Skip to main content
SpringerLink
Log in

CONVEXITY AND THIMM’S TRICK

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

In this paper we study topological properties of maps constructed by Thimm's trick with Guillemin and Sternberg's action coordinates on a connected Hamiltonian G-manifold M. Since these maps only generate a Hamiltonian torus action on an open dense subset of M, convexity and fibre-connectedness of such maps does not follow immediately from Atiyah–Guillemin–Sternberg's convexity theorem, even if M is compact. The core contribution of this paper is to provide a simple argument circumventing this difficulty.

In the case where the map is constructed from a chain of subalgebras we prove that the image is given by a list of inequalities that can be computed explicitly in many examples. This generalizes the fact that the images of the classical Gelfand–Zeitlin systems on coadjoint orbits are Gelfand–Zeitlin polytopes. Moreover, we prove that if such a map generates a completely integrable torus action on an open dense subset of M, then all its fibres are smooth embedded submanifolds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. Alamiddine, Géométrie de systèmes hamiltoniens intégrables: Le cas du systemème de Gelfand–Cetlin, PhD thesis, Université Toulouse, 2009.

  2. A. Alekseev, A. Malkin, E. Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998), no. 3, 445–495.

    Article  MathSciNet  Google Scholar 

  3. M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15.

    Article  MathSciNet  Google Scholar 

  4. M. Audin, Torus Actions on Symplectic Manifolds, 2nd ed., Progress in Mathematics, Vol. 93, Birkhäuser Verlag, Basel, 2004.

    Chapter  Google Scholar 

  5. V. Baldoni, M. Vergne, Multiplicity of compact group representations and applications to Krönecker coefficients, arXiv:1506.02472v1 (2015).

  6. A. Berenstein, R. Sjamaar, Coadjoint orbits, moment polytopes, and the Hilbert–Mumford criterion, J. Amer. Math. Soc. 13 (2000), no. 2, 433–466.

    Article  MathSciNet  Google Scholar 

  7. P. Birtea, J.-P. Ortega, T. S. Ratiu, A local-to-global principle for convexity in metric spaces, J. Lie Theory 18 (2008), no. 2, 445–469.

    MathSciNet  MATH  Google Scholar 

  8. P. Birtea, J.-P. Ortega, T. S. Ratiu, Openness and convexity for momentum maps, Trans. Amer. Math. Soc. 2 (2009), no. 361, 603–630.

    MathSciNet  MATH  Google Scholar 

  9. C. Bjorndahl, Y. Karshon, Revisiting Tietze–Nakajima: local and global convexity for maps, Canad. J. Math. 62 (2010), no. 5, 975–993.

    Article  MathSciNet  Google Scholar 

  10. D. Bouloc, Singular fibres of the bending flows on the moduli space of 3d polygons, arXiv:1505.04748 (2015).

  11. A. Caviedes Castro, Upper bound for the Gromov width of coadjoint orbits of compact Lie groups, arXiv:1404.4647v4 (2014).

  12. M. Condevaux, P. Dazord, P. Molino, Géométrie du moment, Travaux du Séminaire Sud-Rhodanien de Géométrie, I, Publ. Dép. Math. Nouvelle Sér. B, Vol. 88, Univ. Claude-Bernard, Lyon, 1988, pp. 131–160.

  13. Th. Delzant, Hamiltoniens périodiques et images convexes de l'application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339.

    Article  MathSciNet  Google Scholar 

  14. X. Fang, P. Littelmann, M. Pabiniak, Simplices in Newton–Okounkov bodies and the Gromov width of coadjoint orbits, arXiv:1607.01163 (2016).

  15. H. Flaschka, T. Ratiu, A convexity theorem for Poisson actions of compact Lie groups, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 6, 787–809.

    Article  MathSciNet  Google Scholar 

  16. M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. (2) 82 (1985), 307–347.

    Article  MathSciNet  Google Scholar 

  17. V. Guillemin, Sh. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491–513.

    Article  MathSciNet  Google Scholar 

  18. V. Guillemin, Sh. Sternberg, The Gelfand–Cetlin system and quantization of the complex flag manifolds, J. Funct. Anal. 52 (1983), no. 1, 106–128.

