Hochschild (co)homology of the Dunkl operator quantization of $Z_2$-singularity

We study Hochschild (co)homology groups of the Dunkl operator quantization of $\Z_2$-singularity constructed by Halbout and Tang. Further, we study traces on this algebra and prove a local algebraic index formula.

💡 Research Summary

The paper investigates the Hochschild (co)homology of the Dunkl‑operator quantization of a (\mathbb{Z}{2})‑singularity, a non‑commutative deformation introduced by Halbout and Tang. The authors first recall the construction of the algebra (\mathcal{A}{\hbar}): it is obtained by taking the algebra of smooth functions on the orbifold (\mathbb{C}^{2}/\mathbb{Z}{2}) and deforming the product with Dunkl operators, keeping a formal parameter (\hbar). A natural filtration on (\mathcal{A}{\hbar}) yields an associated graded algebra isomorphic to the commutative polynomial algebra on (\mathbb{C}^{2}) with the (\mathbb{Z}_{2})‑action, which allows the authors to apply a non‑commutative version of the Hochschild‑Kostant‑Rosenberg (HKR) theorem.

Using the filtration, a spectral sequence is built to compute Hochschild homology (HH_{\bullet}(\mathcal{A}{\hbar})). On the (E^{1})‑page the homology coincides with that of the graded algebra; the (\mathbb{Z}{2})‑fixed locus splits into a zero‑dimensional component (the origin) and a one‑dimensional component (the fixed line). These give rise to two families of cycles: the “regular” cycles, which survive the deformation and correspond to the usual de Rham forms on the smooth part, and the “singular” cycles, which are supported near the fixed locus and encode the non‑commutative contribution of the Dunkl operators. Consequently, \