ON JORDAN CLASSES FOR VINBERG’S θ-GROUPS

V. L. Popov has recently introduced an analogue of Jordan classes (packets or decomposition classes) for the action of a theta-group (G(0), V), showing that they are finitely-many, locally-closed, irreducible unions of G(0)-orbits of constant dimension partitioning V. We carry out a local study of their closures showing that Jordan classes are smooth and that their closure is a union of Jordan classes. We parametrize Jordan classes and G(0)-orbits in a given class in terms of the action of subgroups of Vinberg's little Weyl group, and include several examples and counterexamples underlying the differences with the symmetric case and the critical issues arising in the theta-situation.