On the existence of $B$-root subgroups on affine spherical varieties

TL;DR

This paper establishes combinatorial conditions for the existence of certain additive group actions on affine spherical varieties, and demonstrates their role in connecting prime divisors with the open orbit.

Contribution

It provides new criteria based on weight combinatorics for the existence of $B$-root subgroups on affine spherical varieties.

Findings

Sufficient conditions for $B$-normalized additive actions are formulated.

Every $G$-stable prime divisor can be connected to the open orbit via such actions.

Abstract

Let $X$ be an irreducible affine algebraic variety that is spherical with respect to an action of a connected reductive group $G$. In this paper we provide sufficient conditions, formulated in terms of weight combinatorics, for the existence of one-parameter additive actions on $X$ normalized by a Borel subgroup $B \subset G$. As an application, we prove that every $G$-stable prime divisor in $X$ can be connected with the open $G$-orbit by means of a suitable $B$-normalized one-parameter additive action.