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 $\{ (x,y) \in GL_n(k) \times GL_n(k) \mid [x,y]=\zeta I \}$ is irreducible of dimension $n^2 +n/d$, where $\zeta$ is a root of unity of order $d$ with $d$ dividing $n$.