The Orbit Method for Compact Connected Lie Groups

TL;DR

This thesis provides a comprehensive overview of Kirillov's Orbit Method for compact connected Lie groups, linking coadjoint orbits, complex structures, and representation theory.

Contribution

It offers a detailed exposition of the Orbit Method, including the construction of irreducible representations via coadjoint orbits and the Borel-Weil Theorem for compact Lie groups.

Findings

Coadjoint orbits are symplectic manifolds with complex structures.

Irreducible unitary representations can be realized as holomorphic sections of line bundles.

The Orbit Method connects geometric structures with representation theory.

Abstract

The goal of this diploma thesis is to give a detailed description of Kirillov's Orbit Method for the case of compact connected Lie groups. The theory of Kirillov aims at finding all irreducible unitary representations of a given Lie group $G$. The first chapter is intended to recall some facts about Lie groups. The most important result is that for a functional $\lambda$ on the Lie algebra $\mathfrak t$ of a maximal torus in a compact connected Lie group the stabilizer is of the form $C(T_1)$ for a smaller torus $T_1$. This allows us to define a complex structure on the coadjoint orbit $\mathcal O_\lambda = G/C(T_1)$. In the second chapter we present the Borel-Weil Theorem which tells us that all irreducible unitary representations of a compact connected Lie group can be realized in the space of holomorphic sections of the bundle $G\times_\tau\mathbb C$, where $\tau$ is a character of a maximal torus. Finally, in the third chapter we show that coadjoint orbits are symplectic manifolds for which prequantization yields a complex line bundle. Then we investigate for which classes of coadjoint orbits one obtains representations in the space of polarized sections of the prequantum bundle. This will lead us back to the result of Borel-Weil.