The signature with local coefficients of locally symmetric spaces
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.
scientific article; zbMATH DE number 3977300
scientific article