  For $\mathfrak{g}=\mathfrak{sl}(n,\mathbb{C})$, let $w$ be an element in the
Weyl group $W=S_n$. We use $L_w$ to denote a highest weight module of
$\mathfrak{sl}(n,\mathbb{C})$ with highest weight $-w\rho-\rho$. In this paper
we will give a characterization for those $w$ such that $L_w$ is a highest
weight Harish-Chandra module and the associated variety of $L_w$ will be
characterized by the information of $w$. We also count the number of those
highest weight Harish-Chandra modules $L_w$ in a given Harish-Chandra cell.
Finally, we give some other highest weight modules with irreducible associated
varieties from the smoothness of Schubert varieties.
