Quantum symmetric functions

We study quantum deformations of Poisson orbivarieties. Given a Poisson manifold $({\mathbb{R}}^m,\alpha )$ we consider the Poisson orbivariety $({\mathbb{R}}^m{)}^n\mathrm{/}{S}_n$. The Kontsevich star product on functions on $({\mathbb{R}}^m{)}^n$ induces a star product on functions on $({\mathbb{R}}^m{)}^n\mathrm{/}{S}_n$. We provide explicit formulae for the case $\mathfrak{h}\times \mathfrak{h}\mathrm{/}\mathcal{W}$, where $\mathfrak{h}$ is the Cartan subalgebra of a classical Lie algebra $\mathfrak{g}$ and $\mathcal{W}$ is the Weyl group of $\mathfrak{h}$. We approach our problem from a fairly general point of view, introducing Polya functors for categories over non-symmetric Hopf operads.