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

TL;DR

This paper demonstrates that in quasiaffine spherical varieties, the regular locus is highly connected via $B$-normalized additive group actions, linking orbits and automorphisms.

Contribution

It establishes the connectivity of $G$-orbits in the regular locus through $B$-normalized additive actions, extending understanding of automorphism groups in spherical varieties.

Findings

Every $G$-orbit in the regular locus can be connected via $B$-normalized additive actions.

The regular locus is transitive under the subgroup generated by $G$ and $B$-normalized additive groups.

Provides a method to connect orbits in spherical varieties using $B$-root subgroups.

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.