Sums of Galois representations and arithmetic homology

Let Γ0(n,N) denote the usual congruence subgroup of type Γ0 and level N in SL(n,Z). Suppose for i = 1, 2 we have an irreducible odd ndimensional Galois representation ρi attached to a homology Hecke eigenclass in H∗(Γ0(n,Ni),Mi), where the level Ni and the weight and nebentype making up Mi are as predicted by the Serre-style conjecture of Ash-Doud-PollackSinnott. We assume that n is odd, N1N2 is squarefree and that ρ1⊕ρ2 is odd. We prove two theorems that assert that ρ1 ⊕ ρ2 is attached to a homology Hecke eigenclass in H∗(Γ0(2n,N),M), where N and M are as predicted by the Serre-style conjecture. The first theorem requires the hypothesis that the highest weights of M1 and M2 are small in a certain sense. The second theorem requires the truth of a conjecture as to what degrees of homology can support Hecke eigenclasses with irreducible Galois representations attached, but no hypothesis on the highest weights of M1 and M2. This conjecture is known to be true for n = 3 so we obtain unconditional results for GL(6). A similar result for GL(4) appeared in an earlier paper.