  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.
