Commuting varieties in bad characteristic

TL;DR

This work determines the irreducibility and exact dimensions of the commuting variety and its nilpotent counterpart for the symplectic Lie algebra $\mathfrak{sp}_{2n}$ over an algebraically closed field of characteristic $2$. The authors compute centralizer dimensions via Liebeck–Seitz's indecomposable module classification and a discrepancy bound, establishing foundational dimension control for the commuting variety. They construct a dense irreducible component using the action on a maximally abelian subalgebra of dimension $2n$ and apply fibre-dimension and Jordan–Chevalley arguments to show all components lie in this closure. For the nilpotent case, they leverage a parabolic subalgebra and a Baranovsky–Premet fibration to the irreducible nilpotent commuting variety of $\mathfrak{gl}_n$, obtaining the precise dimension $\dim(\mathfrak{sp}_{2n})+n-1$ and irreducibility.

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.

Theorems & Definitions (16)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Theorem 1
• Remark