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