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Connecting orbits in quasiaffine spherical varieties via $B$-root subgroups

Roman Avdeev, Vladimir Zhgoon

Abstract

Given a connected reductive algebraic group $G$ with a Borel subgroup $B$ and a quasiaffine spherical $G$-variety $X$, we prove that every $G$-orbit $Y$ contained in the regular locus of $X$ can be connected by a $B$-normalized additive one-parameter group action with any minimal $G$-orbit in $X$ containing $Y$ in its closure. As a consequence, we show that the regular locus of $X$ is transitive for the subgroup in the automorphism group of $X$ generated by $G$ and all $B$-normalized additive one-parameter subgroups.

Connecting orbits in quasiaffine spherical varieties via $B$-root subgroups

Abstract

Given a connected reductive algebraic group with a Borel subgroup and a quasiaffine spherical -variety , we prove that every -orbit contained in the regular locus of can be connected by a -normalized additive one-parameter group action with any minimal -orbit in containing in its closure. As a consequence, we show that the regular locus of is transitive for the subgroup in the automorphism group of generated by and all -normalized additive one-parameter subgroups.

Paper Structure

This paper contains 9 sections, 19 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tools
  4. Local structure theorem
  5. Root subgroups on total spaces of linearized vector bundles
  6. Homogeneous fiber bundles
  7. Main results
  8. General setup
  9. The spherical case

Key Result

Theorem 1.1

Let $X$ be a quasiaffine spherical $G$-variety (not necessarily normal) and let $Y$ be a $G$-orbit contained in the regular locus of $X$. Then for every minimal $G$-orbit $Y' \subset X$ containing $Y$ in its closure there exists a $B$-root subgroup on $X$ that connects $Y$ with $Y'$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 3.1
  • ...and 23 more