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The nilpotent cone of a reductive Lie algebra has a desingularization givenby thecotangent bundle of the flag variety. Analogously, the nullcone of acartesianpower of the algebra has a desingularization given by a vector bundleover theflag variety. As for the nullcone, the subvariety of elements whosecomponentsare in a same Borel subalgebra, has a desingularization given by avector bundle overthe flag variety. In this note, some properties of thesevarieties are given. Forthe study of the commuting variety, the analogousvariety to the flag variety isthe closure in the Grassmannian of the set ofCartan subalgebras. So someproperties of this variety are given. In particular,it is smooth in codimension $1$.We introduce the generalized isospectralcommuting varieties and give some properties.Furthermore, desingularizations ofthese varieties are given by fiber bundles over adesingularization of theclosure in the grassmannian of the set of Cartan subalgebrascontained in agiven Borel subalgebra.

On the Commuting variety of a reductive Lie algebra and other related varieties