Resumen de Equidimensionality of convolution morphisms and applications to saturation problems
Thomas J. Haines
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.
