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

Jean-Yves Charbonnel (1) , Mouchira Zaiter (2)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 UNIVERSITE LIBANAISE Faculté de Sciences II

The nilpotent cone of a reductive Lie algebra has a desingularization given by the cotangent bundle of the flag variety. Analogously, the nullcone of a cartesian power of the algebra has a desingularization given by a vector bundle over the flag variety. As for the nullcone, the subvariety of elements whose components are in a same Borel subalgebra, has a desingularization given by a vector bundle over the flag variety. In this note, some properties of these varieties are given. For the study of the commuting variety, the analogous variety to the flag variety is the closure in the Grassmannian of the set of Cartan subalgebras. So some properties of this variety are given. In particular, it is smooth in codimension $1$. We introduce the generalized isospectral commuting varieties and give some properties. Furthermore, desingularizations of these varieties are given by fiber bundles over a desingularization of the closure in the grassmannian of the set of Cartan subalgebras contained in a given Borel subalgebra.

Mots clés: commuting variety; desingularization; polynomial algebra; complex

Domaines: Théorie des représentations [math.RT]; Géométrie algébrique [math.AG]; Théorie des groupes [math.GR]

Jean-Yves Charbonnel, Mouchira Zaiter. On the Commuting variety of a reductive Lie algebra and other related varieties.. Journal of Algebra, 2016. ⟨hal-00684400v7⟩