Chevalley restriction theorem explained

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:

	assumption	example
G		SLn, the special linear group
ak g	the Lie algebra of G	ak{sl} _{n} , the Lie algebra of matrices with trace zero
C[ ak g] ^{G}	the polynomial functions on ak g which are invariant under the adjoint G-action
ak h	a Cartan subalgebra of ak g	the subalgebra of diagonal matrices with trace 0
W	the Weyl group of G	the symmetric group Sn
C[ ak h] ^{W}	the polynomial functions on ak h which are invariant under the natural action of W	polynomials f on the space \{x _{1,} ... , x _{n} , \sum x _{i} =0 \} which are invariant under all permutations of the xi

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

C[ ak g] ^{G} \cong C[ ak 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 ak g : = G x _{B} ak b \to ak g