Sheets of Symmetric Lie Algebras and Slodowy Slices

M. Bulois

Journal of Lie theory, Tome 21 (2011) no. 1, pp. 1-54

Résumé

\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.

Détail

Classification : 14L30, 17B20, 22E46
Mots-clés : Semisimple Lie algebra, symmetric Lie algebra, sheet, Jordan class, Slodowy slice, nilpotent orbit, root system