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 $\text{mathbbm}k$ 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 $\mathfrak{u}$.