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Journal of Lie Theory 21 (2011), No. 1, 001--054
Copyright Heldermann Verlag 2011

Sheets of Symmetric Lie Algebras and Slodowy Slices

MichaŽl Bulois
77 Rue de Versailles, 91300 Massy, France


\def\g{{\frak g}} \def\k{{\frak k}} \def\l{{\frak l}} \def\p{{\frak p}} \def\N{{\Bbb N}} Let $\theta$ be an involution of the finite dimensional reductive Lie algebra $\g$ and $\g=\k\oplus\p$ be the associated Cartan decomposition. Denote by $K\subset G$ the connected subgroup having $\k$ as Lie algebra. The $K$-module $\p$ is the union of the subsets $\p^{(m)}:=\{x \mid \dim K.x =m\}$, $m \in\N$, and the $K$-sheets of $(\g,\theta)$ are the irreducible components of the $\p^{(m)}$. The sheets can be, in turn, written as a union of so-called Jordan $K$-classes. We introduce conditions in order to describe the sheets and Jordan classes in terms of Slodowy slices. When $\g$ is of classical type, the $K$-sheets are shown to be smooth; if $\g=\g\l_N$ a complete description of sheets and Jordan classes is then obtained.

Keywords: Semisimple Lie algebra, symmetric Lie algebra, sheet, Jordan class, Slodowy slice, nilpotent orbit, root system.

MSC: 14L30, 17B20, 22E46

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