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The Index of Centralizers of Elements in Classical Lie Algebras
O. S. Yakimova Independent University of Moscow
Abstract:
The index of a finite-dimensional Lie algebra
$\mathfrak{g}$ is the minimum of dimensions of the stabilizers
$\mathfrak{g}_\alpha$ over all covectors
$\alpha\in\mathfrak{g}^*$. Let
$\mathfrak{g}$ be a reductive Lie algebra over a field
$\mathbb{K}$ of characteristic
$\ne2$. Élashvili conjectured that the index of
$\mathfrak{g}_\alpha$ is always equal to the index, or, which is the same, the rank of
$\mathfrak{g}$. In this article, Élashvili's conjecture is proved for classical Lie algebras. Furthermore, it is shown that if
$\mathfrak{g}=\mathfrak{gl}_n$ or
$\mathfrak{g}=\mathfrak{sp}_{2n}$ and
$e\in\mathfrak{g}$ is a nilpotent element, then the coadjoint action of
$\mathfrak{g}_e$ has a generic stabilizer. For
$\mathfrak{g}=\mathfrak{so}_n$, we give examples of nilpotent elements
$e\in\mathfrak{g}$ such that the coadjoint action of
$\mathfrak{g}_e$ does not
have a generic stabilizer.
UDC:
512.815.1 Received: 29.06.2004
DOI:
10.4213/faa18
Fulltext:
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