Root subgroups on affine spherical varieties

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 \(T \subseteq B\).