In this paper, we study sheets of symmetric Lie algebras through their Slodowy slices. In particular, we introduce a notion of slice induction of nilpotent orbits which coincides with the parabolic induction in the Lie algebra case. We also study in more detail the sheets of the non-trivial symmetric Lie algebra of type G 2 . We characterize their singular loci and provide a nice desingularization lying in so 7 .
If (G, V ) is a polar representation with Cartan subspace c and Weyl group W , it is shown that there is a natural morphism of Poisson schemes c ⊕ c * /W → V ⊕ V * / / /G. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if (G, V ) is visible. The conjecture is proved for visible stable locally free polar representations and some other examples.
Soit θ une involution de l'algèbre de Lie semi-simple de dimension finie g et g = k ⊕ p la décomposition de Cartan associée. La variété commutante nilpotente de l'algèbre de Lie symétrique (g, θ) est formée des paires d'éléments nilpotents (x, y) de p tels que [x, y] = 0. Il est conjecturé que cette variété est équidimensionnelle et que ses composantes irréductibles sont indexées par les orbites d'éléments p-distingués. Cette conjecture a été démontrée par A. Premet dans le cas (g × g, θ) avec θ(x, y) = (y, x). Dans ce travail, nous la prouvons dans un grand nombre d'autres cas.
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