Structure constants for Hecke and representation rings

In 2002, Kapovich, Leeb, and Millson studied certain structure constants for two related rings: the spherical Hecke algebra of a split connected reductive group over a local non-Archimedean field, and the representation ring of the Langlands dual group. The structure constants of the former are defined relative to characteristic functions of double cosets, and the latter relative to highest weight representations. They proved that the nonvanishing of one of the latter structure constants always implies the nonvanishing of the corresponding former one. For GLn, the reverse implication also holds, and is due to P. Hall. Both proofs are combinatorial in nature. In this paper, we provide geometric proofs for both results, using affine Grassmannians. We also provide some additional results concerning minuscule coweights and the equidimensionality of the fibers of certain Bott-Samelson resolutions of affine Schubert varieties for GLn.