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See detailDelorme’s intertwining conditions for sections of homogeneous vector bundles on two- and three-dimensional hyperbolic spaces
Palmirotta, Guendalina UL; Olbrich, Martin UL

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 ▲]

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See detailInfinite dimensional moment map geometry and closed Fedosov star products
La Fuente-Gravy, Laurent UL

in Annals of Global Analysis and Geometry (2016), 49(1), 1-22

We study the Cahen–-Gutt moment map on the space of symplectic connections of a symplectic manifold. Given a Kähler manifold (M, ω, J ), we define a Calabi-type functional F on the space M of Kähler ... [more ▼]

We study the Cahen–-Gutt moment map on the space of symplectic connections of a symplectic manifold. Given a Kähler manifold (M, ω, J ), we define a Calabi-type functional F on the space M of Kähler metrics in the class [ω]. We study the space of zeroes of F. When (M, ω, J ) has non-negative Ricci tensor and ω is a zero of F, we show the space of zeroes of F near ω has the structure of a smooth finite dimensional submanifold. We give a new motivation, coming from deformation quantization, for the study of moment maps on infinite dimensional spaces. More precisely, we establish a strong link between trace densities for star products (obtained from Fedosov-type methods) and moment map geometry on infinite dimensional spaces. As a byproduct, we provide, on certain Kähler manifolds, a geometric characterization of a space of Fedosov star products that are closed up to order 3. [less ▲]

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