Commuting varieties in bad characteristic

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.

💡 Research Summary

The paper investigates the commuting variety and the commuting nilpotent variety of the symplectic Lie algebra $\mathfrak{sp}_{2n}$ over an algebraically closed field $k$ of characteristic 2, a setting traditionally regarded as “bad” characteristic for type C groups. The main results are two theorems:

Theorem A. The commuting variety \