Towards a symplectic version of the Chevalley restriction theorem

Abstract

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 $\mathfrak{c}\oplus \mathfrak{c}^{\ast }/W\rightarrow V\oplus V^{\ast }/\!\!/\!\!/G$. 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 some other examples.