Geometry and Topology of Coadjoint Orbits of Semisimple Lie Groups

TL;DR

This paper introduces explicit parameterizations, Kähler structures, and cohomology basis forms for coadjoint orbits of classical compact Lie groups, facilitating their calculation and application in various fields.

Contribution

It provides new explicit methods for parameterizing coadjoint orbits, establishing Kähler structures, and defining cohomology basis forms, advancing the geometric understanding of these orbits.

Findings

Explicit stereographic projection parameterization

Kählerian structure on coadjoint orbits

Basis two-forms for cohomology groups

Abstract

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.