2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/172969The connections between the objects mentioned in the title are used to give a short proof of the Cartan--Helgason theorem and a natural construction of the compactifications.Representation TheoryGroup Theory22ECartan--Helgason theorem, Poisson transform, and Furstenberg--Satake compactificationstext