Root subgroups on 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 $T\subseteq B$.

v3: 31 pages, revised according to the referee's suggestions, numbering of sections changed

References in corpus (5)

• Complete toric varieties with reductive automorphism group
• Commutative algebraic monoid structures on affine surfaces
• On extended weight monoids of spherical homogeneous spaces
• On the existence of $B$-root subgroups on affine spherical varieties
• Characterizing quasi-affine spherical varieties via the automorphism group

Cited by in corpus (1)

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