Abstract
Let G be a semisimple Lie group and Γ a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L 2( Γ∖G) associated to test functions f ∈ C c(G).
In this paper we present a cohomological interpretation of the trace formula involving the K-theory of the maximal group C ∗-algebras of G and Γ. As an application, we exploit the role of group C ∗-algebras as recipients of “higher indices” of elliptic differential operators and we obtain the index theoretic version of the Selberg trace formula developed by Barbasch and Moscovici from ours.
In memory of Ronald G. Douglas
B. Mesland and M. H. Şengün gratefully acknowledge support from the Max Planck Institute for Mathematics in Bonn, Germany. H. Wang is supported by NSFC-11801178 and Shanghai Rising-Star Program 19QA1403200.
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Notes
- 1.
Lafforgue uses the notation 〈H, x〉 for our π ∗(x), where H is the representation space of π.
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Mesland, B., Şengün, M.H., Wang, H. (2020). A K-Theoretic Selberg Trace Formula. In: Curto, R.E., Helton, W., Lin, H., Tang, X., Yang, R., Yu, G. (eds) Operator Theory, Operator Algebras and Their Interactions with Geometry and Topology . Operator Theory: Advances and Applications, vol 278. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-43380-2_19
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