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

For T $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ through the Laplace transform.