  Let $\Gamma$ be a lattice in a locally compact group $G$. In earlier work, we
used $KK$-theory to equip the $K$-groups of any $\Gamma$-$C^{*}$-algebra on
which the commensurator of $\Gamma$ acts with Hecke operators. 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.
