Английская Википедия:Chevalley restriction theorem

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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>\mathfrak g</math>	the Lie algebra of G	<math>\mathfrak{sl}_n</math>, the Lie algebra of matrices with trace zero
<math>\mathbb C[\mathfrak g]^G</math>	the polynomial functions on <math>\mathfrak g</math> which are invariant under the adjoint G-action
<math>\mathfrak h</math>	a Cartan subalgebra of <math>\mathfrak g</math>	the subalgebra of diagonal matrices with trace 0
W	the Weyl group of G	the symmetric group Sn
<math>\mathbb C[\mathfrak h]^W</math>	the polynomial functions on <math>\mathfrak h</math> which are invariant under the natural action of W	polynomials f on the space <math>\{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>\mathbb C[\mathfrak g]^{G} \cong \mathbb C[\mathfrak h]^{W}</math>.

Proofs

Шаблон:Harvtxt gives a proof using properties of representations of highest weight. Шаблон:Harvtxt give a proof of Chevalley's theorem exploiting the geometric properties of the map <math>\widetilde \mathfrak g := G \times_B \mathfrak b \to \mathfrak g</math>.

References

• Шаблон:Citation
• Шаблон:Citation