DIRECT SUMS OF MOD p CHARACTERS OF GAL((Q)over-bar/Q) AND THE HOMOLOGY OF GL(n, Z)

We prove the following theorem: Let F be an algebraic closure of a finite field of characteristic p. Let be a continuous homomorphism from the absolute Galois group of to GL(n, F) which is isomorphic to a direct sum of one-dimensional representations (i) where p>n+1 and the product of the conductors of the (i) is squarefree and prime to p. If a certain parity condition holds, then is attached to a Hecke eigenclass in the homology of an arithmetic subgroup of SL(n, Z) with coefficients in a module V, where and V are as predicted by Conjecture 2.2 of [5].