Maximal commutative unipotent subgroups and a characterization of affine spherical varieties

We describe maximal commutative unipotent subgroups of the automorphism group Aut(X) of an irreducible affine variety X. Furthermore, we show that a group isomorphism Aut(X) → Aut(Y) maps unipotent elements to unipotent elements, where Y is irreducible and affine. Using this result, we show that the automorphism group detects sphericity and the weight monoid. As an application, we show that an affine toric variety different from an algebraic torus is determined by its automorphism group among normal irreducible affine varieties, and we show that a smooth affine spherical variety different from an algebraic torus is determined by its automorphism group (up to an automorphism of the base field) among smooth irreducible affine varieties. This generalizes results obtained by Cantat, Kraft, Liendo, Urech, Xie and the authors.