Commuting varieties in bad characteristic

TL;DR

This paper proves the irreducibility and computes the dimensions of the commuting and commuting nilpotent varieties of the symplectic Lie algebra over an algebraically closed field of characteristic 2.

Contribution

It establishes irreducibility and determines the dimensions of these commuting varieties in bad characteristic, specifically for the symplectic Lie algebra.

Findings

Both varieties are irreducible.

Dimensions are $ ext{dim}(rak{sp}_{2n}) + 2n$ and $ ext{dim}(rak{sp}_{2n}) + n - 1$.

Results hold in characteristic 2, a bad characteristic for the algebra.

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.