The Novikov-Shubin invariants for locally symmetric spaces (Q1569017)

Let \(M\) be a closed Riemannian manifold with universal covering \(\widetilde M\). The \(L^2\)-Betti number \(b^{(2)}_p(\widetilde M)\) can be defined as the limit \(\lim_{t\to \infty} \int_{\mathcal F}\text{tr}_\mathbb{R}(e^{-t\Delta_p} (x,x)\bigr) dvol\), where \(e^{-t\Delta_p} (x,y)\) denotes the heat kernel on the universal covering in dimension \(p\) and \({\mathcal F}\) is a fundamental domain for the action of the fundamental group of \(M\) on \(\widetilde M\). The Novikov-Shubin invariants \(\alpha_p (\widetilde M)\) measure how fast this limit is approached. It is a real number or \(\infty\) or takes by definition the symbolic value \(\infty^+\) if the Laplacian acting on \(p\)-forms on \(\widetilde M\) has a gap in its spectrum at zero. These numbers depend only on the homotopy type of \(M\) by results of Dodziuk and Gromov-Shubin. The paper is devoted to explicit computations of the Novikov-Shubin invariants in the case, where \(M\) is a locally symmetric space. The main technical input are the Plancherel formula and \(({\mathfrak g},K)\)-homology. There seems to be a misprint in Proposition 11.1. The sign \(<\) should be replaced by \(\leq\) in assertion (i).