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 [MV1, MV2] to the twisted product of 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 nonvanishing 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 [KT] and Kapovich-Leeb-Millson [KLM]. 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.