  Orbits of coadjoint representations of classical compact Lie groups have a
lot of applications. They appear in representation theory, geometrical
quantization, theory of magnetism, quantum optics etc. As geometric objects the
orbits were the subject of much study. However, they remain hard for
calculation and application. We propose simple solutions for the following
problems: an explicit parameterization of the orbit by means of a generalized
stereographic projection, obtaining a K\"{a}hlerian structure on the orbit,
introducing basis two-forms for the cohomology group of the orbit.
