Given an arithmetic group G, there are two classical Hecke modules associated to G that are of central importance to number theory. Namely, the space of cusp forms associated to G (with fixed weight) and the cohomology of the arithmetic manifold M associated to G (with coefficients in a fixed local system). The Eichler-Shimura isomorphism (and its generalization) relates these two spaces as Hecke modules.

In this talk, we will introduce some new Hecke modules. We will equip the topological K-theory groups of M with Hecke operators and argue that the Chern character which relates topological K-theory to the cohomology is Hecke equivariant. We will then consider the operator K-theory of two noncommutative C*-algebras associated to G and equip them with Hecke operators. We will argue that these new Hecke modules should hold the same “arithmetic information” as the cohomology of M and thus should be quite relevant to number theory.

This is joint work with Bram Mesland (Bonn).