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
${\displaystyle {\mathfrak {g}}}$	the Lie algebra of G	${\displaystyle {\mathfrak {sl}}_{n}}$ , the Lie algebra of matrices with trace zero
${\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}}$	the polynomial functions on ${\displaystyle {\mathfrak {g}}}$  which are invariant under the adjoint G-action
${\displaystyle {\mathfrak {h}}}$	a Cartan subalgebra of ${\displaystyle {\mathfrak {g}}}$	the subalgebra of diagonal matrices with trace 0
W	the Weyl group of G	the symmetric group Sn
${\displaystyle \mathbb {C} [{\mathfrak {h}}]^{W}}$	the polynomial functions on ${\displaystyle {\mathfrak {h}}}$  which are invariant under the natural action of W	polynomials f on the space ${\displaystyle \{x_{1},\dots ,x_{n},\sum x_{i}=0\}}$  which are invariant under all permutations of the xi

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

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

Proofs

Humphreys (1980) gives a proof using properties of representations of highest weight. Chriss & Ginzburg (2010) give a proof of Chevalley's theorem exploiting the geometric properties of the map ${\displaystyle {\widetilde {\mathfrak {g}}}:=G\times _{B}{\mathfrak {b}}\to {\mathfrak {g}}}$ .

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. S2CID 14890248. Zbl 1185.22001.
• Humphreys, James E. (1980). Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics. Vol. 9. Springer. Zbl 0447.17002.