Chevalley restriction theorem

In the mathematical theory of Lie groups, the Chevalley restriction theorem describes functions on a Lie algebra which are invariant under the action of a Lie group in terms of functions on a Cartan subalgebra.

Statement
Chevalley's theorem requires the following notation:

Chevalley's theorem asserts that the restriction of polynomial functions induces an isomorphism
 * $$\mathbb C[\mathfrak g]^{G} \cong \mathbb C[\mathfrak h]^{W}$$.

Proofs
gives a proof using properties of representations of highest weight. give a proof of Chevalley's theorem exploiting the geometric properties of the map $$\widetilde \mathfrak g := G \times_B \mathfrak b \to \mathfrak g$$.