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

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 $mathfrak u$.