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For an algebraic group , Anderson introduced the notion of Mirković-Vilonen (MV) polytopes as images of MV-cycles under the moment map of the affine Grassmannian. It was shown by Kamnitzer that MV-polytopes and their corresponding cycles can be described as solutions of the tropical Plücker relations. Another construction of MV-cycles, by Gaussent and Littelmann, can be given by using LS-galleries, a more discrete version of Littelmann's path model.
This article gives a direct combinatorial construction of the MV-polytopes using LS-galleries. This construction is linked to the retractions of the affine building and the Bott-Samelson variety corresponding to , leading to a type-independent definition of MV-polytopes not involving the tropical Plücker relations.
Let be a unitary group over a totally real field, and let be a Shimura variety associated to . For certain primes of good reduction for , we construct cycles on the characteristic fiber of . These cycles are defined as the loci on which the Verschiebung map has small rank on particular pieces of the Lie algebra of the universal abelian variety on . The geometry of these cycles turns out to be closely related to Shimura varieties for a different unitary group , which is isomorphic to at all finite places but not isomorphic to at archimedean places. More precisely, each cycle has a natural desingularization , which is almost isomorphic to a scheme parameterizing certain subbundles of the Lie algebra of the universal abelian variety over a Shimura variety associated to . We exploit this relationship to construct an injection of the étale cohomology of into that of . This yields a geometric construction of Jacquet-Langlands transfers of automorphic representations of to automorphic representations of .