Methods of Equivariant Quantization

This article is a survey of recent work [15, 6, 7, 13] developing a new approach to quantization based on the equivariance with respect to some Lie group of symmetries. Examples are provided by conformai and projective differential geometry: given a smooth manifold M endowed with a flat conformal/projective structure, we establish a canonical isomorphism between the space of symmetric contravariant tensor fields on M and the space of differential operators on M. This leads to a notion of conformally/projectively invariant star product on T * M.