Sheets of Symmetric Lie Algebras and Slodowy Slices

Michaël Bulois (1)

1 LM - Laboratoire de mathématiques de Brest

Let T be an involution of the finite dimensional complex reductive Lie algebra g and g=k+p be the associated Cartan decomposition. Denote by K the adjoint group of k. The K-module p is the union of the subsets p^{(m)}={x | dim K.x =m}, indexed by integers m, and the K-sheets of (g,T) 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 K-classes in terms of Slodowy slices. When g is of classical type, the K-sheets are shown to be smooth; if g=gl_N a complete description of sheets and Jordan K-classes is then obtained.

Mots clés: slodowy slice; sheets; symmetric Lie algebras; semisimple Lie algebras

Domaines: Théorie des représentations [math.RT]; Géométrie algébrique [math.AG]

Michaël Bulois. Sheets of Symmetric Lie Algebras and Slodowy Slices. Journal of Lie Theory, 2011, 21, pp.1-54. ⟨hal-00464531⟩