On the Gelfand–Kirillov dimension of a discrete series representation

Abstract

Lower bounds to the Gelfand–Kirillov dimension of discrete series are given for semisimple Lie groups with finite center by showing that the K-finite vectors are torsion free with respect to enveloping algebras of certain unipotent subgroups. In particular we prove two folk theorems about the Gelfand–Kirillov dimension. The first is that the holomorphic (or anti-holomorphic) discrete series are the “smallest” and representations with Whittaker models for minimal parabolic subgroups are the “largest” (a more precise result in the quasi-split case is due to Kostant). We also show that if G is quaternionic and not of type A or C, then the quaternionic discrete series is the “smallest”.