A generalization of the Cartan–Helgason theorem for Riemannian symmetric spaces of rank one

Abstract

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 d in U, let µ be the highest restricted weight of d, and let s be the M-type acting in the highest restricted weight subspace of Hd. Fix a K-type t. Earlier we proved that if U / K has rank one, then d|K contains t if and only if t|M contains s and µ in µs,t + ?sph, where ?sph is the set of highest restricted spherical weights and µs,t is a suitable element of a* uniquely determined by s and t. 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.