2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64789We make a fine study of the SO(2,C)-double coset decomposition in SL(2,C) and give a full description of the intersection of the various cells with the complex crown Xi of SL(2,R)/SO(2). A non-linear convexity theorem is proved and applications to analytically continued spherical functions are given.15 pages, 2 figures. Simpler geometric proofs of the results in Section 3 are given. To appear in Math. ZRepresentation Theory22Exxtext