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 .

In this note we present Hironaka's invariants as developped by Giraud: the ridge and the directrix. We give an effective definition and a functorial one and show their equivalence. The fruit is an effective algorithm that computes the additive generators of the "ridge", and the generators of its invariant algebra.

De Concini and Procesi have defined in [1] the wonderful compactification X of a symmetric space X = G/G σ where G is a semisimple adjoint group and G σ the subgroup of fixed points of G by an involution σ. It is a closed subvariety of a grassmannian of the Lie algebra g of G. In this paper, we prove that, when the rank of X is equal to the rank of G, the variety is defined by linear equations. The set of equations expresses the fact that the invariant alternate trilinear form w on g vanishes on the −1-eigenspace of σ.

In this note we present Hironaka's invariants as developped by Giraud: the ridge and the directrix. We give an effective definition and a functorial one and show their equivalence. The fruit is an effective algorithm that computes the additive generators of the "ridge", and the generators of its invariant algebra.

No abstract