Hecke modules for arithmetic groups via bivariant K-theory

TL;DR

This paper develops a framework using bivariant K-theory to study Hecke operators on the K-theory of C*-algebras associated with arithmetic groups, establishing their compatibility with topological invariants and products.

Contribution

It introduces a novel approach connecting Hecke operators with Kasparov products in KK-theory, enabling Hecke modules structure on K-groups of arithmetic groups.

Findings

Hecke operators commute with the Chern character in topological K-theory.

The Shimura product of double cosets corresponds to the Kasparov product.

KK-groups become genuine Hecke modules for arithmetic groups.

Abstract

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.