The commuting variety of ${\mathfrak{pgl}}_n$

Abstract

We are considering the commuting variety of the Lie algebra ${\mathfrak{pgl}}_n$ over an algebraically closed field of characteristic $p>0$, namely the set of pairs $\{(A,B)\in {\mathfrak{pgl}}_n\times {\mathfrak{pgl}}_n\mid [A,B]=0\}$. We prove that if $n=pr$, then there are precisely two irreducible components, of dimensions $n^2+r-1$ and $n^2+n-2$. We also prove that the variety $\{(x,y)\in G{L}_n(k)\times G{L}_n(k)\mid [x,y]=\zeta I\}$ is irreducible of dimension $n^2+n/d$, where $\zeta$ is a root of unity of order $d$ with $d$ dividing $n$.