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Degenerations of spherical subalgebras and spherical roots

Roman Avdeev

Abstract

We obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain one-parameter degenerations of the Lie algebras corresponding to such subgroups. As an application, we exhibit explicit algorithms for computing the set of spherical roots of such a spherical subgroup.

Degenerations of spherical subalgebras and spherical roots

Abstract

We obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain one-parameter degenerations of the Lie algebras corresponding to such subgroups. As an application, we exhibit explicit algorithms for computing the set of spherical roots of such a spherical subgroup.

Paper Structure

This paper contains 31 sections, 63 theorems, 57 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and conventions
  4. Levi roots and their properties
  5. One-parameter degenerations in complete varieties
  6. Additive degenerations of subspaces in simple sl_2-modules
  7. Generalities on spherical varieties
  8. Some combinatorial invariants of spherical varieties
  9. Parabolic induction
  10. Spherical modules
  11. A reduction for spherical modules
  12. Classification of spherical modules and some consequences
  13. Generalities on spherical subgroups
  14. Normalizers of spherical subgroups
  15. Wonderful varieties and Demazure embeddings
  16. ...and 16 more sections

Key Result

Proposition \oldthetheorem

The following assertions hold:

Figures (1)

  • Figure 1:

Theorems & Definitions (121)

  • Proposition \oldthetheorem
  • proof : Proof of (\ref{['prop_properties_of_g(lambda)_c']})
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem: see Ko
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Example \oldthetheorem
  • Proposition \oldthetheorem: see BriP
  • ...and 111 more