Structure Constants for Hecke and Representation Rings
نویسندگان
چکیده
In [KLM] the authors study 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 former are defined relative to characteristic functions of double cosets, and the latter relative to highest weight representations. They prove 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 note, we provide geometric proofs of 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.
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