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_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 = SL(2, R)^d (d ∈ N) and G = SL(2, 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