Invariant deformations of orbit closures in $\mathfrak{sl}_n$

Sébastien Jansou (1) , Nicolas Ressayre (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

We study deformations of orbit closures for the action of a connected semisimple group $G$ on its Lie algebra $\mathfrak{g}$, especially when $G$ is the special linear group. The tools we use are on the one hand the invariant Hilbert scheme and on the other hand the sheets of $\mathfrak{g}$. We show that when $G$ is the special linear group, the connected components of the invariant Hilbert schemes we get are the geometric quotients of the sheets of $\mathfrak{g}$. These quotients were constructed by Katsylo for a general semisimple Lie algebra $\mathfrak{g}$; in our case, they happen to be affine spaces.

Mots clés: sheet; invariant Hilbert scheme

Domaines: Géométrie algébrique [math.AG]; Théorie des représentations [math.RT]

Sébastien Jansou, Nicolas Ressayre. Invariant deformations of orbit closures in $\mathfrak{sl}_n$. 2007. ⟨hal-00157519v2⟩