  We classify completely prime primitive ideals whose associated varieties are
the closure of the minimal nilpotent orbit of
$\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, and classify irreducible
$(\mathfrak{g},\mathfrak{k})$-modules which have those ideals as annihilators.
Moreover, we irreducibly decompose them as $\mathfrak{k}$-modules.
