Hecke modules for arithmetic groups via bivariant K-theory

Bram Mesland

*(Leiden University)*

15:15-16:15, Nov 12, 2019 Science Building A510

__Abstract:__

Let ¦£ be a lattice in a locally compact group G. In this talk I will discuss how KK-theory can be elegantly employed to equip with Hecke operators the K-groups of any ¦£-C?-algebra on which the commensurator of ¦£ acts. When ¦£ is arithmetic, this gives Hecke operators on the K-theory of certain C*-algebras that are naturally associated with ¦£. I will discuss various properties of the Hecke operators, such as commutation with the Chern character in the case of the topological K-theory of the arithmetic manifold associated to ¦£ and the fact that the Shimura product of double cosets naturally corresponds to the Kasparov product. If time allows I will discuss Hecke equivariant maps in KK-theory and apply this to the Borel?Serre compactification as well as various noncommutative compactifications associated with ¦£.

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