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On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilizers
Published online by Cambridge University Press: 01 July 2016
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
$\dim \text{Stab}_{G}\langle v\rangle =\dim \text{Stab}_{\mathfrak{g}}\langle v\rangle$.
MSC classification
- Type
- Research Article
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- Copyright
- © The Author 2016
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