Advertisement

Mathematische Zeitschrift

, Volume 291, Issue 3–4, pp 1093–1132 | Cite as

Hessenberg varieties, Slodowy slices, and integrable systems

  • Hiraku Abe
  • Peter CrooksEmail author
Article
  • 184 Downloads

Abstract

This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group G and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, \(X(H_0)\), is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko–Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to \(X(H_0)\). We also show \(X(H_0)\) to have an open dense symplectic leaf isomorphic to \(G/Z \times S_{\text {reg}}\), where Z is the centre of G and \(S_{\text {reg}}\) is a regular Slodowy slice in the Lie algebra of G. This allows us to invoke results about integrable systems on \(G\times S_{\text {reg}}\), as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties.

Keywords

Hessenberg variety Integrable system Slodowy slice Toda lattice 

Mathematics Subject Classification

Primary 14L30 Secondary 17B80 53D20 

Notes

Acknowledgements

We wish to recognize Ana Bălibanu, Megumi Harada, Takashi Otofuji, and Steven Rayan for helpful conversations. The first author is grateful to Mikiya Masuda for his support and encouragement. He is supported in part by a JSPS Research Fellowship for Young Scientists (Postdoctoral Fellow): 16J04761 and a JSPS Grant-in-Aid for Early-Career Scientists: 18K13413. The second author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada [PDF–516638–2018].

