  For a reductive Lie algbera over an algbraically closed field of
charasteristic zero,we consider a borel subgroup $B$ of its adjoint group, a
Cartan subalgebra contained inthe Lie algebra of $B$ and the closure $X$ of its
orbit under $B$ in the Grassmannian.The variety $X$ plays an important role in
the study of the commuting variety. In thisnote, we prove that $X$ is
Gorenstein with rational singularities.
