  For any special nilpotent orbit, let $\frac{1}{2}h^{\vee}$ be one half of the
semisimple element of a Jacobson-Morozov triple associated to the orbit. In
1985, Barbasch and Vogan defined the notion of special unipotent
representations with infinitesimal character
$(\frac{1}{2}h^{\vee},\frac{1}{2}h^{\vee})$. Some properties of such
representations were discovered when $\frac{1}{2}h^{\vee}$ is integral. In this
manuscript, we give a complete proof of these properties when
$\frac{1}{2}h^{\vee}$ is not integral.
