into their structure.

Findings

Determined the Wallach set for the family of representations.

Realized representations as weighted L2-spaces on the cone.

Applied Laplace transform as a key analytical tool.

Abstract

For T $\Omega$ a Hermitian symmetric tube-type domain, a family ($\pi$ $\mu$) $\mu$$\in$C of holomorphic vector-valued representations is studied. The corresponding Wallach set is determined. The main tool is a realization of the representations as weighted L 2-spaces on the cone $\Omega$ through the Laplace transform.