Inversion of Rankin–Cohen operators via Holographic Transform
[Inversion des opérateurs de Rankin–Cohen par transformation holographique]
Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 2131-2190.

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.

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.

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DOI : https://doi.org/10.5802/aif.3386
Classification : 22E45,  22E46,  32M15,  33C45,  33C80,  43A85
Mots clés : brisure de symétrie, transformation holographique, opérateurs de Rankin–Cohen, opérateurs de Juhl, polynômes orthogonaux, règles de branchement, méthode F
@article{AIF_2020__70_5_2131_0,
     author = {Kobayashi, Toshiyuki and Pevzner, Michael},
     title = {Inversion of Rankin{\textendash}Cohen operators via Holographic Transform},
     journal = {Annales de l'Institut Fourier},
     pages = {2131--2190},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {70},
     number = {5},
     year = {2020},
     doi = {10.5802/aif.3386},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3386/}
}
Kobayashi, Toshiyuki; Pevzner, Michael. Inversion of Rankin–Cohen operators via Holographic Transform. Annales de l'Institut Fourier, Tome 70 (2020) no. 5, pp. 2131-2190. doi : 10.5802/aif.3386. https://aif.centre-mersenne.org/articles/10.5802/aif.3386/

[1] Andrews, Georges E.; Askey, Richard; Roy, Ranjan Special functions, Encyclopedia of Mathematics and Its Applications, 71, Cambridge University Press, 1999 | MR 1688958 | Zbl 0920.33001

[2] Clerc, Jean-Louis Another approach to Juhl’s conformally covariant differential operators from S n to S n-1 , SIGMA, Symmetry Integrability Geom. Methods Appl., Volume 13 (2017), 26 | MR 3635957 | Zbl 1365.58020

[3] Cohen, Henri Sums involving the values at negative integers of L-functions of quadratic characters, Math. Ann., Volume 217 (1975), pp. 271-285 | Article | MR 382192 | Zbl 0311.10030

[4] Debnath, Lokenath; Bhatta, Dambaru Integral transforms and their applications, CRC Press, 2015 | Zbl 1310.44001

[5] Faraut, Jacques; Korányi, Adam Analysis on Symmetric Cones, Oxford Science Publications; Oxford Mathematical Monographs, Clarendon Press; Oxford University Press, 1994 | Zbl 0841.4300

[6] Fefferman, Charles L.; Graham, C. Robin Juhl’s formulae for GJMS operators and Q-curvatures, J. Am. Math. Soc., Volume 26 (2013) no. 4, pp. 1191-1207 | Article | MR 3073887 | Zbl 1276.53042

[7] Gradshteyn, I. S.; Ryzhik, I. M. Table of integrals, series, and products, Elsevier/Academic Press, 2015 (Translated from the Russian. With a preface by Daniel Zwillinger and Victor Moll, Eighth edition. Revised from the seventh edition) | Zbl 1300.65001

[8] Harish-Chandra Representations of semisimple Lie groups on a Banach space, Proc. Natl. Acad. Sci. USA, Volume 37 (1951), pp. 170-173 | Article | Zbl 0042.12602

[9] Juhl, Andreas Families of conformally covariant differential operators, Q-curvature and holography, Progress in Mathematics, 275, Springer, 2009 | Article | MR 2521913 | Zbl 1177.53001

[10] Kashiwara, Masaki; Kowata, Atsutaka; Minemura, Katsuhiro; Okamoto, Kiyosato; Oshima, Toshio; Tanaka, Makoto Eigenfunctions of invariant differential operators on a symmetric space, Ann. Math., Volume 107 (1978), pp. 1-39 | Article | MR 485861 | Zbl 0377.43012

[11] Kobayashi, Toshiyuki Discrete decomposability of the restriction of A 𝔮 (λ) with respect to reductive subgroups II: Micro-local analysis and asymptotic K-support, Ann. Math., Volume 147 (1998) no. 3, pp. 709-729 | MR 1637667 | Zbl 0910.22016

[12] Kobayashi, Toshiyuki Discrete decomposability of the restriction of A 𝔮 (λ) with respect to reductive subgroups III: Restriction of Harish-Chandra modules and associated varieties, Invent. Math., Volume 131 (1998) no. 2, pp. 229-256 | Article | MR 1608642 | Zbl 0907.22016

[13] Kobayashi, Toshiyuki Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs. Representation theory and automorphic forms, Representation Theory and Automorphic Forms (Progress in Mathematics), Volume 255, Springer, 2008, pp. 45-109 | Article | MR 2369547 | Zbl 1304.22013

[14] Kobayashi, Toshiyuki Restrictions of generalized Verma modules to symmetric pairs, Transform. Groups, Volume 17 (2012) no. 2, pp. 523-546 | Article | MR 2921076 | Zbl 1257.22014

[15] Kobayashi, Toshiyuki F-method for symmetry breaking operators, Differ. Geom. Appl., Volume 33 (2014), pp. 272-289 | Article | MR 3159963 | Zbl 1311.22016

[16] Kobayashi, Toshiyuki; Kubo, Toshihisa; Pevzner, Michael Conformal symmetry breaking operators for anti-de Sitter spaces, Geometric methods in physics XXXV (Trends in Mathematics), Birkhäuser/Springer, 2018, pp. 69-85 | Zbl 1403.53081

[17] Kobayashi, Toshiyuki; Oshima, T. Finite multiplicity theorems for induction and restriction, Adv. Math., Volume 248 (2013), pp. 921-944 | Article | MR 3107532 | Zbl 1317.22010

[18] Kobayashi, Toshiyuki; Pevzner, Michael Differential symmetry breaking operators. I. General theory and F-method., Sel. Math., New Ser., Volume 22 (2016) no. 2, pp. 801-845 | Article | MR 3477336 | Zbl 1338.22006

[19] Kobayashi, Toshiyuki; Pevzner, Michael Differential symmetry breaking operators. II. Rankin–Cohen operators for symmetric pairs., Sel. Math., New Ser., Volume 22 (2016) no. 2, pp. 847-911 | Article | MR 3477337 | Zbl 1342.22029

[20] Kobayashi, Toshiyuki; Speh, Birgit Symmetry Breaking for Representations of Rank One Orthogonal Groups, Memoirs of the American Mathematical Society, 238, American Mathematical Society, 2015 | Article | MR 3400768 | Zbl 1334.22015

[21] Maldacena, Juan M. The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys., Volume 2 (1998) no. 2, pp. 231-252 | Article | MR 1633016 | Zbl 0914.53047

[22] Molchanov, Vladimir F. Tensor products of unitary representations of the three-dimensional Lorentz group, Math. USSR, Izv., Volume 15 (1980), pp. 113-143 | Article | Zbl 0448.22010

[23] Repka, Joe Tensor products of holomorphic discrete series representations, Can. J. Math., Volume 31 (1979), pp. 836-844 | Article | MR 540911 | Zbl 0373.22006

[24] Witten, Edward Anti-de-Sitter space and holography, Adv. Theor. Math. Phys., Volume 2 (1998) no. 2, pp. 253-291 | Article | MR 1633012 | Zbl 0914.53048

[25] Zagier, Don Bernard Modular forms and differential operators, Proc. Indian Acad. Sci., Math. Sci., Volume 104 (1994) no. 1, pp. 57-75 | Article | MR 1280058 | Zbl 0806.11022