Matching of Hecke operators for exceptional dual pair correspondences

Let G be a split algebraic group of type E_n defined over a p-adic field. This group contains a dual pair G×G^' where one of the groups is of type G₂. The minimal representation of G, when restricted to the dual pair, gives a correspondence of representations of the two groups in the dual pair. We prove a matching of spherical Hecke algebras of G and G^', when acting on the minimal representation. This implies that the correspondence is functorial, in the sense of Arthur and Langlands, for spherical representations.