A K-theoretic Selberg trace formula

TL;DR

This paper provides a cohomological interpretation of the Selberg trace formula using K-theory of group C*-algebras, linking harmonic analysis, operator algebras, and index theory for semisimple Lie groups.

Contribution

It introduces a novel K-theoretic and cohomological perspective on the Selberg trace formula, connecting it with higher index theory of elliptic operators.

Findings

Cohomological interpretation of the trace formula via K-theory.

Application to index theory of elliptic differential operators.

Derivation of an index-theoretic version of the Selberg trace formula.

Abstract

Let G be a semisimple Lie group and H 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(G/H) associated to test functions. 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 H. 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.