Advertisement

Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties

  • Dmitri I. PanyushevEmail author
  • Oksana S. Yakimova
Article
First Online:
  • 67 Downloads

Abstract

The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra \({\mathfrak {g}}\), we obtain several results on the completeness of homogeneous Poisson-commutative subalgebras of \({\mathscr {S}}({\mathfrak {g}})\) on coadjoint orbits. This concerns, in particular, Gelfand–Tsetlin and Mishchenko–Fomenko subalgebras. Our results reveal the crucial role of nilpotent orbits and sheets in \({\mathfrak {g}}\simeq {\mathfrak {g}}^{*}\).

Keywords

Integrable systems Moment map Coisotropic actions Coadjoint orbits 

Mathematics Subject Classification

17B63 14L30 17B08 17B20 22E46 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

We are grateful to the anonymous referee for the detailed report and important suggestions.

References

  1. 1.
    Akhiezer, D., Panyushev, D.: Multiplicities in the branching rules and the complexity of homogeneous spaces. Mosc. Math. J. 2(1), 17–33 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Avdeev, R.S., Petukhov, A.V.: Spherical actions on flag varieties. Mat. Sb. 205(9), 3–48 (2014). English translation in Sb. Math., 205(9-10), 1223–1263 (2014)Google Scholar
  3. 3.
    Bolsinov, A.: Commutative families of functions related to consistent Poisson brackets. Acta Appl. Math. 24(3), 253–274 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bolsinov, A., Zhang, P.: Jordan–Kronecker invariants of finite-dimensional Lie algebras. Transform. Groups 21(1), 51–86 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Borho, W., Kraft, H.: Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen. Comment. Math. Helv. 54(1), 61–104 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Charbonnel, J.-Y., Moreau, A.: The index of centralizers of elements of reductive Lie algebras. Doc. Math. 15, 387–421 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Colarusso, M., Evens, S.: The complex orthogonal Gelfand–Zeitlin system, arXiv:1808.04424v1 [math.RT]
  8. 8.
    Crooks, P., Rosemann, S., Roeser, M.: Slodowy slices and the complete integrability of Mishchenko–Fomenko subalgebras on regular adjoint orbits. arXiv:1803.04942v1 [math.SG]
  9. 9.
    Dufour, J.-P., Zung, N.T.: Poisson structures and their normal forms. In: Progress in Mathematics, vol. 242. Birkhäuser, Basel (2005)Google Scholar
  10. 10.
    Feigin, B., Frenkel, E., Rybnikov, L.: Opers with irregular singularity and spectra of the shift of argument subalgebra. Duke Math. J. 155(2), 337–363 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Gelfand, I.M., Tsetlin, M.L.: Finite-dimensional representations of the group of unimodular matrices. (Russian). Dokl. Akad. Nauk SSSR (N.S.) 71, 825–828 (1950). English transl. In: Gelfand, I.M. Collected Papers, vol. II, Springer-Verlag, Berlin, pp. 653–656 (1988)Google Scholar
  12. 12.
    Gelfand, I.M., Tsetlin, M.L.: Finite-dimensional representations of groups of orthogonal matrices. (Russian). Dokl. Akad. Nauk SSSR (N.S.) 71, 1017–1020 (1950). English transl. In: Gelfand, I.M. Collected Papers, vol. II, Springer-Verlag, Berlin, pp. 657–661 (1988)Google Scholar
  13. 13.
    de Graaf, W.: Computing with nilpotent orbits in simple Lie algebras of exceptional type. LMS J. Comput. Math. 11, 280–297 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Guillemin, V., Sternberg, S.: The moment map and collective motion. Ann. Physics 127(1), 220–253 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Guillemin, V., Sternberg, S.: The Gelfand–Cetlin system and quantization of the complex flag manifolds. J. Funct. Anal. 52(1), 106–128 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Guillemin, V., Sternberg, S.: On collective complete integrability according to the method of Thimm. Ergod. Theory Dyn. Syst. 3(2), 219–230 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Guillemin, V., Sternberg, S.: Multiplicity-free spaces. J. Differ. Geom. 19(1), 31–56 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Harada, M.: The symplectic geometry of the Gelfand–Cetlin–Molev basis for representations of \({\rm Sp}(2n,{\mathbb{C}})\). J. Symplectic Geom. 4(1), 1–41 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Heckman, G.J.: Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups. Invent. Math. 67(2), 333–356 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Huckleberry, A.T., Wurzbacher, T.: Multiplicity-free complex manifolds. Math. Ann. 286(1–3), 261–280 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Johnson, K.D.: A note on branching theorems. Proc. Am. Math. Soc. 129(2), 351–353 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Knop, F.: Der Zentralisator einer Liealgebra in einer einhüllenden Algebra. J. Reine Angew. Math. 406, 5–9 (1990)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Knop, F.: Weylgruppe und Momentabbildung. Invent. Math. 99, 1–23 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Kostant, B.: Lie group representations on polynomial rings. Am. J. Math. 85, 327–404 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Kostant, B.: Fomenko–Mischenko theory, Hessenberg varieties, and polarizations. Lett. Math. Phys. 90(1–3), 253–285 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Kostant, B., Wallach, N.: Gelfand–Zeitlin theory from the perspective of classical mechanics. I., In: Studies in Lie theory, 319–364, Progr. Math., vol. 243. Birkhäuser Boston (2006)Google Scholar
