Mathematics – Representation Theory
Scientific paper
2006-09-25
Mathematics
Representation Theory
To be published in Annales de l'ENS. French, 78 pages, 13 figures, English Introduction. Typographic errors corrected. Section
Scientific paper
In this article we use the expansion for biquantization described in Cattaneo-Felder [math.QA/0309180] for the case of symmetric spaces. We introduce a function of two variables $E(X,Y)$ 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 $E$ function. We prove that Cattaneo and Felder's star product coincides with Rouviere'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 $\sigma$-stable polarizations.
Cattaneo Alberto S.
Torossian Charles
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