In 2002, Arkhipov and Bezrukavnikov established an equivalence between the Iwahori-equivariant derived category of constructible \ell-adic sheaves on the affine flag variety, the so-called Iwahori-Whittaker category on the affine flag variety, and the Langlands dual group equivariant derived category of coherent sheaves on the Langlands dual Springer resolution for a connected reductive group over \bar{\mathbb{F}}_p. In this talk, we discuss this equivalence for p-adic groups by studying a mixed-characteristic Gaitsgory's central functor. This is a joint work with J. Ansch\"utz, J. Louren\c{c}o, and Z. Wu.