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

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Date
2012-12-04MFO Scientific Program
Research in Pairs 2011Series
Oberwolfach Preprints;2012,14Author
Goodwin, Simon M.
Röhrle, Gerhard
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Abstract
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $\mathbb{k}$ 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 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 $\mathfrak{u}$.