Irregular and Singular Loci of Commuting Varieties

Abstract

Let ${\user1{\mathcal{C}}}$ be the commuting variety of the Lie algebra $\mathfrak{g}$ of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let ${\user1{\mathcal{C}}}^{{{\text{sing}}}}$ be the singular locus of ${\user1{\mathcal{C}}}$ and let ${\user1{\mathcal{C}}}^{{{\text{irr}}}}$ be the locus of points whose G-stabilizers have dimension > rk G. We prove that: (a) ${\user1{\mathcal{C}}}^{{{\text{sing}}}}$ is a nonempty subset of ${\user1{\mathcal{C}}}^{{{\text{irr}}}}$; (b) ${\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{irr}}}} = 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)}$ where the maximum is taken over all simple ideals $\mathfrak{a}$ of $\mathfrak{g}$ and $l{\left( \mathfrak{a} \right)}$ is the “lacety” of $\mathfrak{a}$; and (c) if $\mathfrak{t}$ is a Cartan subalgebra of $\mathfrak{g}$ and $\alpha \in \mathfrak{t}^{*}$ root of $\mathfrak{g}$ with respect to $\mathfrak{t}$, then $\overline{{G{\left( {{\text{Ker}}\,\alpha \times {\text{Ker }}\alpha } \right)}}}$ is an irreducible component of ${\user1{\mathcal{C}}}^{{{\text{irr}}}}$ of codimension 4 in ${\user1{\mathcal{C}}}$. This yields the bound ${\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 5 - {\text{max}}\,l{\left( \mathfrak{a} \right)}$ and, in particular, ${\text{codim}}_{{\user1{\mathcal{C}}}} \,{\user1{\mathcal{C}}}^{{{\text{sing}}}} \geqslant 2$. The latter may be regarded as an evidence in favor of the known longstanding conjecture that ${\user1{\mathcal{C}}}$ is always normal. We also prove that the algebraic variety ${\user1{\mathcal{C}}}$ is rational.