Vol. 3, No. 4, 2018

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Hecke modules for arithmetic groups via bivariant $K\mskip-2mu$-theory

Bram Mesland and Mehmet Haluk Şengün

Vol. 3 (2018), No. 4, 631–656

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

KK-theory, Hecke operators, arithmetic groups
Mathematical Subject Classification 2010
Primary: 11F32, 11F75, 19K35, 55N20
Received: 23 October 2017
Revised: 4 March 2018
Accepted: 22 March 2018
Published: 18 December 2018
Bram Mesland
Mathematik Zentrum
University of Bonn
Mehmet Haluk Şengün
School of Mathematics and Statistics
University of Sheffield
United Kingdom