Direct Sums of Mod p Characters of Gal and the Homology of GL(n, ℤ)

Abstract

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