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We introduce and study deformations of finite-dimensional modules over rational Cherednik algebras. Our main tool is a generalization of usual harmonic polynomials for each Coxeter group --- the so-called quasiharmonic polynomials. A surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all Coxeter groups. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]