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

Jean-Yves Charbonnel, Mouchira Zaiter

Published 2012-04-02, updated 2014-12-23Version 3

In this note, we discuss some varieties which are constructed analogously to the isospectral commuting varieties. These varieties are subvarieties of varieties having very simple desingularizations. For instance, this is the case of the nullcone of any cartesian power of a reductive Lie algebra and we prove that it is normal. Moreover, as a byproduct of these investigations and a Ginzburg's result, we get that the normalizations of the isospectral commuting variety and the commuting variety have rational singularities.