The trace formula for Hecke operators over rank one lattices
Let \(G\) be a real reductive group and let \(\Gamma \subset G\) be a lattice of rank one, in the sense of [OW] [\textit{M. Osborne} and \textit{G. Warner}, J. Reine Angew. Math. 324, 1--113 (1981; Zbl 0469.22007)]. Let \(\pi\) denote the left regular representation of \(G\) on \(L^ 2(G/\Gamma)\) and let \(L^ 2_ d(G/\Gamma)\) denote the closure of the sum of the \(G\)-invariant irreducible subspaces of \(L^ 2(G/\Gamma)\). If \(\alpha \in C_ c^{\infty}(G)\) is \(K\)-finite and \(T\) is a Hecke operator, the operator \(\pi(\alpha)T\) is trace-class, by a result of Donelly (recently W. Müller has proved this without restriction on the rank). The purpose of the present paper is to arrive at an explicit expression for its trace, under an assumption analogous to that in [OW], in the case of \(\pi(\alpha)\). The trace is given in terms of (weighted) orbital integrals and the usual intertwining and residual terms. The exposition is self- contained and the proofs follow the lines of those in [OW], in the case of \(\pi (\alpha)\).
scientific article; zbMATH DE number 4146292