  We show that untwisted respectively twisted conjugacy classes of a compact
and simply connected Lie group which satisfy a certain integrality condition
correspond naturally to irreducible highest weight representations of the
corresponding affine Lie algebra. Along the way, review the classification of
twisted conjugacy classes of a simply connected compact Lie group $G$ and give
a description of their stabilizers in terms of the Dynkin diagram of the
corresponding twisted affine Lie algebra.
