On a variety related to the commuting variety of a reductive Lie algebra

TL;DR

This paper investigates a specific algebraic variety associated with a reductive Lie algebra, proving it has Gorenstein properties and rational singularities, which are important in the context of commuting varieties.

Contribution

It establishes that the variety X related to the Borel subgroup orbit closure in the Grassmannian is Gorenstein with rational singularities.

Findings

X is Gorenstein

X has rational singularities

The variety plays a key role in commuting variety studies

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 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.