Component Groups of Centralizers of Nilpotents in Complex Symmetric Spaces

Let G be the adjoint group of a simple Lie algebra , and let KC → Aut( C) be the complexified isotropy representation at the identity coset of the corresponding symmetric space. If e C is nilpotent, we consider the centralizer of e in KC. We show that the conjugacy classes of the component group of this centralizer can be described in terms generalizing the Bala–Carter classification of nilpotent orbits in the complexification of .