COHERENT STATES, HOLOMORPHIC EXTENSIONS, AND HIGHEST WEIGHT REPRESENTATIONS

Let G be a connected finite dimensional Lie group. In this paper we co nsider 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 ex tendability is that the unitary representation of G is a highest weigh t representation. This result provides a direct bridge from representa tion theory to coadjoint orbits in g, where g is the Lie algebra of G . Namely the moment map associated naturally to a unitary representati on maps the orbit of the highest weight ray (the coherent state orbit) to a coadjoint orbit in g which has many interesting geometric prope rties such as certain convexity properties and an invariant complex st ructure. In this paper we use the interplay between the orbit picture and representation theory to obtain a classification of all irreducibl e holomorphic representations of the semigroups S mentioned above and a classification 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 representation s to separate the points.