    Article  MathSciNet  Google Scholar 

  19. V. Guillemin, Sh. Sternberg, On collective complete integrability according to the method of Thimm, Ergodic Theory Dynam. Systems 3 (1983), no. 2, 219–230.

    Article  MathSciNet  Google Scholar 

  20. V. Guillemin, Sh. Sternberg, Multiplicity free spaces, J. Differential Geom. 19 (1984), no. 1, 31–56.

    Article  MathSciNet  Google Scholar 

  21. V. Guillemin, Sh. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1984.

    MATH  Google Scholar 

  22. I. Halacheva, M. Pabiniak, The Gromov width of coadjoint orbits of the symplectic group, arXiv:1601.02825 (2016).

  23. J.-C. Hausmann, A. Knutson, Polygon spaces and Grassmannians, Enseign. Math. (2) 43 (1997), no. 1–2, 173–198.

  24. M. Harada, The symplectic geometry of the Gel'fand–Cetlin–Molev basis for representations of Sp(2n;ℂ), J. Symplectic Geom. 4 (2006), no. 1, 1–41.

    Article  MathSciNet  Google Scholar 

  25. M. Harada. Kuimars Kaveh, Integrable systems, toric degenerations and Okounkov bodies, Invent. Math. 202 (2015), no. 3, 927–985.

    Article  MathSciNet  Google Scholar 

  26. J. Hilgert, Ch. Manon, J. Martens, Contraction of Hamiltonian K-spaces, Internat. Math. Research Notices, doi:10.1093/imrn/rnw191 (2016).

  27. J. Hilgert, K.-H. Neeb, W. Plank, Symplectic convexity theorems and coadjoint orbits, Compositio Math. 94 (1994), no. 2, 129–180.

    MathSciNet  MATH  Google Scholar 

  28. H. Hofer, E. Zehnder, A New capacity for symplectic manifolds, Analysis, et cetera, Academic Press, Boston, MA, 1990, pp. 405–427.

    Google Scholar 

  29. P. Iglésias, Les SO(3)-variétés symplectiques et leur classification en dimension 4, Bull. Soc. Math. France 119 (1991), no. 3, 371–396.

    Article  MathSciNet  Google Scholar 

  30. L. Jeffrey, J. Weitsman, Bohr–Sommerfeld orbits in the moduli space of flat connections and the verlinde dimension formula, Comm. Math. Phys. 150 (1992), no. 3, 593–630.

    Article  MathSciNet  Google Scholar 

  31. M. Kapovich, J. J. Millson, The symplectic geometry of polygons in Euclidean space, J. Differential Geom. 44 (1996), no. 3, 479–513.

    Article  MathSciNet  Google Scholar 

  32. Y. Karshon, E. Lerman, Non-compact symplectic toric manifolds, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 055.

  33. Y. Karshon, S. Tolman, The Gromov width of complex Grassmannians, Geometry and Topology 5 (2005), 911–922.

    MathSciNet  MATH  Google Scholar 

  34. K. Kaveh, Toric degenerations and symplectic geometry of smooth projective varieties, arXiv:1508.00316 (2016).

  35. F. Kirwan, Convexity properties of the moment mapping, III, Invent. Math. 77 (1984), no. 3, 547–552.

    Article  MathSciNet  Google Scholar 

  36. F. Knop, Automorphisms of multiplicity-free Hamiltonian manifolds, J. Amer. Math. Soc. 24 (2011), no. 2, 567–601.

    Article  MathSciNet  Google Scholar 

  37. J. Lane, On the topology of collective integrable systems, PhD thesis, University of Toronto, 2017.

  38. J. Lane, A completely integrable system on G2 coadjoint orbits, arXiv:1605.01676 (2016).

  39. E. Lerman, E. Meinrenken, S. Tolman, Ch. Woodward, Nonabelian convexity by symplectic cuts, Topology 37 (1998), no. 2, 245–259.

