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On Jordan classes for Vinberg's theta-groups

Giovanna Carnovale, Francesco Esposito, Andrea Santi

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.

On Jordan classes for Vinberg's theta-groups

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.

Paper Structure

This paper contains 15 sections, 43 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Vinberg's -groups and Jordan classes
  3. Graded Lie algebras
  4. The Jordan decomposition
  5. The Cartan subspace
  6. Dimensions of centralizers and regularity conditions
  7. Jordan classes and sheets for -groups
  8. Closure of a $G_0$-Jordan class
  9. Closure of $G_0$-Jordan classes: the semisimple parts
  10. A local study of the closure of a $G_0$-Jordan class
  11. Regularity questions
  12. Slice-induction and parametrization of orbits and classes
  13. Slice-induction
  14. Parametrization of orbits and classes
  15. Cartan, Levi and parabolic subalgebras in $\mathbb Z_m$-graded Lie algebras

Key Result

Lemma 2.1

There exists an adapted bilinear form $\kappa$ on $\mathfrak{g}$ if and only if $\mathfrak{z}(\mathfrak{g})$ is symmetrically graded, i.e., $\dim\mathfrak{z}(\mathfrak{g})_{l}=\dim\mathfrak{z}(\mathfrak{g})_{-l}$ for all $l\in\mathbb Z_m$. In this case $\dim\mathfrak{g}_{l}=\dim\mathfrak{g}_{-l}$ fo

Theorems & Definitions (83)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Theorem 2.6
  • Proposition 2.7
  • ...and 73 more