The commuting variety of pgln

Abstract

We are considering the commuting variety of the Lie algebra ${\mathrm{pgl}}_n$ over an algebraically closed field of characteristic $p>0$, namely the set of pairs $\left\{\right(A,B)\in {\mathrm{pgl}}_n\times {\mathrm{pgl}}_n|[A,B]=0\}$. 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 $\left\{\right(x,y)\in G{L}_n(k)\times G{L}_n(k\left)\right|[x,y]=\zeta I\}$ is irreducible of dimension $n^2+n/d$, where ζ is a root of unity of order d with d dividing n.