On some families of smooth affine spherical varieties of full rank

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Let G be a complex connected reductive group. I. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify: (a) all such varieties when G is SL(2) × C× and (b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and F. Knop's classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of C. Woodward's result that every reflective Delzant polytope is the moment polytope of such a manifold.