**Extreme points and convexity in representation theory**

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.

*Karl-Herrmann Neeb*

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