On the nullcone and the variety $\chi_\mathfrak{g}$ of a semisimple lie algebra

TL;DR

This paper studies the nullcone of a semisimple Lie algebra, showing it is a closed, irreducible variety with a normalization that has rational singularities and is bijective.

Contribution

It proves the nullcone is a closed, irreducible subvariety with a normalization having rational singularities and a bijective normalization morphism.

Findings

Nullcone is a closed, irreducible subvariety of g×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\timesg such that x and y are nilpotents and are in the same Borel subalgebra. The main result of this paper is that N is a closed and irreducible subvariety of g \times g whose normalization has rational singularities and such that the normalization morphism is bijective.