\(L^ 2\)-index and the Selberg trace formula (Q792685)

A method is developed for computing the \(L^ 2\)-index of a locally symmetric elliptic differential operator \(D_{\Gamma}\), acting on a locally symmetric manifold \(M_{\Gamma}=\Gamma \backslash G/K\) with G semisimple of real-rank 1 and \(\Gamma\) of finite co-volume, based on applying the Selberg trace formula to the difference of the two heat kernels associated to \(D_{\Gamma}\). The applications include an extension of the Osborne-Warner multiplicity formule to certain non- integrable discrete series and showing the existence, in some cases, of non-invariant \(L^ 2\)-cohomology classes in the middle dimension.