Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group

Let G be a reductive group, and let G be its Langlands dual group. Arkhipov and Bezrukavnikov proved that the Whittaker category on the affine flags {Fl}G is equivalent to the category of G -equivariant quasi-coherent sheaves on the Springer resolution of the nilpotent cone. This paper proves this theorem in the quantum case. We show that the twisted Whittaker category on {Fl}G and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category O is equivalent to the twisted Whittaker category on {Fl}G in the generic case. The strong version of our main theorem claims a motivic equivalence between the Whittaker category on {Fl}G and a factorization module category, which holds in the de Rham setting, the Betti setting, and the -adic setting.