Inversion of Rankin–Cohen operators via Holographic Transform
Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 2131-2190.

The analysis of branching problems for restriction of representations brings the concept of symmetry breaking transform and holographic transform. Symmetry breaking operators decrease the number of variables in geometric models, whereas holographic operators increase it. Various expansions in classical analysis can be interpreted as particular occurrences of these transforms. From this perspective we investigate two remarkable families of differential operators: the Rankin–Cohen operators and the holomorphic Juhl conformally covariant operators. Then we establish for the corresponding symmetry breaking transforms the Parseval–Plancherel type theorems and find explicit inversion formulæ with integral expression of holographic operators.

The proof uses the F-method which provides a duality between symmetry breaking operators in the holomorphic model and holographic operators in the L 2 -model, leading us to deep links between special orthogonal polynomials and branching laws for infinite-dimensional representations of real reductive Lie groups.

L’analyse des problèmes de branchement des restrictions des représentations fait émerger le concept de transformation de brisure de symétrie et celui de transformation holographique. Les opérateurs de brisure de symétrie diminuent le nombre de variables dans les modèles géométriques tandis que les opérateurs holographiques l’augmentent. Plusieurs développements en série ou intégrale de l’analyse classique sont des cas particuliers de telles transformations.

Dans cette perspective, nous étudions deux familles remarquables d’opérateurs différentiels : les opérateurs de Rankin–Cohen et les opérateurs conformément covariants de Juhl. Nous établissons alors des théorèmes de type Parseval–Plancherel pour les transformations de brisure de symétrie associées et trouvons des formules intégrales explicites pour les opérateurs holographiques correspondants.

Published online:
DOI: 10.5802/aif.3386
Classification: 22E45, 22E46, 32M15, 33C45, 33C80, 43A85
Keywords: Symmetry breaking, holographic transform, Rankin–Cohen operators, Juhl operators, orthogonal polynomials, branching rules, F-method
Mot clés : brisure de symétrie, transformation holographique, opérateurs de Rankin–Cohen, opérateurs de Juhl, polynômes orthogonaux, règles de branchement, méthode F
Kobayashi, Toshiyuki 1; Pevzner, Michael 2

1 Graduate School of Mathematical Sciences The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 (Japan) and Kavli Institute for the Physics and Mathematics of the Universe (WPI), 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, (Japan)
2 Laboratoire de Mathématiques de Reims, Université de Reims-Champagne-Ardenne, UMR 9008 du CNRS, F-51687, Reims, (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Kobayashi, Toshiyuki; Pevzner, Michael. Inversion of Rankin–Cohen operators via Holographic Transform. Annales de l'Institut Fourier, Volume 70 (2020) no. 5, pp. 2131-2190. doi : 10.5802/aif.3386.

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