Affine spherical homogeneous spaces with good quotient by a maximal   unipotent subgroup
  For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies some simple condition. 
Roman Avdeev
math.AG math.RT