
Journal of Lie Theory 21 (2011), No. 1, 001054 Copyright Heldermann Verlag 2011 Sheets of Symmetric Lie Algebras and Slodowy Slices Michaël Bulois 77 Rue de Versailles, 91300 Massy, France michael.bulois@univangers.fr [Abstractpdf] \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 socalled 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 [ Fulltextpdf (798 KB)] for subscribers only. 