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.

Received:
Revised:
Accepted:
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
@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/}
}
TY  - JOUR
AU  - Kobayashi, Toshiyuki
AU  - Pevzner, Michael
TI  - Inversion of Rankin–Cohen operators via Holographic Transform
JO  - Annales de l'Institut Fourier
PY  - 2020
SP  - 2131
EP  - 2190
VL  - 70
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3386/
DO  - 10.5802/aif.3386
LA  - en
ID  - AIF_2020__70_5_2131_0
ER  - 
%0 Journal Article
%A Kobayashi, Toshiyuki
%A Pevzner, Michael
%T Inversion of Rankin–Cohen operators via Holographic Transform
%J Annales de l'Institut Fourier
%D 2020
%P 2131-2190
%V 70
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3386/
%R 10.5802/aif.3386
%G en
%F AIF_2020__70_5_2131_0
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. 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 | Zbl

[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 | Zbl

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

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

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

[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 | DOI | MR | Zbl

[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

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

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

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

[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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

Cited by Sources: