  In this paper we prove, for G a connected reductive algebraic group
satisfying a technical assumption, that the Satake category of G (with
coefficients in a finite field, a finite extension of Q_l, or the ring of
integers of such a field) can be described via Iwahori-Whittaker perverse
sheaves on the affine Grassmannian. As an application, we confirm a conjecture
of Juteau-Mautner-Williamson describing the tilting objects in the Satake
category.
