Centers of centralizers of nilpotent elements in exceptional Lie superalgebras

Abstract

Let $\mathfrak{g}={\mathfrak{g}}_{\bar{0}}\oplus {\mathfrak{g}}_{\bar{1}}$ be a finite-dimensional simple Lie superalgebra of type $D(2,1;\alpha )$, $G(3)$ or $F(4)$ over $\mathbb{C}$. Let $G$ be the simply connected semisimple algebraic group over $\mathbb{C}$ such that $\mathrm{Lie}(G)={\mathfrak{g}}_{\bar{0}}$. Suppose $e\in {\mathfrak{g}}_{\bar{0}}$ is nilpotent. We describe the centralizer ${\mathfrak{g}}^e$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}({\mathfrak{g}}^e)$ especially. We also determine the labelled Dynkin diagram for $e$. We prove theorems relating the dimension of ${\left(\mathfrak{z}({\mathfrak{g}}^e)\right)}^{G^e}$ and the labelled Dynkin diagram.

31 pages, 14 figures, accepted by Journal of algebra and its applications