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We consider the Newton stratification on Iwahori double cosets in the loopgroup of a reductive group. We describe a group-theoretic condition on thegeneric Newton point, called cordiality, under which the Newton poset (i.e. theindex set for non-empty Newton strata) is saturated and Grothendieck'sconjecture on closures of the Newton strata holds. Finally, we give severallarge classes of Iwahori double cosets for which this condition is satisfied bystudying certain paths in the associated quantum Bruhat graph.

Generic Newton points and the Newton poset in Iwahori double cosets