Let U∕K be a compact
Riemannian symmetric space with U simply connected and K connected. Let
G∕K be the noncompact dual space, with G and U analytic subgroups of
the simply connected complexification Gℂ. Let G = KAN be an Iwasawa
decomposition of G, and let M be the centralizer of A in K. For δ ∈U, let μ
be the highest restricted weight of δ, and let σ be the M-type acting in
the highest restricted weight subspace of Hδ. Fix a K-type τ. Earlier we
proved that if U∕K has rank one, then δ|K contains τ if and only if τ|M
contains σ and μ ∈ μσ,τ+ Λsph, where Λsph is the set of highest restricted
spherical weights and μσ,τ is a suitable element of a∗ uniquely determined by
σ and τ. In this paper we obtain an explicit formula for this element in
the case of U∕K = Sn, Pn(ℂ), Pn(ℍ). This gives a generalization of the
Cartan–Helgason theorem to arbitrary K-types on these rank one symmetric
spaces.