References

  1. 1.
    Abbaspour, H., Moskowitz, M.: Basic Lie Theory. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    Abe, H., Crooks, P.: Hessenberg varieties for the minimal nilpotent orbit. Pure. Appl. Math. Q. 12(2), 183–223 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Abe, H., DeDieu, L., Galetto, F., Harada, M.: Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies. Select. Math. (N.S.) 24(3), 2129–2163 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Abe, H., Fujita, N., Zheng, H.: Geometry of regular Hessenberg varieties. arXiv:1712.09269, 26 pp. (2017)
  5. 5.
    Abe, H., Harada, M., Horiguchi, T., Masuda, M.: The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A. Int. Math. Res. Notes.  https://doi.org/10.1093/imrn/rnx275, pp 73 (2017)
  6. 6.
    Abe, H., Horiguchi, T., Masuda, M.: The cohomology rings of regular semisimple Hessenberg varieties for \(h=(h(1),n,\ldots ,n)\). J. Comb. 24 pp. arXiv:1704.00934 (2017) (to appear)
  7. 7.
    Abe, T., Horiguchi, T., Masuda, M., Murai, S., Sato, T.: Hessenberg varieties and hyperplane arrangements. arxiv:1611.00269, pp 45 (2016)
  8. 8.
    Adler, M.: On a trace functional for formal pseudo differential operators and the symplectic structure of the Korteweg-de Vries type equations. Invent. Math. 50(3), 219–248 (1978/79)Google Scholar
  9. 9.
    Adler, M., van Moerbeke, P.: The Toda lattice, Dynkin diagrams, singularities and abelian varieties. Invent. Math. 103(2), 223–278 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Adler, M., van Moerbeke, P., Vanhaecke, P.: Algebraic integrability, Painlevé geometry and Lie algebras, vol. 47 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics. Springer, Berlin (2004)Google Scholar
  11. 11.
    Anderson, D., Tymoczko, J.: Schubert polynomials and classes of Hessenberg varieties. J. Algebra 323(10), 2605–2623 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ayzenberg, A., Buchstaber, V.: Manifolds of isospectral matrices and Hessenberg varieties. arXiv:1803.01132, 17 pp. (2018)
  13. 13.
    Bălibanu, A.: The partial compactification of the universal centralizer. arXiv:1710.06327, 13 pp. (2017)
  14. 14.
    Bălibanu, A.: The Peterson variety and the wonderful compactification. Rep. Theory 21, 132–150 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bielawski, R.: Hyperkähler structures and group actions. J. Lond. Math. Soc. (2) 55(2), 400–414 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bloch, A.M., Brockett, R.W., Ratiu, T.S.: A new formulation of the generalized Toda lattice equations and their fixed point analysis via the momentum map. Bull. Am. Math. Soc. (N.S.) 23(2), 477–485 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Bolsinov, A.V.: Compatible Poisson brackets on Lie algebras and the completeness of families of functions in involution. Izv. Akad. Nauk SSSR Ser. Mat. 55(1), 68–92 (1991)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Brosnan, P., Chow, T.Y.: Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties. Adv. Math. 329, 955–1001 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cheong, D.: Quantum cohomology rings of Lagrangian and orthogonal Grassmannians and total positivity. Trans. Am. Math. Soc. 361(10), 5505–5537 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chriss, N., Ginzburg, V.: Representation theory and complex geometry. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston, MA (2010). Reprint of the 1997 editionGoogle Scholar
  21. 21.
    Ciocan-Fontanine, I.T.: Quantum cohomology of flag varieties. Int. Math. Res. Notes 6, 263–277 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Crooks, P.: Complex adjoint orbits in Lie theory and geometry. Exp. Math. (2018).  https://doi.org/10.1016/j.exmath.2017.12.001
  23. 23.
    Crooks, P.: An equivariant description of certain holomorphic symplectic varieties. Bull. Aust. Math. Soc. 97(2), 207–214 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Crooks, P., Rayan, S.: Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices. To appear in Math. Res. Lett. 17 pp. arXiv:1706.05819 (2017)
  25. 25.
    De Mari, F., Procesi, C., Shayman, M.A.: Hessenberg varieties. Trans. Am. Math. Soc. 332(2), 529–534 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    De Mari, F., Shayman, M.A.: Generalized Eulerian numbers and the topology of the Hessenberg variety of a matrix. Acta. Appl. Math. 12(3), 213–235 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ercolani, N.M., Flaschka, H., Singer, S.: The geometry of the full Kostant-Toda lattice. In: Integrable systems (Luminy, 1991), vol. 115 of Progr. Math. Birkhäuser Boston, Boston, MA, pp. 181–225 (1993)Google Scholar
  28. 28.
    Fernandes, R., Laurent-Gengoux, C., Vanhaecke, P.: Global action-angle variables for non-commutative integrable systems. J. Symplect. Geom. 44 pp. arXiv:1503.0084 (2015) (to appear)
  29. 29.
    Flaschka, H.: The Toda lattice. II. Existence of integrals. Phys. Rev. B 9(4), 1924–1925 (1974)Google Scholar
  30. 30.
    Gesztesy, F., Holden, H., Simon, B., Zhao, Z.: On the Toda and Kac-van Moerbeke systems. Trans. Am. Math. Soc. 339(2), 849–868 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ginzburg, V.: Principal nilpotent pairs in a semisimple Lie algebra I (with an appendix by A. Elashvili and D. Panyushev). Invent. Math. 140(3), 511–561 (2000)Google Scholar
  32. 32.
    Givental, A., Kim, B.: Quantum cohomology of flag manifolds and Toda lattices. Commun. Math. Phys. 168(3), 609–641 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Grothendieck, A., Raynaud, M.: Revêtements étales et groupe fondamental. Lecture Notes in Mathematics, Vol. 224. Springer, Berlin (1971). Séminaire de Géométrie Algébrique du Bois Marie 1960–1961 (SGA 1)Google Scholar
  34. 34.
    Guay-Paquet, M.: A second proof of the Shareshian-Wachs conjecture, by way of a new Hopf algebra, 36 pp. arXiv:1601.05498 (2016)
