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Acta Mathematica Sinica, English Series

, Volume 34, Issue 3, pp 563–596 | Cite as

On Some Families of Smooth Affine Spherical Varieties of Full Rank

  • Kay Paulus
  • Guido Pezzini
  • Bart Van Steirteghem
Article
  • 32 Downloads

Abstract

Let G be a complex connected reductive group. 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 for G = 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 Knop’s classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of Woodward’s result that every reflective Delzant polytope is the moment polytope of such a manifold.

Keywords

Affine spherical variety weight monoid multiplicity free Hamiltonian manifold moment polytope 

MR(2010) Subject Classification

14M27 20G05 53D20 

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Notes

Acknowledgements

The authors wish to thank Michel Brion and Baohua Fu for organizing the International Conference on Spherical Varieties at the Tsinghua Sanya International Mathematics Forum in November 2016. Van Steirteghem presented some of the results in this paper during his talk at the Conference, and is grateful for the invitation.

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Kay Paulus
    • 1
  • Guido Pezzini
    • 2
  • Bart Van Steirteghem
    • 1
    • 3
  1. 1.Department MathematikFAU Erlangen-NürnbergErlangenGermany
  2. 2.Dipartimento di Matematica“Sapienza” Università di RomaRomaItaly
  3. 3.Department of MathematicsMedgar Evers College-City University of New YorkBrooklynUSA

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