Rankin–Cohen brackets and representations of conformal Lie groups

Michael Pevzner

doi:10.5802/ambp.319

Rankin–Cohen brackets and representations of conformal Lie groups
[Crochets de Rankin-Cohen et représentations des groupes de Lie conformes]

Michael Pevzner ¹

¹ University of Reims – FR 3399 CNRS Moulin de la Housse, BP 1037 F-51687, Reims France

Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 2, pp. 455-484.

Résumé
Abstract

Ce texte est une version étendue d’un cours donné par l’auteur lors de l’école d’été Formes quasimodulaires et applications qui s’est tenue à Besse en juin 2010.

L’objectif principal de ce travail est de présenter les crochets de Rankin-Cohen dans le cadre de la théorie des représentations unitaires des groupes de Lie conformes et d’expliquer des résultats récents sur leurs analogues pour des groupes de Lie de rang supérieur. Diverses identités que vérifient de tels opérateurs bi-différentiels covariants seront expliquées en terme de l’associativité d’un produit non commutatif induit sur l’ensemble des formes modulaires holomorphes par la quantification covariante de l’espace symétrique para-hermitien associé.

This is an extended version of a lecture given by the author at the summer school “Quasimodular forms and applications” held in Besse in June 2010.

The main purpose of this work is to present Rankin-Cohen brackets through the theory of unitary representations of conformal Lie groups and explain recent results on their analogues for Lie groups of higher rank. Various identities verified by such covariant bi-differential operators will be explained by the associativity of a non-commutative product induced on the set of holomorphic modular forms by a covariant quantization of the associate para-Hermitian symmetric space.

MR Zbl

DOI : 10.5802/ambp.319

Classification : 11F11, 22E46, 47L80
Keywords: Rankin-Cohen brackets, Unitary representations, Conformal groups, Covariant quantization
Mot clés : Crochets de Rankin-Cohen, représentations unitaires, groupes conformes, quantisation covariante

Affiliations des auteurs :

Michael Pevzner ¹

¹ University of Reims – FR 3399 CNRS Moulin de la Housse, BP 1037 F-51687, Reims France

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Michael Pevzner. Rankin–Cohen brackets and representations of conformal Lie groups. Annales mathématiques Blaise Pascal, Tome 19 (2012) no. 2, pp. 455-484. doi : 10.5802/ambp.319. https://ambp.centre-mersenne.org/articles/10.5802/ambp.319/

Bibliographie
Cité par

[1] Katsuma Ban On Rankin-Cohen-Ibukiyama operators for automorphic forms of several variables, Comment. Math. Univ. St. Pauli, Volume 55 (2006) no. 2, pp. 149-171 | MR | Zbl

[2] Y. Choie; B. Mourrain; P. Solé Rankin-Cohen brackets and invariant theory, J. Algebraic Combin., Volume 13 (2001) no. 1, pp. 5-13 | DOI | MR | Zbl

[3] Henri Cohen Sums involving the values at negative integers of $L$ -functions of quadratic characters, Math. Ann., Volume 217 (1975) no. 3, pp. 271-285 | DOI | MR | Zbl

[4] Paula Beazley Cohen; Yuri Manin; Don Zagier Automorphic pseudodifferential operators, Algebraic aspects of integrable systems (Progr. Nonlinear Differential Equations Appl.), Volume 26, Birkhäuser Boston, Boston, MA, 1997, pp. 17-47 | MR | Zbl

[5] Alain Connes; Henri Moscovici Modular Hecke algebras and their Hopf symmetry, Mosc. Math. J., Volume 4 (2004) no. 1, p. 67-109, 310 | MR | Zbl

[6] Alain Connes; Henri Moscovici Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Mosc. Math. J., Volume 4 (2004) no. 1, p. 111-130, 311 | MR | Zbl

[7] Gerrit van Dijk; Michael Pevzner Ring structures for holomorphic discrete series and Rankin-Cohen brackets, J. Lie Theory, Volume 17 (2007) no. 2, pp. 283-305 | MR | Zbl

[8] Wolfgang Eholzer; Tomoyoshi Ibukiyama Rankin-Cohen type differential operators for Siegel modular forms, Internat. J. Math., Volume 9 (1998) no. 4, pp. 443-463 | DOI | MR | Zbl

[9] Amine M. El Gradechi The Lie theory of the Rankin-Cohen brackets and allied bi-differential operators, Adv. Math., Volume 207 (2006) no. 2, pp. 484-531 | DOI | MR | Zbl

