Stable Group Homology and Hecke Operators

About 30 years ago Ruth Charney proved that the homology of the general linear group stabilized. In the case of GL(n,Z) this means that if M is any constant coefficient module and i is fixed, H_i(GL(n,Z),M) doesn't depend on n if n is sufficiently large. Recently, Frank Calegari and Akshay Venkatesh asked: how do the Hecke operators act on the stable homology? I will answer this question and show that the Hecke eigenvalues correspond to Galois representations which are isomorphic to direct sums of powers of the cyclotomic character.