Gaussian weighted unreduced \(L_2\) cohomology of locally symmetric spaces
Let \((M,g)\) be a locally symmetric manifold of nonpositive sectional curvature and of finite volume. The universal cover \(X=G/K\) is a globally symmetric space of noncompact type. The \(w\) weighted unreduced \(L_2\) cohomology is an usual unreduced \(L_2\) cohomology, where \(w: M \rightarrow (0,\infty)\) is a weighting function and dvol is replaced by the measure \(w\) dvol. The weights for the de Rham cohomology of exponential decay have been constructed by \textit{A. Borel} [Duke Math. J. 50, 605-623 (1983; Zbl 0528.22010)]. Later \textit{J. Franke} constructed weighting functions on the Siegel sets in the almost fundamental domain of the fundamental theorem of reduction theory [Ann. Sci. Éc. Norm. Supér. (4) 31, 181-279 (1998; Zbl 0938.11026)]. The Gaussian weight has different characteristics in terms of coefficient bundles. In the present paper the author proves that if \(M = \Gamma / G \setminus K\) is a locally symmetric space of nonpositive sectional curvature and arbitrary rank, the Gaussian weighted unreduced \(L_2\) cohomology is isomorphic to the usual de Rham cohomology. In this case the isomorphism extends to the standard coefficient bundles.
scientific article; zbMATH DE number 1893534