  35. 35.
    Guillemin, V., Sternberg, S.: On collective complete integrability according to the method of Thimm. Ergodic Theory Dyn. Syst. 3(2), 219–230 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Harada, M., Precup, M.: The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture, 40 pp. arXiv:1709.06736 (2017)
  37. 37.
    Hénon, M.: Integrals of the Toda lattice. Phys. Rev. B 9(4), 1921–1923 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Horiguchi, T.: The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials, 9 pp. arXiv:1801.07930 (2018)
  39. 39.
    Humphreys, J.E.: Conjugacy Classes in Semisimple Algebraic Groups, vol. 43 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1995)Google Scholar
  40. 40.
    Insko, E., Precup, M.: The singular locus of semisimple Hessenberg varieties, 24 pp. arXiv:1709.05423 (2017)
  41. 41.
    Insko, E., Yong, A.: Patch ideals and Peterson varieties. Transf. Groups 17(4), 1011–1036 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Kamvissis, S.: On the long time behavior of the doubly infinite Toda lattice under initial data decaying at infinity. Commun. Math. Phys. 153(3), 479–519 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Kim, B.: Quantum cohomology of flag manifolds \(G/B\) and quantum Toda lattices. Ann. Math. (2) 149(1), 129–148 (1999)Google Scholar
  44. 44.
    Kodama, Y., Williams, L.: The full Kostant-Toda hierarchy on the positive flag variety. Commun. Math. Phys. 335(1), 247–283 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Kostant, B.: The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Am. J. Math. 81, 973–1032 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Kostant, B.: Lie group representations on polynomial rings. Am. J. Math. 85, 327–404 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Kostant, B.: On Whittaker vectors and representation theory. Invent. Math. 48(2), 101–184 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Kostant, B.: The solution to a generalized Toda lattice and representation theory. Adv. Math. 34(3), 195–338 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Kostant, B.: Flag manifold quantum cohomology, the Toda lattice, and the representation with highest weight \(\rho \). Select. Math. (N.S.) 2(1), 43–91 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Krüger, H., Teschl, G.: Long-time asymptotics for the Toda lattice in the soliton region. Math. Z. 262(3), 585–602 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Laurent-Gengoux, C., Pichereau, A., Vanhaecke, P.: Poisson Structures. Grundlehren der Mathematischen Wissenschaften, vol. 347. Springer, Heidelberg (2013)zbMATHGoogle Scholar
  52. 52.
    Marsden, J.E., Ratiu, T., Raugel, G.: Symplectic connections and the linearisation of Hamiltonian systems. Proc. R. Soc. Edinburgh Sect. A 117(3–4), 329–380 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Mishchenko, A.S., Fomenko, A.T.: Euler equations on finite-dimensional Lie groups. Izv. Akad. Nauk SSSR Ser. Mat. 42(2), 396–415, 417 (1978)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Moser, J.: Finitely many mass points on the line under the influence of an exponential potential—an integrable system. Lect. Notes Phys. 38, 467–497 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Ortega, J.-P., Ratiu, T.S.: Momentum Maps and Hamiltonian Reduction. Progress in Mathematics, vol. 222. Birkhäuser Boston Inc, Boston, MA (2004)CrossRefzbMATHGoogle Scholar
  56. 56.
    Panyushev, D.I., Yakimova, O.S.: The argument shift method and maximal commutative subalgebras of Poisson algebras. Math. Res. Lett. 15(2), 239–249 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Peterson, D.: Quantum cohomology of \({G}/{P}\). Lecture series at Massachusetts Institute of Technology. Spring term 1997 (unpublished)Google Scholar
  58. 58.
    Precup, M.: Affine pavings of Hessenberg varieties for semisimple groups. Select. Math. (N.S.) 19(4), 903–922 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Precup, M.: The Betti numbers of regular Hessenberg varieties are palindromic. Transf. Groups 23(2), 491–499 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Rietsch, K.: Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties. J. Am. Math. Soc. 16(2), 363–392 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Shareshian, J., Wachs, M.L.: Chromatic quasisymmetric functions. Adv. Math. 295, 497–551 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Symes, W.W.: Systems of Toda type, inverse spectral problems, and representation theory. Invent. Math. 59(1), 13–51 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Tauvel, P., Yu, R.W.T.: Lie Algebras and Algebraic Groups. Springer Monographs in Mathematics. Springer, Berlin (2005)zbMATHGoogle Scholar
  64. 64.
    Toda, M.: Vibration of a chain with nonlinear interaction. J. Phys. Soc. Jpn. 22(2), 431–436 (1967)CrossRefGoogle Scholar
  65. 65.
    Toda, M.: Wave propagation in anharmonic lattices. J. Phys. Soc. Jpn. 23(3), 501–506 (1967)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Tymoczko, J.S.: Decomposing Hessenberg Varieties Over Classical Groups. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–Princeton University (2003). https://arxiv.org/pdf/math/0211226.pdf
  67. 67.
    Tymoczko, J.S.: Hessenberg varieties are not pure dimensional. Pure Appl. Math. Q. 2, 3, Special Issue: In honor of Robert D. MacPherson. Part 1, 779–794 (2006)Google Scholar
  68. 68.
    Varadarajan, V.S.: On the ring of invariant polynomials on a semisimple Lie algebra. Am. J. Math. 90, 308–317 (1968)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Osaka City University Advanced Mathematical InstituteOsakaJapan
  2. 2.Department of MathematicsNortheastern UniversityBostonUSA

Personalised recommendations