Affine Weyl groups and their parabolic quotients are used extensively asindexing sets for objects in combinatorics, representation theory, algebraicgeometry, and number theory. Moreover, in the classical Lie types we canconveniently realize the elements of these quotients via intuitive geometricand combinatorial models such as abaci, alcoves, coroot lattice points, corepartitions, and bounded partitions. Berg, Jones, and Vazirani described abijection between n-cores with first part equal to k and (n-1)-cores with firstpart less than or equal to k, and they interpret this bijection in terms ofthese other combinatorial models for the quotient of the affine symmetric groupby the finite symmetric group. In this paper we discuss how to generalize thebijection of Berg-Jones-Vazirani to parabolic quotients of affine Weyl groupsin type C. We develop techniques using the associated affine hyperplanearrangement to interpret this bijection geometrically as a projection ofalcoves onto the hyperplane containing their coroot lattice points. We arethereby able to analyze this bijective projection in the language of variousadditional combinatorial models developed by Hanusa and Jones, such as abaci,core partitions, and canonical reduced expressions in the Coxeter group.
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