L^2-invariants of locally symmetric spaces

TL;DR

This paper provides a uniform method to compute $L^2$-invariants of compact locally symmetric spaces using harmonic analysis and Lie algebra cohomology, revealing their dependence on the fundamental rank of the isometry group.

Contribution

It introduces a unified approach to calculate $L^2$-Betti numbers, Novikov-Shubin invariants, and $L^2$-torsion, extending previous results and highlighting the role of fundamental rank.

Findings

Nonvanishing of $L^2$-torsion when fundamental rank equals 1

Invariants are determined by the fundamental rank of the isometry group

Provides a uniform computational framework for $L^2$-invariants

Abstract

We explain how the Harish-Chandra Plancherel Theorem and results in relative Lie algebra cohomology can be used in order to compute in a uniform way the $L^2$-Betti numbers, the Novikov-Shubin invariants, and the $L^2$-torsion of compact locally symmetric spaces thus completing results previously obtained by Borel, Lott, Mathai, Hess and Schick. It turns out that the behaviour of these invariants is essentially determined by the fundamental rank of the group of isometries of the corresponding globally symmetric space. In particular, we show the nonvanishing of the $L^2$-torsion whenever the fundamental rank is equal to 1.