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

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

2007-06-26

arxiv.org/abs/0706.3828v2

Mathematics

Algebraic Geometry

16 pages

Scientific paper

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.

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