Root subgroups on affine spherical varieties

TL;DR

This paper studies $B$-normalized additive group actions on affine spherical varieties, providing a construction that generalizes toric cases and classifying such actions for horospherical and rank-one cases.

Contribution

It introduces a generalized construction of normalized additive actions and fully classifies them for affine horospherical and rank-one affine spherical varieties.

Findings

Complete description of $G$-normalized actions on affine horospherical varieties.

Connection of the open orbit with all $G$-stable divisors via these actions.

Classification of actions for rank-one affine spherical varieties.

Abstract

Given a connected reductive algebraic group $G$ and a Borel subgroup $B \subseteq G$, we study $B$-normalized one-parameter additive group actions on affine spherical $G$-varieties. We establish basic properties of such actions and their weights and discuss many examples exhibiting various features. We propose a construction of such actions that generalizes the well-known construction of normalized one-parameter additive group actions on affine toric varieties. Using this construction, for every affine horospherical $G$-variety $X$ we obtain a complete description of all $G$-normalized one-parameter additive group actions on $X$ and show that the open $G$-orbit in $X$ can be connected with every $G$-stable prime divisor via a suitable choice of a $B$-normalized one-parameter additive group action. Finally, when $G$ is of semisimple rank $1$, we obtain a complete description of all $B$-normalized one-parameter additive group actions on affine spherical $G$-varieties having an open orbit of a maximal torus $T \subseteq B$.