On Jordan classes for Vinberg's theta-groups

TL;DR

This paper studies Jordan classes for Vinberg's theta-groups, showing their structure, smoothness, and how their closures decompose, with parametrizations and examples highlighting differences from symmetric cases.

Contribution

It extends the concept of Jordan classes to theta-groups, providing a detailed local analysis, parametrizations, and examples that reveal new structural insights.

Findings

Jordan classes are smooth and their closures are unions of Jordan classes.

Parametrization of Jordan classes and orbits via Vinberg's little Weyl group.

Examples illustrating differences from symmetric cases and key issues in theta-situations.

Abstract

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.