Geodesic
Birational Darboux coordinates on nilpotent coadjoint orbits classical complex Lie groups, Jordan blocks $2\times2$
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 24, Tome 465 (2017), pp. 5-12 Cet article a éte moissonné depuis la source Math-Net.Ru

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A problem of the constructing of the birational Darboux coordinates on the nilpotent coadjoint orbits of the complex Lie groups $SO(N,\mathbb C)$ and $Sp(N,\mathbb C)$ is considered. The nilpotent case is the most difficult case of the orbits. The difficulties arise if the Jordan blocks of the different parities of the sizes present in the Jordan form of the matrices from the orbit. The desired coordinates has been found on the orbits consisting of the matrices with the Jordan blocks of the sizes one and two. The explicit formulae for the coordinates are presented. 
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     title = {Birational {Darboux} coordinates on nilpotent coadjoint orbits classical complex {Lie} groups, {Jordan} blocks~$2\times2$},
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M. V. Babich. Birational Darboux coordinates on nilpotent coadjoint orbits classical complex Lie groups, Jordan blocks $2\times2$. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 24, Tome 465 (2017), pp. 5-12. http://geodesic.mathdoc.fr/item/ZNSL_2017_465_a0/

[1] M. V. Babich, “Birational Darboux Coordinates on (Co)Adjoint Orbits of $GL(N,\mathbb C)$”, Functional Analysis and Its Applications, 50:1 (2016), 17–30 | DOI | MR | Zbl

[2] M. V. Babich, “On birational Darboux coordinates on coadjoint orbits of classical complex Lie groups”, Zap. Nauchn. Semin. POMI, 432, 2015, 36–57 ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v432/p036.pdf | MR | Zbl

[3] M. V. Babich, “Young tableaux and stratification of space of complex square matrices”, Zap. Nauchn. Semin. POMI, 433, 2015, 41–64 ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v433/p041.pdf | MR

[4] S. E. Derkachov, A. N. Manashov, “$R$-matrix and Baxter $Q$-operators for the noncompact $SL(N;C)$ invariant spin chain”, SIGMA, 2 (2006), 084 ; arXiv: nlin.SI/0612003 | DOI | MR | Zbl

[5] M. V. Babich, S. E. Derkachov, “On rational symplectic parametrization of coajoint orbit of $GL(N)$. Diagonalizable case”, Algebra and Analiz, 22:3 (2010), 16–31 | MR | Zbl

[6] M. V. Babich, “About Coordinates on the Phase Spaces of the Schlesinger System and the Garnier-Painlevé 6 System”, Doklady Mathematics, 75:1 (2007), 71 | DOI | MR | Zbl

[7] A. G. Reiman, M. A. Semenov-Tyan'-Shanskii, Integrable systems. Group-theoretic approach, Sovremennaya Matematika, Inst. for Computer Research, Moscow–Izhevsk, 2003

[8] I. M. Gelfand, M. I. Najmark, Unitary representation of classical groups, Tr. Mat. Inst. Steklov, 36, 1950 | MR | Zbl