A K-theoretic Selberg trace formula (Q2220166)

Given a semisimple Lie group \(G\) and a cocompact lattice \(\Gamma\) in \(G\), any compactly supported function on \(G\) defines a trace class operator on the Hilbert space \(L^2(\Gamma\backslash G)\) by convolution. The Selberg trace formula is an equality arising by computing its trace in two different ways: the ``spectral side'' of the equality sums up the traces of the operator's diagonal components with respect the decomposition of the right regular representation \(R^\Gamma\) on \(L^2(\Gamma\backslash G)\) into irreducible representations \(\pi\in\widehat G\) of \(G\) and the ``geometric side'' is a weighted sum over the conjugacy classes \((\gamma)\in\langle\Gamma\rangle\) of \(\Gamma\) of orbital integrals over \(G_\gamma\backslash G\) of functions obtained by decomposing the diagonal of the operator's kernel. The paper under review develops a \(K\)-theoretic analogue, whose spectral side is the decomposition of the induced map \(R^\Gamma_*\colon K_0(C^*(G))\to\mathbb{Z}\) as the sum of the maps \(\pi_*\colon K_0(C^*(G))\to\mathbb{C}\) counted with the multiplicity of \(\pi\in\widehat G\) in \(R^\Gamma\) and whose geometric side decomposes the map \(\mathbf{1}_*\colon K_0(L^1(\Gamma))\to\mathbb{C}\) induced by the trivial representation \(\mathbf{1}\) of \(\Gamma\) into a sum indexed by the conjugacy classes \((\gamma)\in\langle\Gamma\rangle\) of \(\Gamma\). Equality between the two sides holds for elements corresponding to each other under \(K_0(C^*(G))\xrightarrow{\operatorname{res}}K_0(C^*(\Gamma))\leftarrow K_0(L^1(\Gamma))\), where \(\operatorname{res}\) denotes the restriction homomorphism given by the Mackey--Rieffel theory of imprimitivity bimodules. Furthermore, it is shown that this \(K\)-theoretic version implies the index theoretic version of the Selberg trace formula developed in [\textit{D. Barbasch} and \textit{H. Moscovici}, J. Funct. Anal. 53, 151--201 (1983; Zbl 0537.58039)]. For the entire collection see [Zbl 1455.47001].