On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let $\mathfrak{g}$ be a reductive Lie algebra over an algebraically closed field of characteristic zero and $\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1$ an arbitrary $\mathbb{Z}_2$-grading. We consider the variety $\mathfrak{C}_1=\{(x,y)\mid[x,y]=0\}\subset\mathfrak{g}_1\times\mathfrak{g}_1$, which is called the commuting variety associated with the $\mathbb{Z}_2$-grading. Earlier it was proved by the author that $\mathfrak{C}_1$ is irreducible, if the $\mathbb{Z}_2$-grading is of maximal rank. Now we show that $\mathfrak{C}_1$ is irreducible for $(\mathfrak{g},\mathfrak{g}_0)=(\mathfrak{sl}_{2n},\mathfrak{sp}_{2n})$ and $(\textrm{E}_6,\textrm{F}_4)$. In the case of symmetric pairs of rank one, we show that the number of irreducible components of $\mathfrak{C}_1$ is equal to that of nonzero non-$\vartheta$-regular nilpotent $G_0$-orbits in $\mathfrak{g}_1$. We also discuss a general problem of the irreducibility of commuting varieties.