Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra, $G$ the adjoint group and $W$ the Weyl group. The Chevalley restriction theorem says that the natural map $$\Bbb C[\mathfrak{g}]^G\to \Bbb C[\mathfrak{h}]^W$$ is a ring isomorphism.

Now, let $$M=\{(x,y)\in\mathfrak{g}\times\mathfrak{g}:[x,y]=0\}$$ be the variety of commuting pairs and let $G$ act on $M$ diagonally. Is $$\Bbb C[M]^G\to\Bbb C[\mathfrak{h}\times\mathfrak{h}]^W$$ also a ring isomorphism?