Non-uniqueness of the natural and projectively equivariant quantization

In [C. Duval, V. Ovsienko, Projectively equivariant quantization and symbol calculus: Noncommutative hypergeometric functions, Lett. Math. Phys. 57 (1) (2001) 61–67], the authors showed the existence and the uniqueness of a sl(m+1,R)-equivariant quantization in non-critical situations. The curved generalization of the sl(m+1,R)-equivariant quantization is the natural and projectively equivariant quantization. In [M. Bordemann, Sur l’existence d’une prescription d’ordre naturelle projectivement invariante (submitted for publication). math.DG/0208171] and [Pierre Mathonet, Fabian Radoux, Natural and projectively equivariant quantizations by means of Cartan connections, Lett. Math. Phys. 72 (3) (2005) 183–196], the existence of such a quantization was proved in two different ways. In this paper, we show that this quantization is not unique.