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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?

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Apparently the answer to your underlying question is yes, due to a short 1997 note (mostly in English) by Tony Joseph here. See especially his Theorem 2.9. You may also want to consult some of his earlier references. Since it's quite natural, this kind of question has certainly been raised much earlier, though the differences in notation may have led me to misunderstand something about the comparison between Joseph's formulation and yours.

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