On the dual of complex Olshanskii semigroups

Let $G$ be a connected Lie group and $\hat G$ its unitary dual. We are interested in the part $\Lambda\subset\hat G$ which corresponds to the unitary highest weight representations of $G$. Then there are several topologies on $\Lambda$: The euclidean topology $T_E$ which comes from the identification of $\Lambda$ with the set of highest weights, the induced topology $T_I$ induced from the Fell topology on $\hat G$ and finally a natural topology $T_S$ which comes from the hull kernel topology of certain CCR C^*-algebras which are related to the holomorphic extemsion of unitary highest weight representations to complex Olshanskii semigroups $S$. One of the main results in this paper is the inclusion chain $T_S\subset T_I\subset T_E$. Further we exhibit very large interesting subspaces of $\Lambda$ where these topologies coincide. Finally we show that the Borel structures on $\Lambda$ induced from the three different topologies coincide.