Cores of Symplectic Double Groupoids via Reduction

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We use symplectic reduction to give a new construction of the core C of a symplectic double groupoid D as the common leaf space of characteristic foliations associated to various coisotropic submanifolds of D. In the case of the cotangent double groupoid of a Lie groupoid G, the canonical relations arising from this process turn out to be cotangent lifts of structure maps associated to G. We also show that under this reduction procedure the double groupoid structure on D descends to a groupoid structure on the leaf space above, recovering the core groupoid structure on C of Brown and Mackenzie.