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Generalized Mullineux involution and perverse equivalences

We define a generalization of the Mullineux involution on multipartitions using the theory of crystals for higher level Fock spaces. Our generalized Mullineux involution turns up in representation theory via two important derived functors on cyclotomic Cherednik category $\mathcal{O}$: Losev's "$κ=0$" wall-crossing, and the Ringel duality.