On birational Darboux coordinates on coadjoint orbits of classical complex Lie groups
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 \textit{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 [\textit{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)]. \textit{I. M. Gel'fand} and \textit{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.
scientific article; zbMATH DE number 6501791