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Zap. Nauchn. Sem. POMI, 2015, Volume 432, Pages 36–57 (Mi znsl6109)  

This article is cited in 3 scientific papers (total in 3 papers)

On birational Darboux coordinates on coadjoint orbits of classical complex Lie groups

M. V. Babichab

a St. Petersburg Department of Steklov Mathematical Institute, St. Petersburg, Russia
b St. Petersburg State University, St. Petersburg, Russia

Abstract: 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.

Key words and phrases: coadjoint orbit, classical Lie groups, Lie algebra, Lie–Poisson–Kirillov–Kostant form, symplectic fibration, rational Darboux coordinates.

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English version:
Journal of Mathematical Sciences (New York), 2015, 209:6, 830–844

UDC: 512.643.8+514.164.1+517.912
Received: 22.12.2014

Citation: M. V. Babich, “On birational Darboux coordinates on coadjoint orbits of classical complex Lie groups”, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Zap. Nauchn. Sem. POMI, 432, POMI, St. Petersburg, 2015, 36–57; J. Math. Sci. (N. Y.), 209:6 (2015), 830–844

Citation in format AMSBIB
\by M.~V.~Babich
\paper On birational Darboux coordinates on coadjoint orbits of classical complex Lie groups
\inbook Representation theory, dynamical systems, combinatorial methods. Part~XXIV
\serial Zap. Nauchn. Sem. POMI
\yr 2015
\vol 432
\pages 36--57
\publ POMI
\publaddr St.~Petersburg
\jour J. Math. Sci. (N. Y.)
\yr 2015
\vol 209
\issue 6
\pages 830--844

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    This publication is cited in the following articles:
    1. M. V. Babich, “Birational Darboux Coordinates on (Co)Adjoint Orbits of $\operatorname{GL}(N,\mathbb C)$”, Funct. Anal. Appl., 50:1 (2016), 17–30  mathnet  crossref  crossref  mathscinet  isi  elib
    2. M. V. Babich, “Birational Darboux coordinates on nilpotent coadjoint orbits classical complex Lie groups, Jordan blocks $2\times2$”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 24, Zap. nauchn. sem. POMI, 465, POMI, SPb., 2017, 5–12  mathnet
    3. M. V. Babich, “On parametrization of symplectic quotient of Cartesian product of coadjoint orbits of complex general linear group with respect to its diagonal action”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 25, K 70-letiyu M. A. Semenova-Tyan-Shanskogo, Zap. nauchn. sem. POMI, 473, POMI, SPb., 2018, 7–16  mathnet
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