Let G be a connected finite
dimensional Lie group. In this paper we consider the problem of extending irreducible
unitary representations of G to holomorphic representations of certain semigroups S
containing G and a dense open submanifold on which the semigroup multiplication is
holomorphic. We show that a necessary and sufficient condition for extendability is
that the unitary representation of G is a highest weight representation. This result
provides a direct bridge from representation theory to coadjoint orbits in g∗, where g
is the Lie algebra of G. Namely the moment map associated naturally to a
unitary representation maps the orbit of the highest weight ray (the coherent
state orbit) to a coadjoint orbit in g∗ which has many interesting geometric
properties such as certain convexity properties and an invariant complex
structure.
In this paper we use the interplay between the orbit picture and representation
theory to obtain a classification of all irreducible holomorphic representations of the
semigroups S mentioned above and a classication of unitary highest weight
representations of a rather general class of Lie groups. We also characterize the class
of groups and semigroups having sufficiently many highest weight representations to
separate the points.
