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Journal of Lie Theory 33 (2023), No. 1, 133--148
Copyright Heldermann Verlag 2023

Unitarizable Vector-Valued Holomorphic Discrete Series and the Laplace Transform: an Example

Jean-Louis Clerc
Université de Lorraine, Institut E. Cartan, Nancy, France


For $T_\Omega$ a Hermitian symmetric tube-type domain, a family $(\pi_\mu)_{\mu\in \mathbb{C}}$ 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.

Keywords: Tube-type domains, Euclidean Jordan algebras, holomorphic discrete series, weighted Bergman spaces, Laplace transform, Wallach set.

MSC: 22E46, 32M15, 44A10.

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