Star-Representations sur des sous-varietes co-isotropes

Abstract

For a coisotropic (or first-class) submanifold C of a Poisson manifold X we consider star-products for which the vanishing ideal I of C becomes a left ideal in the deformed algebra thus defining a left module structure on the space of smooth functions on C. We show how this can be deduced from a formality conjecture a la Tamarkin generalized to cochains compatible with C. To this end we first prove a theorem a la Hochschild-Kostant-Rosenberg between the space of compatible multivector fields and compatible multidifferential operators. We then equip the latter with a G-infinity structure, and prove that the obstructions to the formality are controlled by certain cohomology groups which we reduce in the case of C being a subvectorspace of the vectorspace M. In codimension 1 we conjecture -encouraged by low-dimensional examples and Gloessner's representation theorem (1998)- that formality holds. For higher codimensions it is not impossible that obstructions occur which in the symplectic case are linked to the Atiyah-Molino class of a regular foliation.