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
$\mathfrak{𝔤}$	the Lie algebra of G	${\mathfrak{𝔰}\mathfrak{𝔩}}_n$, the Lie algebra of matrices with trace zero
$\mathbb{ℂ}[\mathfrak{𝔤}{]}^G$	the polynomial functions on $\mathfrak{𝔤}$ which are invariant under the adjoint G-action
$\mathfrak{𝔥}$	a Cartan subalgebra of $\mathfrak{𝔤}$	the subalgebra of diagonal matrices with trace 0
W	the Weyl group of G	the symmetric group Sn
$\mathbb{ℂ}[\mathfrak{𝔥}{]}^W$	the polynomial functions on $\mathfrak{𝔥}$ 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

$\mathbb{ℂ}[\mathfrak{𝔤}{]}^G\cong \mathbb{ℂ}[\mathfrak{𝔥}{]}^W$.

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 $\widetilde{\mathfrak{𝔤}}:=G{\times }_B\mathfrak{𝔟}\to \mathfrak{𝔤}$.

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.