On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilizers

Abstract

Let$G$be a simple simply connected exceptional algebraic group of type$G_{2}$,$F_{4}$,$E_{6}$or$E_{7}$over an algebraically closed field$k$of characteristic$p>0$with$\mathfrak{g}=\text{Lie}(G)$. For each nilpotent orbit$G\cdot e$of$\mathfrak{g}$, we list the Jordan blocks of the action of$e$on the minimal induced module$V_{\text{min}}$of$\mathfrak{g}$. We also establish when the centralizers$G_{v}$of vectors$v\in V_{\text{min}}$and stabilizers$\text{Stab}_{G}\langle v\rangle$of$1$-spaces$\langle v\rangle \subset V_{\text{min}}$are smooth; that is, when$\dim G_{v}=\dim \mathfrak{g}_{v}$or$\dim \text{Stab}_{G}\langle v\rangle =\dim \text{Stab}_{\mathfrak{g}}\langle v\rangle$.