Chevalley restriction theorem
# 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 | $SL_n,$ the special linear group |
| ${\mathfrak {g}}$ | the Lie algebra of G | ${\mathfrak {sl}}_{n}$, 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 $S_n$ |
| $\mathbb {C} [{\mathfrak {h}}]^{W}$ | 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 $x_i$ |

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 (2010) give a proof of Chevalley's theorem exploiting the geometric properties of the map ${\widetilde {\mathfrak {g}}}:=G\times _{B}{\mathfrak {b}}\to {\mathfrak {g}}$.

