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The description of the Paley-Wiener space for compactly supported smoothfunctions $C^\infty_c(G)$ on a semi-simple Lie group $G$ involves certainintertwining conditions that are difficult to handle. In the present paper, wemake them completely explicit for $G=\mathbf{SL}(2,\mathbb{R})^d$ ($d\in\mathbb{N}$) and $G=\mathbf{SL}(2,\mathbb{C})$. Our results are based on adefining criterion for the Paley-Wiener space, valid for general groups of realrank one, that we derive from Delorme's proof of the Paley-Wiener theorem. In aforthcoming paper, we will show how these results can be used to studysolvability of invariant differential operators between sections of homogeneousvector bundles over the corresponding symmetric spaces.

Delorme's intertwining conditions for sections of homogeneous vector bundles on two and three dimensional hyperbolic spaces