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Let $G$ and $\check{G}$ be Langlands dual connected reductive groups. Weestablish a monoidal equivalence of $\infty$-categories between equivariantquasicoherent sheaves on the formal neighborhood of the nilpotent cone in $G$and Steinberg-Whittaker D-modules on the loop group of $\check{G}$, asconjectured by Bezrukavnikov. More generally, we establish equivalences betweenvarious spectral and automorphic realizations of affine Hecke categories andtheir modules, confirming conjectures of Bezrukavnikov.

A Langlands dual realization of coherent sheaves on the nilpotent cone