Hecke modules for arithmetic groups via bivariant $K$-theory

Abstract

Let $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$-$C^{\ast }$-algebra on which the commensurator of $\mathrm{\Gamma }$ acts. When $\mathrm{\Gamma }$ is arithmetic, this gives Hecke operators on the $K$-theory of certain $C^{\ast }$-algebras that are naturally associated with $\mathrm{\Gamma }$. In this paper, we first study the topological $K$-theory of the arithmetic manifold associated to $\mathrm{\Gamma }$. 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 $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$. Along the way we discuss the relation between the $K$-theory and the integral cohomology of low-dimensional manifolds as Hecke modules.