On related varieties to the commuting variety of a semisimple Lie algebra

TL;DR

This paper studies the nullcone of a semisimple Lie algebra, proving it is a closed, irreducible variety with rational singularities and a bijective normalization morphism, advancing understanding of its geometric properties.

Contribution

It establishes the geometric structure and singularity properties of the nullcone associated with a semisimple Lie algebra.

Findings

Nullcone is a closed, irreducible subvariety of g x g

Normalization of the nullcone has rational singularities

Normalization morphism is bijective

Abstract

Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x,y) in g x g such that x and y are nilpotents and are in the same Borel sualgebra. The main result of this paper is that N is a closed and irreducible subvariety of g x g, its normalisation has rational singularities and its normalization morphism is bijective.