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:

	assumption	example
G	complex connected semisimple Lie group	SLn, the special linear group
$𝔤$	the Lie algebra of G	${𝔰𝔩}_n$, the Lie algebra of matrices with trace zero
$ℂ[𝔤{]}^G$	the polynomial functions on $𝔤$ which are invariant under the adjoint G-action
$𝔥$	a Cartan subalgebra of $𝔤$	the subalgebra of diagonal matrices with trace 0
W	the Weyl group of G	the symmetric group Sn
$ℂ[𝔥{]}^W$	the polynomial functions on $𝔥$ which are invariant under the natural action of W	polynomials f on the space $\{x_1,\dots ,x_n,\sum x_i=0\}$ which are invariant under all permutations of the xi

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

$ℂ[𝔤{]}^G\cong ℂ[𝔥{]}^W$.

Proofs

Template:Harvtxt gives a proof using properties of representations of highest weight. Template:Harvtxt give a proof of Chevalley's theorem exploiting the geometric properties of the map $\widetilde{𝔤}:=G{\times }_B𝔟\to 𝔤$.

References

• Template:Citation
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