PROJECTIONS OF MEASURES ON NILPOTENT ORBITS AND ASYMPTOTIC MULTIPLICITIES OF K-TYPES IN RINGS OF REGULAR FUNCTIONS .2.

摘要

Let G be the adjoint group of a real semi-simple Lie algebra g and let K be a maximal compact subgroup of G. K-c, the complexification of K, acts on p(c)*, the complexified cotangent space of G/K at eK. If O is a nilpotent K-c orbit in p(c)*, we study the asymototic behavior of the multiplicaties of K-types in the module R([O]) over bar, the regular functions on the Zariski closure of O. Sekiguchi has shown that each such orbit O corresponds naturally to a nilpotent G orbit Omega in g*. We show that if the split rank of g equals one, then the asymptotic behavior of K-types is determined precisely by beta(Omega), the canonical Lionville measure on Omega. David Vogan has conjectured that this relationship is true in general. We show that when g is complex, this conjecture can be reduced to the case in which O is not induced from a nilpotent orbit of a proper Levi-subalgebra of g. We also relate this conjecture to a recent result of Schmid and Vilonen that links the characteristic cycle of a Harish Chandra module to its asymototic support. (C) 1996 Academic Press, Inc. [References: 23]