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

• Autores: Roberto Camporesi
• Localización: Pacific journal of mathematics, ISSN 0030-8730, Vol. 222, Nº 1, 2005, págs. 1-28
• Idioma: inglés
• DOI: 10.2140/pjm.2005.222.1
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• Resumen

  • 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.