Rankin–Cohen brackets and representations of conformal Lie groups

Pevzner, Michael

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

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.

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

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