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

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.

This paper has been withdrawn and replaced by arXiv:1105.5921v1