Equidimensionality of convolution morphisms and applications to saturation problems

Fix a split connected reductive group G over a field k, and a positive integer r. For any r-tuple of dominant coweights μi of G, we consider the restriction mμ
• of the r-fold convolution morphism of Mirkovic–Vilonen to the twisted product of affine Schubert varieties corresponding to μ
• . We show that if all the coweights μi are minuscule, then the fibers of mμ
• are equidimensional varieties, with dimension the largest allowed by the semi-smallness of mμ
• . We derive various consequences: the equivalence of the non-vanishing of Hecke and representation ring structure constants, and a saturation property for these structure constants, when the coweights μi are sums of minuscule coweights. This complements the saturation results of Knutson–Tao and Kapovich–Leeb–Millson. We give a new proof of the P-R-V conjecture in the “sums of minuscules” setting. Finally, we generalize and reprove a result of Spaltenstein pertaining to equidimensionality of certain partial Springer resolutions of the nilpotent cone for GLn.