Characterizing smooth affine spherical varieties via the automorphism group

Let G be a connected reductive algebraic group. We prove that for a quasi-affine G-spherical variety the weight monoid is determined by the weights of its non-trivial G_a-actions that are homogeneous with respect to a Borel subgroup of G. As an application we get that a smooth affine spherical variety that is non-isomorphic to a torus is determined by its automorphism group (considered as an ind-group) inside the category of smooth affine irreducible varieties.