$L^2$-invariants of locally symmetric spaces

Abstract

Let $X=G/K$ be a Riemannian symmetric space of the noncompact type, $\Gamma \subset G$ a discrete, torsion-free, cocompact subgroup, and let $Y=\Gamma \backslash X$ be the corresponding locally symmetric space. In this paper we explain how the Harish-Chandra Plancherel Theorem for $L^2(G)$ and results on $(\mathfrak{g},K)$-cohomology can be used in order to compute the $L^2$-Betti numbers, the Novikov–Shubin invariants, and the $L^2$-torsion of $Y$ in a uniform way thus completing results previously obtained by Borel, Lott, Mathai, Hess and Schick, Lohoue and Mehdi. It turns out that the behaviour of these invariants is essentially determined by the fundamental rank $m={\mathrm{rk}}_{\mathbb{C}}G-{\mathrm{rk}}_{\mathbb{C}}K$ of $G$. In particular, we show the nonvanishing of the $L^2$-torsion of $Y$ whenever $m=1$.