Geometry and Topology of Coadjoint Orbits of Semisimple Lie Groups

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.