On the classification of unitary highest weight modules

In the 1980s, Enright, Howe and Wallach [EHW] and independently Jakobsen [J] gave a complete classification of the unitary highest weight modules. In this paper we give a more direct and elementary proof of the same result for the (universal covers of the) Lie groups $Sp(2n,\mathbb{R}),S{O}^{\ast }(2n)$ and $SU(p,q)$. We also show how to describe the set of unitary highest weight modules with a given infinitesimal character.