The centralisers of nilpotent elements in classical Lie algebras

Publication date: 5 July 2004

Abstract: The index of a finite-dimensional Lie algebra $g$ is the minimum of dimensions of stabilisers $g_alpha$ of elements $alphain g^*$. Let $g$ be a reductive Lie algebra and $z(x)$ a centraliser of a nilpotent element $xin g$. Elashvili has conjectured that the index of the centraliser $z(x)$ equals the index of $g$, i.e., the rank of $g$. Here Elashvili's conjecture is proved for reductive Lie algebras of classical type. It is shown that in cases $g=gl_n$ and $g=sp_{2n}$ the coadjoint action of $z(x)$ has a generic stabiliser. Also, we give an example of a nilpotent element $xin so_8$ such that the coadjoint action of $z(x)$ has no generic stabiliser.

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