Galois representations attached to tensor products of arithmetic cohomology

We compute the action of Hecke operators on tensor products of cohomology classes of lower congruence subgroups of SL(n,Z)SL(n,Z) in trivial weight. We use this computation to prove that if each representation in a collection of Galois representations is attached to a cohomology class of a lower congruence subgroup in trivial weight, then a sum of certain twists of the representations consistent with the main conjectures of [5] and [10] is also attached to such a cohomology class.