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 FlG 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 FlG 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 FlG in the generic case. The strong version of our main theorem claims a motivic equivalence between the Whittaker category on FlG and a factorization module category, which holds in the de Rham setting, the Betti setting, and the -adic setting.