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

Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x,y) in g×g 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×g and its normalization morphism is bijective.