Root subgroups on affine spherical varieties

Given a connected reductive algebraic group G and a Borel subgroup B ⊆ 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 ⊆ B .