Geodesic symmetries and invariant star products on Kähler symmetric spaces (Q1093222)

Starting from work by F. A. Berezin, an earlier paper by the author obtained an invariant star product on every nonexceptional symmetric Kähler space. This would be a generalization to those spaces of the star product on \({\mathbb{R}}^{2n}\) corresponding to Wick quantization. In this letter we consider, via geometric quantization, the unitary operators corresponding to geodesic symmetries, and we define a Weyl quantization (first defined by Berezin on rank 1 spaces) in a way similar to the way in which the Weyl quantization can be obtained from the Wick quantization of \({\mathbb{R}}^{2n}\). We then calculate every Hochschild 2- cochain of another invariant star product, equivalent to the Wick one, which would be a generalization to those spaces of the Moyal star product on \({\mathbb{R}}^{2n}\). M. Cahen and S. Gutt have already provided a theorem of existence and essential unicity of an invariant star product on every irreducible Kähler symmetric space.