Hecke modules for arithmetic groups via bivariant K-theory

Abstract

Let Gamma be a lattice in a locally compact group G. In another work, we used KK-theory to equip with Hecke operators the K-groups of any Gamma-C*-algebra on which the commensurator of Gamma acts. When Gamma is arithmetic, this gives Hecke operators on the K-theory of certain C*-algebras that are naturally associated with Gamma. In this paper, we first study the topological K-theory of the arithmetic manifold associated to Gamma. 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 Gamma 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 Gamma. Along the way we discuss the relation between the K-theory and the integral cohomology of low-dimensional manifolds as Hecke modules.