Some varieties related to the commuting variety of a complex reductive ie algebra

Recently, V. Ginzburg has wrote an article about the isospectral commuting variety. In my talk, I will discuss about some varieties which are constructed analogously to the isospectral commuting varieties. These varieties are subvarieties of varieties having very simple desingularizations. For instance, in a certain sense, the nullcone of the square cartesian of a reductive Lie algebra is related to the isospectral commuting variety. I will try to give some properties of the normalizations of these varities. This a joint work with Mouchira Zaiter and it is still in progress.