Poisson deformations of symplectic quotient singularities

We establish a connection between smooth symplectic resolutions and symplectic deformations of a (possibly singular) affine Poisson variety. In particular, let V be a finite-dimensional complex symplectic vector space and G subset of Sp(V) a finite subgroup. Our main result says that the so-called Calogero-Moser deformation of the orbifold V/G is, in an appropriate sense, a versal Poisson deformation. That enables us to determine the algebra structure on the cohomology H-.(X, C) of any smooth symplectic resolution X --> V/G (multiplicative McKay correspondence). We prove further that if G subset of GL(h) is an irreducible Weyl group and V = h circle plus h*, then no smooth symplectic resolution of V/G exists unless G is of types A, B, C. (C) 2003 Elsevier Inc. All rights reserved.