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 $\Gamma$-$C^{\ast }$-algebra on which the commensurator of $\Gamma$ acts. When $\Gamma$ is arithmetic, this gives Hecke operators on the $K$-theory of certain $C^{\ast }$-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.

Keywords

Mathematical Subject Classification 2010

Milestones

Authors