  A unitary representation of a, possibly infinite dimensional, Lie group $G$
is called semi-bounded if the corresponding operators $i\dd\pi(x)$ from the
derived representations are uniformly bounded from above on some non-empty open
subset of the Lie algebra $\g$. In the first part of the present paper we
explain how this concept leads to a fruitful interaction between the areas of
infinite dimensional convexity, Lie theory, symplectic geometry (momentum maps)
and complex analysis. Here open invariant cones in Lie algebras play a central
role and semibounded representations have interesting connections to
$C^*$-algebras which are quite different from the classical use of the group
$C^*$-algebra of a finite dimensional Lie group. The second half is devoted to
a detailed discussion of semibounded representations of the diffeomorphism
group of the circle, the Virasoro group, the metaplectic representation on the
bosonic Fock space and the spin representation on fermionic Fock space.
