Darboux Coordinates on K-Orbits and the Spectra of Casimir Operators on Lie Groups

We propose an algorithm for obtaining the spectra of Casimir (Laplace) operators on Lie groups. We prove that the existence of the normal polarization associated with a linear functional on the Lie algebra is necessary and sufficient for the transition to local canonical Darboux coordinates $(p,q)$ on the coadjoint representation orbit that is linear in the "momenta." We show that the $\la$-representations of Lie algebras (which are used, in particular, in integrating differential equations) result from the quantization of the Poisson bracket on the coalgebra in canonical coordinates.