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On birational Darboux coordinates on coadjoint orbits of classical complex Lie groups. (English. Russian original) Zbl 1327.22008

J. Math. Sci., New York 209, No. 6, 830-844 (2015); translation from Zap. Nauchn. Semin. POMI 432, 36-57 (2015).
Given a Lie group \(G\), the famous Kirillov-Kostant-Souriau’s theorem asserts that each co-adjoint orbit of \(\lambda\in {\mathfrak g}^*\) carries a natural symplectic structure that is invariant under \(G\).
The importance of co-adjoint representations was emphasized by A. A. Kirillov [Bull. Am. Math. Soc., New Ser. 36, No. 4, 433–488 (1999; Zbl 0940.22013); Lectures on the orbit method. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1229.22003)], Kostant, Auslander, Pukánszky and others. Also see [D. A. Vogan Jr., “Book review of: A. A. Kirillov, Lectures on the orbit method”, Bull. Am. Math. Soc. 42, No. 4, 535–544 (2005)]. I. M. Gel’fand and M. A. Naĭmark [Tr. Mat. Inst. Steklova 36, 1–288 (1950; Zbl 0041.36206)] introduced the method of canonical parametrization of co-adjoint orbits of the general linear group.
In the paper under review, the method is extended to matrix groups preserving a bilinear quadratic form. The method is insensitive to the Jordan form of the orbit and at each step it turns 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.

MSC:

22E10 General properties and structure of complex Lie groups
17B08 Coadjoint orbits; nilpotent varieties
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