Centres of centralizers of nilpotent elements in Lie superalgebras $\mathfrak{sl}(m|n)$ or $\mathfrak{osp}(m|2n)$

Abstract

Let $\bar{G}$ be the simple algebraic supergroup $\mathrm{SL}(m|n)$ or $\mathrm{OSp}(m|2n)$ over $\mathbb{C}$. Let $\mathfrak{g}=\mathrm{Lie}(\bar{G})={\mathfrak{g}}_{\bar{0}}\oplus {\mathfrak{g}}_{\bar{1}}$ and let $G=\bar{G}(\mathbb{C})$ where $\mathbb{C}$ is considered as a superalgebra concentrated in even degree. Suppose $e\in {\mathfrak{g}}_{\bar{0}}$ is nilpotent. We describe the centralizer ${\mathfrak{g}}^e$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}({\mathfrak{g}}^e)$. In particular, we give bases for ${\mathfrak{g}}^e$, $\mathfrak{z}({\mathfrak{g}}^e)$ and ${\left(\mathfrak{z}({\mathfrak{g}}^e)\right)}^{G^e}$. We also determine the labelled Dynkin diagram $\Delta$ with respect to $e$ and subsequently describe the relation between ${\left(\mathfrak{z}({\mathfrak{g}}^e)\right)}^{G^e}$ and $\Delta$.

28 pages, 21 figures, accepted by Journal of Algebra and Its Applications in 2021