The Hecke category (part II—Satake equivalence)

Theorem 1. The convolution ∗ admits a commutativity constraint making Sph into a rigid tensor category. There exists a faithful, exact tensor “fiber” functor Sat : Sph → Vect inducing an equivalence (modulo a sign in the commutativity constraint) of Sph with Rep(G) as tensor categories, where G is the Langlands dual group of the reductive group G, whose weights are the coweights of G and vice versa. More generally, there is an equivalence Satn : Sphn → Repn(G) which is monoidal for each n, respects the factorizable structures on both sides as n varies (including the Sn-equivariance), and for n = 1, agrees with Sat when restricted to each point of X.