  We study the fibers of Mirkovic-Vilonen convolution morphisms. We prove their
equidimensionality when all the coweights in question are minuscule, and some
related statements. We give applications to saturation problems for structure
constants of Hecke and representation rings. An erratum has been added to
correct insufficiencies in the proofs of Lemma 9.4 and Corollary 9.5 as they
appeared in the published version.