[10] Jacques Faraut; Adam Korányi Analysis on symmetric cones, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1994 (Oxford Science Publications) | MR | Zbl

[11] Mogens Flensted-Jensen Discrete series for semisimple symmetric spaces, Ann. of Math. (2), Volume 111 (1980) no. 2, pp. 253-311 | DOI | MR | Zbl

[12] P. Gordan Vorlesungen über Invariantentheorie. Herausgegeben von G. Kerschensteiner. Zweiter Band: Binäre Formen. 360 S., Leipzig. Teubner, 1887 | MR

[13] S. Gundelfinger Zur Theorie der binären Formen., J. Reine Angew. Math (1887), pp. 413-424

[14] Sigurdur Helgason Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, 80, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978 | MR | Zbl

[15] Roger Howe; Eng-Chye Tan Nonabelian harmonic analysis, Universitext, Springer-Verlag, New York, 1992 Applications of $SL (2, ℝ)$ | DOI | MR | Zbl

[16] Soji Kaneyuki; Masato Kozai Paracomplex structures and affine symmetric spaces, Tokyo J. Math., Volume 8 (1985) no. 1, pp. 81-98 | DOI | MR | Zbl

[17] Toshiyuki Kobayashi Discrete series representations for the orbit spaces arising from two involutions of real reductive Lie groups, J. Funct. Anal., Volume 152 (1998) no. 1, pp. 100-135 | DOI | MR | Zbl

[18] Bertram Kostant On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc., Volume 75 (1969), pp. 627-642 | DOI | MR | Zbl

[19] G. Ólafsson; B. Ørsted The holomorphic discrete series for affine symmetric spaces. I, J. Funct. Anal., Volume 81 (1988) no. 1, pp. 126-159 | DOI | MR | Zbl

[20] Peter J. Olver Classical invariant theory, London Mathematical Society Student Texts, 44, Cambridge University Press, Cambridge, 1999 | DOI | MR | Zbl

[21] Peter J. Olver; Jan A. Sanders Transvectants, modular forms, and the Heisenberg algebra, Adv. in Appl. Math., Volume 25 (2000) no. 3, pp. 252-283 | DOI | MR | Zbl

[22] Toshio Ōshima; Toshihiko Matsuki A description of discrete series for semisimple symmetric spaces, Group representations and systems of differential equations (Tokyo, 1982) (Adv. Stud. Pure Math.), Volume 4, North-Holland, Amsterdam, 1984, pp. 331-390 | MR | Zbl

[23] Lizhong Peng; Genkai Zhang Tensor products of holomorphic representations and bilinear differential operators, J. Funct. Anal., Volume 210 (2004) no. 1, pp. 171-192 | DOI | MR | Zbl

[24] M. Pevzner Analyse conforme sur les algèbres de Jordan, J. Aust. Math. Soc., Volume 73 (2002) no. 2, pp. 279-299 | DOI | MR | Zbl

[25] Michael Pevzner Rankin-Cohen brackets and associativity, Lett. Math. Phys., Volume 85 (2008) no. 2-3, pp. 195-202 | DOI | MR | Zbl

[26] Joe Repka Tensor products of holomorphic discrete series representations, Canad. J. Math., Volume 31 (1979) no. 4, pp. 836-844 | DOI | MR | Zbl

[27] Ichirô Satake Algebraic structures of symmetric domains, Kanô Memorial Lectures, 4, Iwanami Shoten, Tokyo, 1980 | MR | Zbl

[28] Wilfried Schmid Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math., Volume 9 (1969/1970), pp. 61-80 | DOI | MR | Zbl

[29] Robert S. Strichartz Harmonic analysis on hyperboloids, J. Functional Analysis, Volume 12 (1973), pp. 341-383 | DOI | MR | Zbl

[30] André Unterberger; Julianne Unterberger Algebras of symbols and modular forms, J. Anal. Math., Volume 68 (1996), pp. 121-143 | DOI | MR | Zbl

[31] Don Zagier Modular forms and differential operators, Proc. Indian Acad. Sci. Math. Sci., Volume 104 (1994) no. 1, pp. 57-75 (K. G. Ramanathan memorial issue) | DOI | MR | Zbl

[32] Genkai Zhang Rankin-Cohen brackets, transvectants and covariant differential operators, Math. Z., Volume 264 (2010) no. 3, pp. 513-519 | DOI | MR | Zbl

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