On a Variety Related to the Commuting Variety of a Reductive Lie Algebra

Abstract

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 in the 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 this note, we prove that $X$ is Gorenstein with rational singularities.