On commuting varieties of nilradicals of Borel subalgebras of reductive Lie algebras

TL;DR

This paper studies the structure of commuting varieties of nilradicals in Lie algebras of Borel subgroups, identifying conditions for equidimensionality and describing irreducible components in specific cases.

Contribution

It characterizes when the commuting variety is equidimensional and describes its irreducible components based on Borel orbit structures.

Findings

Commuting variety is equidimensional when Borel acts with finitely many orbits.

Irreducible components correspond to distinguished Borel orbits.

In cases with infinitely many orbits, the structure of components is explicitly determined.

Abstract

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $\mathbbm k$ of characteristic zero. We consider the commuting variety $\mathcal C(\mathfrak u)$ of the nilradical $\mathfrak u$ of the Lie algebra $\mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $\mathfrak u$ with only a finite number of orbits, we verify that $\mathcal C(\mathfrak u)$ is equidimensional and that the irreducible components are in correspondence with the {\em distinguished} $B$-orbits in $\mathfrak u$. We observe that in general $\mathcal C(\mathfrak u)$ is not equidimensional, and determine the irreducible components of $\mathcal C(\mathfrak u)$ in the minimal cases where there are infinitely many $B$-orbits in $\mathfrak u$.