Quantification pour les paires symétriques et diagrammes de Kontsevich

Abstract

In this article we use the expansion for biquantization described in [7] for the case of symmetric spaces. We introduce a function of two variables pE (X,Y ) for any symmetric pairs. This function has an expansion in terms of Kontsevich’s diagrams. We recover most of the known results though in a more systematic way by using some elementary properties of this pE function. We prove that Cattaneo and Felder’s star product coincides with Rouvière’s for any symmetric pairs. We generalize some of Lichnerowicz’s results for the commutativity of the algebra of invariant differential operators and solve a long standing problem posed by M. Duflo for the expression of invariant differential operators on any symmetric spaces in exponential coordinates. We describe the Harish-Chandra homomorphism in the case of symmetric spaces by using all these constructions. We develop a new method to construct characters for algebras of invariant differential operators. We apply these methods in the case of pσ -stable polarizations.