Projections of measures on nilpotent orbits and asymptotic multiplicities ofK-types in rings of regular functions. I

Abstract

Let G be the adjoint group of a real semi-simple Lie algebra g and let Kbea maximal compact subgroup of G. Kc, the complexification of K, acts on p£, the complexified cotangent space of G/K at eK. If O is a nilpotent KQ orbit inp£, we study the asymptotic behavior of the K-types in the module ϋ[O], the regular functions on the Zariski closure of O. We show that in many cases this asymptotic behavior is determined precisely by the canonical Liouville measure on a nilpotent G orbit in g* which is naturally associated to O. We provide evidence for a conjecture of Vogan stating that this relationship is true in general. Vogan's conjecture is consistent with the philosophy of the orbit method for representations of real reductive groups.