Hecke modules for arithmetic groups via bivariant K -theory

Let Γ be a lattice in a locally compact group G. In earlier work, we used KK-theory to equip the K-groups of any Γ-C∗-algebra on which the commensurator of Γ acts with Hecke operators. When Γ is arithmetic, this gives Hecke operators on the K-theory of certain C∗-algebras that are naturally associated with Γ. In this paper, we first study the topological K-theory of the arithmetic manifold associated to Γ. We prove that the Chern character commutes with Hecke operators. Afterwards, we show that the Shimura product of double cosets naturally corresponds to the Kasparov product and thus that the KK-groups associated to an arithmetic group Γ become true Hecke modules. We conclude by discussing Hecke equivariant maps in KK-theory in great generality and apply this to the Borel-Serre compactification as well as various noncommutative compactifications associated with Γ. Along the way we discuss the relation between the K-theory and the integral cohomology of low-dimensional manifolds as Hecke modules.