Projections of Measures on Nilpotent Orbits and Asymptotic Multiplicities ofK-Types in Rings of Regular Functions II
✍ Scribed by Donald R King
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 1009 KB
- Volume
- 138
- Category
- Article
- ISSN
- 0022-1236
No coin nor oath required. For personal study only.
✦ Synopsis
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 GÂK at eK. If O is a nilpotent K C orbit in p* C , we study the asymptotic behavior of the multiplicities of K-types in the module R[O ], the regular functions on the Zariski closure of O. Sekiguchi has shown that each such orbit O corresponds naturally to a nilpotent G orbit 0 in g*. We show that if the split rank of g equals one, then the asymptotic behavior of K-types is determined precisely by ; 0 , the canonical Liouville measure on 0. 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 asymptotic support.