Double MV cycles and the Naito–Sagaki–Saito crystal

The theory of MV cycles associated to a complex reductive group GG 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 (2006) [7] crystal structure on double MV cycles, generalizing a finite-dimensional result of Baumann and Gaussent (2008) [2]. As an application, we give a geometric construction of the Naito–Sagaki–Saito [23] crystal via the action of SL̂n 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(∞)B(∞) crystal. In particular, our geometric proof works in the previously unknown case of sl̂2.