On the Dimension of the Sheets of a Reductive Lie Algebra

A. Moreau

Journal of Lie Theory, Tome 18 (2008) no. 3, pp. 671-696

Résumé

\def\g{{\frak g}} \def\l{{\frak l}} Let $\g$ be a complex finite dimensional Lie algebra and $G$ its adjoint group. Following a suggestion of A. A. Kirillov, we investigate the dimension of the subset of linear forms $f\in\g^*$ whose coadjoint orbit has dimension $2m$, for $m\in\mathbb{N}$. In this paper we focus on the reductive case. In this case the problem reduces to the computation of the dimension of the sheets of $\g$. These sheets are known to be parameterized by the pairs $(\l, {\cal O}_\l)$, up to $G$-conjugacy class, consisting of a Levi subalgebra $\l$ of $\g$ and a rigid nilpotent orbit ${\cal O}_\l$ in $\l$. By using this parametrization, we provide the dimension of the above subsets for any $m$.

Détail

Classification : 14A10, 14L17, 22E20, 22E46
Mots-clés : Reductive Lie algebra, coadjoint orbit, sheet, index, Jordan class, induced nilpotent orbit, rigid nilpotent orbit