Abstract
Let be a lattice in a locally compact group . In another work, we used -theory to equip with Hecke operators the -groups of any --algebra on which the commensurator of acts. When is arithmetic, this gives Hecke operators on the -theory of certain -algebras that are naturally associated with . In this paper, we first study the topological -theory of the arithmetic manifold associated to . 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 -groups associated to an arithmetic group become true Hecke modules. We conclude by discussing Hecke equivariant maps in -theory in great generality and apply this to the Borel–Serre compactification as well as various noncommutative compactifications associated with . Along the way we discuss the relation between the -theory and the integral cohomology of low-dimensional manifolds as Hecke modules.
Citation
Bram Mesland. Mehmet Haluk Şengün. "Hecke modules for arithmetic groups via bivariant $K$-theory." Ann. K-Theory 3 (4) 631 - 656, 2018. https://doi.org/10.2140/akt.2018.3.631
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