# On some families of smooth affine spherical varieties of full rank

## TL;DR

This paper classifies certain smooth affine spherical varieties of full rank for specific groups using combinatorial methods, and connects these classifications to moment polytopes of multiplicity free Hamiltonian manifolds.

## Contribution

It provides a combinatorial classification of smooth affine spherical varieties of full rank for specific groups and relates these to Hamiltonian manifold moment polytopes.

## Key findings

- Classified all smooth affine spherical varieties for G=SL(2)×C× and for simple G with full rank monoids.
- Established a new proof that reflective Delzant polytopes are moment polytopes of multiplicity free Hamiltonian manifolds.

## 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.