  We consider a reductive dual pair (G, G') in the stable range with G' the
smaller member and of Hermitian symmetric type. We study the theta lifting of
nilpotent K'_C-orbits, where K' is a maximal compact subgroup of G' and we
describe the precise K_C-module structure of the regular function ring of the
closure of the lifted nilpotent orbit of the symmetric pair (G, K). As an
application, we prove sphericality and normality of the closure of certain
nilpotent K_C-orbits obtained in this way. We also give integral formulas for
their degrees.
