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

TL;DR

This paper explores the geometric properties and desingularizations of various varieties associated with reductive Lie algebras, including the commuting variety and its generalizations, revealing their structure and smoothness characteristics.

Contribution

It introduces new properties and desingularizations of commuting and related varieties, extending understanding of their geometric and algebraic structures.

Findings

The nullcone of a cartesian power has a vector bundle desingularization.

The subvariety with components in the same Borel subalgebra is smooth in codimension 1.

Generalized isospectral commuting varieties are introduced with specific properties.

Abstract

The nilpotent cone of a reductive Lie algebra has a desingularization given by thecotangent bundle of the flag variety. Analogously, the nullcone of a cartesianpower of the algebra has a desingularization given by a vector bundle over theflag variety. As for the nullcone, the subvariety of elements whose componentsare in a same Borel subalgebra, has a desingularization given by a vector bundle overthe flag variety. In this note, some properties of these varieties are given. Forthe study of the commuting variety, the analogous variety to the flag variety isthe closure in the Grassmannian of the set of Cartan subalgebras. So someproperties 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 adesingularization of the closure in the grassmannian of the set of Cartan subalgebrascontained in a given Borel subalgebra.