  27. 27.
    Krämer, M.: Multiplicity free subgroups of compact connected Lie groups. Arch. Math. 27, 28–36 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Kraft, H.: Parametrisierung von Konjugationsklassen in \({\mathfrak{sl}}_n\). Math. Ann. 234(3), 209–220 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Losev, I.V.: Algebraic Hamiltonian actions. Math. Z. 263(3), 685–723 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Luna, D.: Sur les orbites fermées des groupes algébriques réductifs. Invent. Math. 16, 1–5 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Mishchenko, A.S., Fomenko, A.T.: Euler equation on finite-dimensional Lie groups. Math. USSR Izv. 12, 371–389 (1978)zbMATHCrossRefGoogle Scholar
  32. 32.
    Molev, A., Yakimova, O.: Quantisation and nilpotent limits of Mishchenko–Fomenko subalgebras. arXiv:1711.03917v1 [math.RT]
  33. 33.
    Panyushev, D.: Complexity and nilpotent orbits. Manuscr. Math. 83, 223–237 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Panyushev, D.: The index of a Lie algebra, the centraliser of a nilpotent element, and the normaliser of the centraliser. Math. Proc. Camb. Phil. Soc. 134(1), 41–59 (2003)CrossRefGoogle Scholar
  35. 35.
    Panyushev, D., Premet, A., Yakimova, O.: On symmetric invariants of centralisers in reductive Lie algebras. J. Algebra 313, 343–391 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Panyushev, D., Yakimova, O.: The argument shift method and maximal commutative subalgebras of Poisson algebras. Math. Res. Lett. 15(2), 239–249 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Panyushev, D., Yakimova, O.: Poisson-commutative subalgebras of \(\mathscr {S}({\mathfrak{g}})\) associated with involutions. arXiv:1809.00350v1 [math.RT]
  38. 38.
    Tarasov, A.A.: The maximality of certain commutative subalgebras in the Poisson algebra of a semisimple Lie algebra. Russ. Math. Surv. 57(5), 1013–1014 (2002)zbMATHCrossRefGoogle Scholar
  39. 39.
    Thompson, R.C.: Pencils of complex and real symmetric and skew matrices. Linear Algebra Appl. 147, 323–371 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Vinberg, E.B.: Some commutative subalgebras of a universal enveloping algebra. Math. USSR-Izv 36, 1–22 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Vinberg, E.B.: Commutative homogeneous spaces and co-isotropic symplectic actions. Uspekhi Mat. Nauk 56(1(337)), 3–62 (2001). English translation in Russian Math. Surveys, 56(1), 1–60 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Vinberg, E.B., Kimelfeld, B.N.: Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups. Funktsional. Anal. i Pril. 12(3), 12–19 (1978). English translation in Functional Anal. Appl., 12(3), 168–174 (1978)MathSciNetGoogle Scholar
  43. 43.
    Винберг, Э.Б., Попов, В.Л. “Теория Инвариантов”, В: Соврем. пробл. математики. Фундаментальные направл., т. 55, стр. 137–309. Москва: ВИНИТИ 1989 (Russian). English translation: Popov, V.L., Vinberg, E.B.: “Invariant theory”, In: Algebraic Geometry IV, Encyclopaedia Math. Sci., vol. 55, pp.123–284. Springer, New York (1994)Google Scholar
  44. 44.
    Vinberg, E.B., Yakimova, O.S.: Complete families of commuting functions for coisotropic Hamiltonian actions. Adv. Math. 348, 523–540 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Yakimova, O.: The centralisers of nilpotent elements in classical Lie algebras. Funct. Anal. Appl. 40(1), 42–51 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Yakimova, O.: Surprising properties of centralisers in classical Lie algebras. Ann. Inst. Fourier (Grenoble) 59(3), 903–935 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Yakimova, O.: One-parameter contractions of Lie–Poisson brackets. J. Eur. Math. Soc. 16, 387–407 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Zorin, A.A.: On the commutativity of the centralizer of a subalgebra in a universal enveloping algebra. Funktsional. Anal. i Pril. 43(2), 47–63 (2009). English translation in Funct. Anal. Appl., 43(2), 119–131 (2009)Google Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute for Information Transmission Problems of the R.A.S.MoscowRussia
  2. 2.Mathematisches InstitutUniversität zu KölnCologneGermany

Personalised recommendations