Journal of Lie Theory 18 (2008), No. 3, 671--696
Copyright Heldermann Verlag 2008
On the Dimension of the Sheets of a Reductive Lie Algebra
Anne Moreau
ETH Zürich, Departement Mathematik, Rämistrasse 101, HG G66.4, 8092 Zürich, Switzerland
anne.moreau@math.ethz.ch
[Abstract-pdf]
\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 $m$.
Keywords: Reductive Lie algebra, coadjoint orbit, sheet, index, Jordan class, induced nilpotent orbit, rigid nilpotent orbit.
MSC: 14A10, 14L17, 22E20, 22E46
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