Matching of Hecke operators for exceptional dual pair correspondences

Let G be a split algebraic group of type EnEn defined over a p  -adic field. This group contains a dual pair G×G′G×G′ where one of the groups is of type G2G2. 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′G′, when acting on the minimal representation. This implies that the correspondence is functorial, in the sense of Arthur and Langlands, for spherical representations.