If A is a unital C^*-algebra, then its states form a compact convex set whose extreme points correspond to irreducible representations of A. If A is the algebra C(X) of continuous functions on a compact space, then its states are probability measures on X and its extreme points are Dirac measures. In this lecture we explain how the philosophy underlying these facts can be adapted to the representation theory of Lie groups. This leads to moment sets of unitary representations, Kähler structures on coadjoint orbits and in particular to a complete classification of a particularly nice class of representation in very geometric terms. Moreover, it provides a geometric interpretation of Heisenberg's Uncertainty Principle.