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

  • Dmitri I. PanyushevEmail author
  • Oksana S. Yakimova


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 \({\mathfrak {g}}\simeq {\mathfrak {g}}^{*}\).


Integrable systems Moment map Coisotropic actions Coadjoint orbits 

Mathematics Subject Classification

17B63 14L30 17B08 17B20 22E46 



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


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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

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