Noncommutative Dunkl operators and braided Cherednik algebras

Abstract.

We introduce braided Dunkl operators \(\underline{\nabla}_1,\ldots,\underline{\nabla}_n\) that act on a q-symmetric algebra \(S_{\bf q}({\mathbb{C}}^n)\) and q-commute. Generalising the approach of Etingof and Ginzburg, we explain the q-commutation phenomenon by constructing braided Cherednik algebras \(\underline{{\mathcal{H}}}\) for which the above operators form a representation. We classify all braided Cherednik algebras using the theory of braided doubles developed in our previous paper. Besides ordinary rational Cherednik algebras, our classification gives new algebras \(\underline{{\mathcal{H}}}(W_+)\) attached to an infinite family of subgroups of even elements in complex reflection groups, so that the corresponding braided Dunkl operators \(\underline{\nabla}_i\) pairwise anticommute. We explicitly compute these new operators in terms of braided partial derivatives and W₊-divided differences.