Let G be the adjoint group
of a real semi-simple Lie algebra g and let K be a maximal compact subgroup of G.
KC, the complexification of K, acts on pC∗, the complexified cotangent space of
G∕K at eK. If 𝒪 is a nilpotent KC orbit in pC∗, we study the asymptotic behavior
of the K-types in the module R[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.
