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

in Annals of Global Analysis and Geometry (2022), 63(9),

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 ... [more ▼]

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. [less ▲]

A topological Paley-Wiener-Schwartz Theorem for sections of homogeneous vector bundles on G/K

E-print/Working paper (2022)

We study the Fourier transform for compactly supported distributional sections of complex homogeneous vector bundles on symmetric spaces of non-compact type X = G/K. We prove a characterisation of their ... [more ▼]

We study the Fourier transform for compactly supported distributional sections of complex homogeneous vector bundles on symmetric spaces of non-compact type X = G/K. We prove a characterisation of their range. In fact, from Delorme’s Paley-Wiener theorem for compactly supported smooth functions on a real reductive group of Harish-Chandra class, we deduce topological Paley-Wiener and Paley-Wiener- Schwartz theorems for sections. [less ▲]