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Rankin-Cohen brackets and representations of conformal Lie groups. (Crochets de Rankin-Cohen et représentations des groupes de Lie conformes.) (English. French summary) Zbl 1283.11072
Summary: 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.

MSC:
11F11 Holomorphic modular forms of integral weight
22E46 Semisimple Lie groups and their representations
47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
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