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

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Chevalley's theorem requires the following notation:

| assumption | example

|
|

G | complex connected semisimple Lie group | SLn, the special linear group
g {\displaystyle {\mathfrak {g}}} | the Lie algebra of G |  s l n {\displaystyle {\mathfrak {sl}}_{n}} , the Lie algebra of matrices with trace zero
C [ g ] G {\displaystyle \mathbb {C} [{\mathfrak {g}}]^{G}} | the polynomial functions on  g {\displaystyle {\mathfrak {g}}} which are invariant under the adjoint G-action |
h {\displaystyle {\mathfrak {h}}} | a Cartan subalgebra of  g {\displaystyle {\mathfrak {g}}} | the subalgebra of diagonal matrices with trace 0
W | the Weyl group of G | the symmetric group Sn
C [ h ] W {\displaystyle \mathbb {C} [{\mathfrak {h}}]^{W}} | the polynomial functions on  h {\displaystyle {\mathfrak {h}}} which are invariant under the natural action of W | polynomials f on the space  { x 1 , … , x n , ∑ x i = 0 } {\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

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

## Proofs

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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  g ~ := G × B b -> g {\displaystyle {\widetilde {\mathfrak {g}}}:=G\times _{B}{\mathfrak {b}}\to {\mathfrak {g}}} .

## References

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* 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