# Bram Mesland: A Hecke module structure on the KK-theory of arithmetic groups

C*-algebra seminar talk by Bram Mesland (Leiden)

Title: A Hecke module structure on the KK-theory of arithmetic groups

Abstract: Let G be a locally compact group, H a discrete subgroup and C(G,H) the commensurator of H in G. The cohomology of H is a module over the Shimura Hecke ring of the pair (H,C(G,H)). This construction plays a central role in the cohomology of arithmetic groups and automorphic representation theory. For instance, it recovers the action of the Hecke operators on modular forms for SL(2,Z) as a particular case.

In this talk I will discuss a general KK-theoretic version of this construction. In particular the Shimura Hecke ring of a pair (H, C(G,H)) maps into the KK-ring associated to an arbitrary H-C*-algebra. From this we obtain a variety of K-theoretic Hecke modules. In particular, we will see how, in the case of arithmetic manifolds, all the Hecke operators arise as the simplest case of a topological correspondence in the sense of Connes-Skandalis. The Chern character provides a Hecke equivariant transformation into cohomology, which is an isomorphism in low dimensions.

We proceed to discuss the compatilbility of the Hecke operators with strong Morita equivalences arising from free and proper actions. We then discuss a fundamental result on Hecke-equivariant maps and deduce that the Baum-Connes assembly commutes with Hecke operators.

This is joint work with M.H. Sengun (Sheffield).

This will be a Zoom talk. Contact Sergey Neshveyev if you want to join.