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

Gelfand-Kirillov dimensions and associated varieties of highest weight modules