Gelfand-Kirillov dimensions and associated varieties of highest weight modules

In this paper, we present a uniform formula of Lusztig's $\mathbf{a}$-functions on classical Weyl groups. Then we obtain an efficient algorithm for the Gelfand-Kirillov dimensions of simple highest weight modules of classical Lie algebras, whose highest weight is not necessarily regular or integral. To deal with type $D$, we prove an interesting property about domino tableaux by introducing an invariant, called the hollow tableau. As an application, the associated varieties of all the simple highest weight Harish-Chandra modules are explicitly determined, including the exceptional cases.