Gelfand–Kirillov Dimensions and Associated Varieties of Highest Weight Modules

Abstract In this paper, we present a uniform formula of Lusztig’s $ \textbf {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 associated with Weyl group elements 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.