Towards a symplectic version of the Chevalley restriction theorem

TL;DR

This paper extends the Chevalley restriction theorem to a symplectic setting for polar representations, establishing a natural Poisson scheme morphism and proving it is an isomorphism in specific cases.

Contribution

It introduces a symplectic analogue of the Chevalley restriction theorem and proves the conjecture for certain classes of polar representations.

Findings

Established a Poisson scheme morphism for polar representations.

Proved the morphism is an isomorphism for visible stable locally free cases.

Conjectured the isomorphism holds generally for visible representations.

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 $\mathfrak c \oplus {\mathfrak c}^*/W \to V\oplus V^*/\!\!/\!\!/ G$. 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.