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

M. V. Babich

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 36-57

Résumé

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.