  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.
