  Let G be a connected linear semisimple Lie group with Lie algebra g and
maximal compact subgroup K. Let K_C -> Aut(p_C) be the complexified isotropy
representation at the identity coset of the corresponding symmetric space.
Suppose that O is a nilpotent K_C-orbit in p_C, and bar(O} is its Zariski
closure in p_C. We study the K-type decomposition of the ring of regular
functions on bar(O} when O is spherical and ``small''. We also show that this
decomposition gives the asymptotic directions of K-types in any irreducible
(g_C, K)-module whose associated variety is bar(O).
