The index of a Lie algebra, the centralizer of a nilpotent element, and the normalizer of the centralizer

Abstract

Let ${\rm q}$ be a Lie algebra and ξ a linear form on ${\rm q}$. Denote by ${\rm q}_{\xi}$ the set of all $x \in {\rm q}$ such that $\xi([{\rm q},x]) = 0$. In other words, ${\rm q}_{\xi}=\{x\in {\rm q} \mid({\rm ad}^*x)\cdot = 0\}$, where ${\rm ad}^*: {\rm q}\rightarrow {\rm g I}({\rm q}^*)$ is the coadjoint representation of ${\rm q}$. The index of ${\rm q}$, ind ${\rm q}$, is defined by \[ {\rm ind\, q} = \min_{\xi\in {\rm q}^*}\dim {\rm q}\xi. \]