L 2 -Invariants of Locally Symmetric Spaces

Let X = G=K be a Riemannian symmetric space of the noncompact type, i‰ G a discrete, torsion-free, cocompact subgroup, and let Y = inX be the corresponding locally symmetric space. In this paper we explain how the Harish-Chandra Plancherel Theorem for L 2 (G) and results on (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 previ- ously obtained by Borel, Lott, Mathai, Hess and Schick, Lohoue and Mehdi. It turns out that the behaviour of these invariants is essen- tially determined by the fundamental rank m = rkCGirkCK of G. In particular, we show the nonvanishing of the L 2 -torsion of Y whenever m = 1.