The signature with local coefficients of locally symmetric spaces
type: Item
instance of: scholarly article
title: The signature with local coefficients of locally symmetric spaces
zbMATH Open document ID: 0605.22009
author: Henri Moscovici
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1985-01-01T00:00:00Z
review text: Let G be a linear connected semisimple Lie group, \(\Gamma\) a discrete subgroup of G and F a finite-dimensional complex irreducible G-module. Further, let K be a maximal compact subgroup of G. Let us consider the case that G possesses a Cartan subgroup contained in K. Then dim G/K is even. We set \(m=(\dim G/K)/2\). The author shows that the cohomology group \(H^ m(\Gamma; F)\) is always nonzero, provided that \(\Gamma\) is torsion- free and cocompact and G is such that the compact dual Y of \(X=G/K\) has odd Euler characteristic. From the introduction: ''Our approach is to use the existence of a G- invariant nondegenerate Hermitian metric on F to construct a flat Hermitian metric on the associated flat vector bundle F over \(X=\Gamma \setminus G/K\), and then to obtain an explicit formula for the signature with coefficients in the corresponding local system \({\mathcal F}\). Since \(H^*(\Gamma; F)=H^*(X_{\Gamma}; {\mathcal F})\), the nonvanishing of this signature implies, of course, the nonvanishing of \(H^ m(\Gamma; F).''\) Some generalizations for the case that \(\Gamma\) is not cocompact but has finite covolume are also given.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 17B56
Mathematics Subject Classification ID: 53B35
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 3977300
zbMATH Keywords: locally symmetric spaces
zbMATH Keywords: signature formula
zbMATH Keywords: nonvanishing signature
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: discrete subgroup
zbMATH Keywords: complex irreducible G-module
zbMATH Keywords: Cartan subgroup
zbMATH Keywords: cohomology group
zbMATH Keywords: Hermitian metric
zbMATH Keywords: local system
reviewed by: Gregory Margulis
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228591
OpenAlex ID: W1971166398
cites work: The index of elliptic operators. III
cites work: \(L^ 2\)-index and the Selberg trace formula
cites work: \(L^ 2\)-cohomology of locally symmetric manifolds of finite volume
cites work: Q3885948
cites work: Limit formulas for multiplicities in \(L^2(\Gamma\setminus G)\)
cites work: A dimension formula for a certain space of automorphic forms of \(\text{SU}(p,1)\)
cites work: Novikov's higher signature and families of elliptic operators
cites work: 𝐿²-index of elliptic operators on locally symmetric spaces of finite volume
cites work: Dirac operator and the discrete series
cites work: Discrete Series Multiplicities in L 2 (Γ\G)
cites work: L//2 cohomology of warped products and arithmetic groups
DOI: 10.2748/TMJ/1178228591
description: scientific article; zbMATH DE number 3977300