Cartan connections and natural and projectively equivariant quantizations

In this paper, the question of existence of a natural and projectively equivariant symbol calculus is analysed using the theory of projective Cartan connections. A close relationship is established between the existence of such a natural symbol calculus and the existence of an sl( m + 1, ℝ)‐equivariant calculus over ℝ m . Moreover, it is shown that the formulae that hold in the non‐critical situations over Rm for the sl( m + 1,ℝ)‐equivariant calculus can be directly generalized to an arbitrary manifold by simply replacing the partial derivatives by invariant differentiations with respect to a Cartan connection.