New Hecke modules for arithmetic groups via operator algebras

Haluk Sengun (University of Sheffield)

Thursday 31st May, 2018 16:00-17:00 Maths 116


Let G be a Lie group and H an arithmetic subgroup. Then the commensurator C(G,H) of H in G is a dense subgroup of G. The Hecke ring T(H) associated to H is the  convolution ring of finitely supported complex valued functions on the double coset space H\C(G,H)/H. This ring acts on various spaces of associated to H, in particular, on the cohomology groups of the arithmetic manifold M associated to H. This Hecke module structure on the cohomology of M plays a central role in the Langlands programme. 
In this talk I will discuss how the Hecke ring T(H) maps into the KK-ring associated to an arbitrary H-C*-algebra. From this we obtain a variety of K-theoretic Hecke modules of interest.  In particular, we equip the topological K-theory of M with a Hecke module structure and show that the Chern character provides a Hecke equivariant transformation into the rational cohomology of M. We also consider the case of two, quite well-known, noncommutative C*-algebras associated to H and study their K-groups as Hecke modules. Time permitting, I will also discuss some ongoing work relating to connections with automorphic forms and to the assembly map.

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

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