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