The Gelfand-Kirillov dimension of a unitary highest weight module

Abstract

During the last decade, a great deal of activity has been devoted to the calculation of the Hilbert-Poincare series of unitary highest weight representations and related modules in algebraic geometry. However, uniform formulas remain elusive—even for more basic invariants such as the Gelfand-Kirillov dimension or the Bernstein degree, and are usually limited to families of representations in a dual pair setting. We use earlier work by Joseph to provide an elementary and intrinsic proof of a uniform formula for the Gelfand-Kirillov dimension of an arbitrary unitary highest weight module in terms of its highest weight. The formula generalizes a result of Enright and Willenbring (in the dual pair setting) and is inspired by Wang’s formula for the dimension of a minimal nilpotent orbit.