Dunkl operator and quantization of ${\mathbb{Z}}_2$-singularity

Abstract

Let $(X,\omega )$ be a symplectic orbifold which is locally like the quotient of a ${\mathbb{Z}}_2$ action on ${\mathbb{R}}^n$. Let $A_X^{((\hslash ))}$ be a deformation quantization of $X$ constructed via the standard Fedosov method with characteristic class being $\omega$. In this paper, we construct a universal deformation of the algebra $A_X^{((\hslash ))}$ parametrized by codimension 2 components of the associated inertia orbifold $\tilde{X}$. This partially confirms a conjecture of Dolgushev and Etingof in the case of ${\mathbb{Z}}_2$ orbifolds. To do so, we generalize the interpretation of Moyal star-product as a composition of symbol 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.

24 pages

References in corpus (1)

• Deformation quantization of Poisson manifolds, I

Cited by in corpus (2)

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