Journal of Lie Theory

no. 3
On the Dimension of the Sheets of a Reductive Lie Algebra
Journal of Lie Theory, Volume 18 (2008) no. 3, pp. 671-696
\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 $m$.
DOI: 10.5802/jolt.518
Classification: 14A10, 14L17, 22E20, 22E46
Keywords: Reductive Lie algebra, coadjoint orbit, sheet, index, Jordan class, induced nilpotent orbit, rigid nilpotent orbit 
@article{JOLT_2008_18_3_a11,
     author = {A. Moreau},
     title = {On the {Dimension} of the {Sheets} of a {Reductive} {Lie} {Algebra}},
     journal = {Journal of Lie Theory},
     pages = {671--696},
     year = {2008},
     volume = {18},
     number = {3},
     doi = {10.5802/jolt.518},
     zbl = {1155.22010},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.518/}
}
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A. Moreau. On the Dimension of the Sheets of a Reductive Lie Algebra. Journal of Lie Theory, Volume 18 (2008) no. 3, pp. 671-696. doi: 10.5802/jolt.518

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