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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

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Chevalley's theorem requires the following notation:

assumption example
G complex connected semisimple Lie group SLn, the special linear group
the Lie algebra of G , the Lie algebra of matrices with trace zero
the polynomial functions on which are invariant under the adjoint G-action
a Cartan subalgebra of the subalgebra of diagonal matrices with trace 0
W the Weyl group of G the symmetric group Sn
the polynomial functions on which are invariant under the natural action of W polynomials f on the space which are invariant under all permutations of the xi

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

.

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 .

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