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
[math]\displaystyle{ \mathfrak g }[/math]	the Lie algebra of G	[math]\displaystyle{ \mathfrak{sl}_n }[/math], the Lie algebra of matrices with trace zero
[math]\displaystyle{ \mathbb C[\mathfrak g]^G }[/math]	the polynomial functions on [math]\displaystyle{ \mathfrak g }[/math] which are invariant under the adjoint G-action
[math]\displaystyle{ \mathfrak h }[/math]	a Cartan subalgebra of [math]\displaystyle{ \mathfrak g }[/math]	the subalgebra of diagonal matrices with trace 0
W	the Weyl group of G	the symmetric group Sn
[math]\displaystyle{ \mathbb C[\mathfrak h]^W }[/math]	the polynomial functions on [math]\displaystyle{ \mathfrak h }[/math] which are invariant under the natural action of W	polynomials f on the space [math]\displaystyle{ \{x_1, \dots, x_n , \sum x_i =0 \} }[/math] which are invariant under all permutations of the xi

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

[math]\displaystyle{ \mathbb C[\mathfrak g]^{G} \cong \mathbb C[\mathfrak h]^{W} }[/math].

Proofs

(Humphreys 1980) gives a proof using properties of representations of highest weight. (Chriss Ginzburg) give a proof of Chevalley's theorem exploiting the geometric properties of the map [math]\displaystyle{ \widetilde \mathfrak g := G \times_B \mathfrak b \to \mathfrak g }[/math].

References

• Chriss, Neil; Ginzburg, Victor (2010), Representation theory and complex geometry., Birkhäuser, doi:10.1007/978-0-8176-4938-8, ISBN 978-0-8176-4937-1 
• Humphreys, James E. (1980), Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, 9, Springer