Champs de vecteurs invariants sur une algèbre de Lie réductive complexe. (Invariant vector fields on a reductive complex Lie algebra)
Let \({\mathfrak g}\) be a simple complex Lie algebra, \({\mathfrak h}\) be a Cartan subalgebra of \({\mathfrak g}\), W be the Weyl group of the pair (\({\mathfrak g},{\mathfrak h})\), and G be the adjoint group of \({\mathfrak g}\). A classical result of Chevalley asserts that there exists an isomorphism of the algebra of G-invariant polynomial functions on \({\mathfrak g}\) onto the algebra of W-invariant polynomial functions on \({\mathfrak h}\), induced by restriction. \textit{J. Sekiguchi} [J. Math. Soc. Japan 36, 147-159 (1984; Zbl 0539.17005)] proved that there exists a similar isomorphism between the Lie algebra of G-invariant vector fields on \({\mathfrak g}\) and a certain Lie algebra of vector fields on \({\mathfrak h}/W.\) In the paper under review, one shows that Sekiguchi's result (and some generalizations) can be obtained from a theorem of \textit{G. Schwarz} [Publ. Math., Inst. Hautes Etud. Sci. 51, 37-135 (1980; Zbl 0449.57009)].
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