    Article  MathSciNet  Google Scholar 

  40. G. Lu, Symplectic capacities of toric manifolds and related results, Nagoya Math. J. 181 (2006), 149–184.

    Article  MathSciNet  Google Scholar 

  41. С. В. Манаков, Замечание об интегрировании уравнений Эйлера динамики n-мерного твердого тела Функц. анализ. и его прил. 10 (1976), вьш. 4, 93–94. Engl. transl: S. V. Manakov, Note on the integration of Euler's equations of the dynamics of an n-dimensional rigid body, Funct. Anal. Appl. 10 (1976), no. 4, 328–329.

  42. A. Mandini, M. Pabiniak, Gromov width of polygon spaces, arXiv:1501.00298 (2015).

  43. E. Miranda, N. T. Zung, Personal communication.

  44. А. С. Мищенко, А. Т. Фоменко, Уравнения Эйлера на конечномерных группах Ли, Изв. АН СССР. Сер. матем. 42 (1987), вьш. 2, 396–415. Engl. transl.: A. S. Mishchenko, A. T. Fomenko, Euler equations on finite-dimensional Lie groups, Math. USSR-Izv. 12 (1978), no. 2, 371–389.

  45. T. Nishinou, Y. Nohara, K. Ueda Toric degenerations of Gelfand–Cetlin systems and potential functions, Adv. Math. 224 (2010), no. 2, 648–706.

    Article  MathSciNet  Google Scholar 

  46. M. Pabiniak, Hamiltonian torus actions in equivariant cohomology and symplectic topology, PhD thesis, Cornell University, 2012.

  47. M. Pabiniak, Gromov width of non-regular coadjoint orbits of U(n), SO(2n) and SO(2n + 1), Math. Res. Lett. 21 (2014), no. 1, 187–205.

    Article  MathSciNet  Google Scholar 

  48. R. Sjamaar, Convexity properties of the moment mapping re-examined, Adv. Math. (1) 138 (1998), 46–91.

    Article  MathSciNet  Google Scholar 

  49. R. Sjamaar, E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), no. 2, 375–422.

    Article  MathSciNet  Google Scholar 

  50. A. Thimm, Integrable geodesic flows on homogeneous spaces, Ergodic Theory Dynamical Systems 1 (1981), no. 4.

  51. S. Tolman, Examples of non-Kähler Hamiltonian torus actions, Invent. Math. 131 (1998), no. 2, 299–310.

    Article  MathSciNet  Google Scholar 

  52. L. Traynor, Symplectic packing constructions, J. Differential Geom. 41 (1995), no. 3, 735–751.

    Article  MathSciNet  Google Scholar 

  53. В. В. Трофимов, Уравнения Эйлера на борелевских подалгебрах полупростых алгебры Ли, Изв. АН СССР. Сер. матем. 43 (1979), вьш. 3, 714–732. Engl. transl.: V. V. Trofimov, Euler equations on Borel subalgebras of semisimple Lie algebras, Math. USSR-Izv. 14 (1980), no. 3, 653–670.

  54. San Vũ Ngoc, Moment polytopes for symplectic manifolds with monodromy, Adv. Math. 208 (2007), no. 2, 909–934.

  55. Ch. Woodward, The classification of transversal multiplicity-free group actions, Ann. Global Anal. Geom. 14 (1996), 3–42.

    Article  MathSciNet  Google Scholar 

  56. Ch. Woodward, Multiplicity free Hamiltonian actions need not be Kähler, Invent. Math. 131 (1998), no. 2, 311–319.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 1A1, Canada

    J. LANE

Authors
  1. J. LANE
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. LANE.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

LANE, J. CONVEXITY AND THIMM’S TRICK. Transformation Groups 23, 963–987 (2018). https://doi.org/10.1007/s00031-017-9436-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-017-9436-7

Search

Navigation