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Double MV Cycles and the Naito-Sagaki-Saito Crystal

The theory of MV cycles associated to a complex reductive group $G$ has proven to be a rich source of structures related to representation theory. We investigate double MV cycles, which are analogues of MV cycles in the case of an affine Kac-Moody group. We prove an explicit formula for the Braverman-Finkelberg-Gaitsgory crystal structure on double MV cycles, generalizing a finite-dimensional result of Baumann and Gaussent. As an application, we give a geometric construction of the Naito-Sagaki-Saito crystal via the action of $\hat{SL}_n$ on Fermionic Fock space. In particular, this construction gives rise to an isomorphism of crystals between the set of double MV cycles and the Naito-Sagaki-Saito crystal. As a result, we can independently prove that the Naito-Sagaki-Saito crystal is the $B(\infty)$ crystal. In particular, our geometric proof works in the previously unknown case of $\hat{\mathfrak{sl}}_2$.