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

The description of the Paley–Wiener space for compactly supported smooth functions \(C^\infty _c(G)\) on a semi-simple Lie group G involves certain intertwining conditions that are difficult to handle. In the present paper, we make them completely explicit for \(G=\textbf{SL}(2,\mathbb {R})^d\) (\(d\in \mathbb {N}\)) and \(G=\textbf{SL}(2,\mathbb {C})\). Our results are based on a defining criterion for the Paley–Wiener space, valid for general groups of real rank one, that we derive from Delorme’s proof of the Paley–Wiener theorem. In a forthcoming paper, we will show how these results can be used to study solvability of invariant differential operators between sections of homogeneous vector bundles over the corresponding symmetric spaces.