  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.
