Commuting varieties in bad characteristic

Abstract

Let $k$ be an algebraically closed field of characteristic $2$. We consider the commuting variety and the commuting nilpotent variety of the Lie algebra ${\mathfrak{sp}}_{2n}$, namely the sets ${\mathcal{C}}_2({\mathfrak{sp}}_{2n})=\{(x,y)\in {\mathfrak{sp}}_{2n}\times {\mathfrak{sp}}_{2n}\mid [x,y]=0\}$ and ${\mathcal{C}}_2^{\text{nil}}({\mathfrak{sp}}_{2n})=\{(x,y)\in {\mathfrak{sp}}_{2n}\times {\mathfrak{sp}}_{2n}\mid x,y\text{ nilpotent, }[x,y]=0\}$ and prove that they are both irreducible, of dimensions $\dim ({\mathfrak{sp}}_{2n})+2n$ and $\dim ({\mathfrak{sp}}_{2n})+n-1$, respectively.