On some families of smooth affine spherical varieties of full rank

Abstract

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 $\mathrm{SL}(2)\times {\mathbb{C}}^{\times }$ 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.

v1: 31 pages. v2: 33 pages, implemented changes requested by referees, changed title, to appear in Acta Mathematica Sinica

References in corpus (2)

• Multiplicity free quasi-Hamiltonian manifolds
• New and old results on spherical varieties via moduli theory