HIGHLY REDUCIBLE GALOIS REPRESENTATIONS ATTACHED TO THE HOMOLOGY OF GL(n,Z) AVNER ASH AND DARRIN DOUD

Let n ≥ 1 and F an algebraic closure of a finite field of characteristic p > n + 1. Let ρ : GQ → GL(n,F) be a Galois representation that is isomorphic to a direct sum of a collection of characters and an odd mdimensional representation τ . We assume that m = 2 or m is odd, and that τ is attached to a homology class in degree m(m−1)/2 of a congruence subgroup of GL(m,Z) in accordance with the main conjecture of [4]. We also assume a certain compatibility of τ with the parity of the characters and that the Serre conductor of ρ is square-free. We prove that ρ is attached to a Hecke eigenclass in Ht(Γ,M), where Γ is a subgroup of finite index in SL(n,Z), t = n(n− 1)/2 and M is an FΓ-module. The particular Γ and M are as predicted by the main conjecture of [4]. The method uses modular cosymbols, as in [1].