Geodesic
On birational Darboux coordinates on coadjoint orbits of classical complex Lie groups
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 36-57 
M. V. Babich. On birational Darboux coordinates on coadjoint orbits of classical complex Lie groups. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 36-57. http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a2/
@article{ZNSL_2015_432_a2,
     author = {M. V. Babich},
     title = {On birational {Darboux} coordinates on coadjoint orbits of classical complex {Lie} groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {36--57},
     year = {2015},
     volume = {432},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a2/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Any coadjoint orbit of the general linear group can be canonically parameterized using an iteration method, where at each step we turn from the matrix of a transformation $A$ to the matrix of the transformation that is the projection of $A$ parallel to an eigenspace of this transformation to a coordinate subspace. We present a modification of the method applicable to the groups $\mathrm{SO}(N,\mathbb C)$ and $\mathrm{Sp}(N,\mathbb C)$. One step of the iteration consists of two actions, namely, the projection parallel to a subspace of an eigenspace and the simultaneous restriction to a subspace containing a co-eigenspace. The iteration gives a set of couples of functions $p_k,q_k$ on the orbit such that the symplectic form of the orbit is equal to $\sum_kdp_k\wedge dq_k$. No restrictions on the Jordan form of the matrices forming the orbit are imposed. A coordinate set of functions is selected in the important case of the absence of nontrivial Jordan blocks corresponding to the zero eigenvalue, which is the case $\dim\ker A=\dim\ker A^2$. This case contains the case of general position, the general diagonalizable case, and many others.

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