Root subgroups on horospherical varieties

Abstract

Given a connected reductive algebraic group $G$ and a spherical $G$-variety $X$, a $B$-root subgroup on $X$ is a one-parameter additive group of automorphisms of $X$ normalized by a Borel subgroup $B \subset G$. We obtain a complete description of all $B$-root subgroups on a certain open subset of $X$. When $X$ is horospherical, we extend the construction of standard $B$-root subgroups introduced earlier by Arzhantsev and Avdeev for affine $X$ and obtain a complete description of all standard $B$-root subgroups, which naturally generalizes the well-known description of root subgroups on toric varieties. As an application, for horospherical $X$ that is either complete or contains a unique closed $G$-orbit, we determine all $G$-stable prime divisors in $X$ that can be connected with the open $G$-orbit via the action of a suitable $B$-root subgroup. For horospherical $X$, we also find sufficient conditions for the existence of $B$-root subgroups on $X$ that preserve the open $B$-orbit in $X$. Finally, when $G$ is of semisimple rank $1$ and $X$ is horospherical and complete, we determine all $B$-root subgroups on $X$, which enables us to describe the Lie algebra of the connected automorphism group of $X$.