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

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

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Acknowledgements

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

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Authors and Affiliations

  1. Institute for Information Transmission Problems of the R.A.S., Bolshoi Karetnyi per. 19, Moscow, 127051, Russia

    Dmitri I. Panyushev

  2. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931, Cologne, Germany

    Oksana S. Yakimova

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  1. Dmitri I. Panyushev
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  2. Oksana S. Yakimova
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Correspondence to Dmitri I. Panyushev.

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To the memory of Bertram Kostant

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The research of the first author was supported by the Russian Foundation for Sciences. The second author is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—project number 330450448.

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Panyushev, D.I., Yakimova, O.S. Poisson-commutative subalgebras and complete integrability on non-regular coadjoint orbits and flag varieties. Math. Z. 295, 101–127 (2020). https://doi.org/10.1007/s00209-019-02357-y

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