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Delorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces Palmirotta, Guendalina ; Olbrich, Martin 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 ▲] Detailed reference viewed: 65 (8 UL)Solvability of systems of invariant differential equations on H2 and beyond Palmirotta, Guendalina ; Olbrich, Martin E-print/Working paper (2022) We show how the Fourier transform for distributional sections of vector bundles over symmetric spaces of non-compact type G/K can be used for questions of solvability of systems of invariant differential ... [more ▼] We show how the Fourier transform for distributional sections of vector bundles over symmetric spaces of non-compact type G/K can be used for questions of solvability of systems of invariant differential equations in analogy to Hörmander’s proof of the Ehrenpreis-Malgrange theorem. We get complete solvability for the hyperbolic plane H2 and partial results for products H^2 × · · · × H^2 and the hyperbolic 3-space H^3. [less ▲] Detailed reference viewed: 59 (0 UL)A topological Paley-Wiener-Schwartz Theorem for sections of homogeneous vector bundles on G/K Palmirotta, Guendalina ; Olbrich, Martin 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 ▲] Detailed reference viewed: 65 (8 UL)Conference 'SL2R Days' Palmirotta, Guendalina ; ; Olbrich, Martin Report (2019) Detailed reference viewed: 123 (8 UL) |
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