Towards a symplectic version of the Chevalley restriction theorem

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak{c}$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak{c}\oplus {\mathfrak{c}}^{\ast }\mathrm{/}W\to V\oplus V^{\ast }\mathrm{/}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{/}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{/}G$. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if $(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and certain further examples.