Dunkl operator and quantization of ℤ₂-singularity

Let (X,ω) be a symplectic orbifold which is locally like the quotient of a ℤ₂ action on ℝⁿ. Let A_X ^((ℏ)) be a deformation quantization of X constructed via the standard Fedosov method with characteristic class being ω. In this paper, we construct a deformation of the algebra A _X^((ℏ)) parametrized by codimension 2 components of the associated inertia orbifold X̃. This partially confirms a conjecture of Dolgushev and Etingof in the case of ℤ₂ orbifolds. To do so, we generalize the interpretation of the Moyal star-product as a composition of symbols of pseudodifferential operators in the case where partial derivatives are replaced with Dunkl operators. The star-products we obtain can be seen as globalizations of symplectic reflection algebras.