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:
assumptionexample
Gcomplex connected semisimple Lie groupSLn, the special linear group
the Lie algebra of G, the Lie algebra of matrices with trace zero
the polynomial functions on which are invariant under the adjoint G-action
a Cartan subalgebra ofthe subalgebra of diagonal matrices with trace 0
Wthe Weyl group of Gthe symmetric group Sn'
the polynomial functions on which are invariant under the natural action of Wpolynomials f on the space which are invariant under all permutations of the xi''

Chevalley's theorem asserts that the restriction of polynomial functions induces an isomorphism

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.