Projections of Measures on Nilpotent Orbits and Asymptotic Multiplicities ofK-Types in Rings of Regular Functions II

LetGbe the adjoint group of a real semi-simple Lie algebragand letKbe a maximal compact subgroup ofG.KC, the complexification ofK, acts onp*C, the complexified cotangent space ofG/KateK. If O is a nilpotentKCorbit inp*C, we study the asymptotic behavior of the multiplicities ofK-types in the moduleR[O], the regular functions on the Zariski closure of O. Sekiguchi has shown that each such orbit O corresponds naturally to a nilpotentGorbitΩing*. We show that if the split rank ofgequals one, then the asymptotic behavior ofK-types is determined precisely byβΩ, the canonical Liouville measure onΩ. David Vogan has conjectured that this relationship is true in general. We show that whengis complex, this conjecture can be reduced to the case in which O is not induced from a nilpotent orbit of a proper Levi-subalgebra ofg. We also relate this conjecture to a recent result of Schmid and Vilonen that links thecharacteristic cycleof a Harish Chandra module to itsasymptotic support.