Hecke Operators and the Stable Homology of GL(n)

Abstract

Let R be a field of any characteristic and A a principal ideal domain. We make a conjecture that asserts that any Hecke operator T acts punctually on any Hecke eigenclass in the stable homology with trivial coefficients R of a principal congruence subgroup Γ in GL(n, A), i.e., as multiplication by the number of single cosets contained in T. In the case where A = ℤ, this conjecture implies that the Galois representation consisting of the sum of the 0th, 1st, ⋅, n − 1st powers of the cyclotomic character is attached to any stable Hecke eigenclass. When A = ℤ and Γ =GL(n, ℤ), this conjecture was already made by Calegari and Venkatesh. We obtain partial results giving evidence for the conjecture. These results imply that some part of the Hecke algebra acts punctually. If the characteristic of R is 0, they show that the entire Hecke algebra acts punctually, which was already known in a completely different way using a result of A. Borel.