Weighted cohomology
Let \(G\) be a semisimple algebraic group defined over \(\mathbb{Q}\). Let \(K\) be a maximal compact subgroup of the group \(G(\mathbb{R})\) of real points of \(G\), and denote by \(X= G(\mathbb{R})/ K\) the associated symmetric space. Let \((\tau, E)\) be a finite-dimensional (complex) rational representation of \(G (\mathbb{R})\). Given an arithmetic torsion free subgroup \(\Gamma\) of \(G (\mathbb{Q})\) there are various cohomology groups attached to the locally symmetric space \(\Gamma \backslash X\); these are defined in analytical, topological or algebraic terms. In this paper the weighted cohomology groups of \(\Gamma\) are introduced. The weighted cohomology groups \(W^p H^i (\Gamma, E)\) depend on an auxiliary parameter \(p\) called a weight profile. The group \(W^p H^i (\Gamma, E)\) is a cohomology theory on a compactification \(\widetilde {\Gamma \backslash X}\) of \(\Gamma \backslash X\) which is called the reductive Borel-Serre compactification; it is the cohomology of a complex of sheaves \(W^p C^\bullet (E)\) on \(\widetilde {\Gamma \backslash X}\). This complex may be thought of as the sheaf \(E\) on \(\Gamma \backslash X< \widetilde {\Gamma \backslash X}\) extended to \(\widetilde {\Gamma \backslash X}\) with certain ``support conditions'' at \((\widetilde {\Gamma \backslash X}) \backslash (\Gamma \backslash X)\), depending on the weight profile. The compactification \(\widetilde {\Gamma \backslash X}\) is ``almost nonsingular'': it has singularities that are particularly amenable to computations. The collection of weighted cohomology groups of an arithmetic group should be seen as analogous to the collection of intersection cohomology groups of a variety. In fact, if \(\Gamma \backslash X\) is Hermitian, for a weight profile \(\mu\) called the middle weight profile there is a canonical isomorphism, \(W^\mu H^i (\Gamma, E)\cong IH^i (\widehat {\Gamma \backslash X}, E)\) where \(IH^i (\widehat {\Gamma \backslash X}, E)\) is the intersection cohomology of the Baily-Borel compactification \(\widehat {\Gamma \backslash X}\) of \(\Gamma \backslash X\). Thus the weighted cohomology of \(\Gamma\) is a generalization of the intersection cohomology group; it makes sense for all \(\Gamma\), whether or not \(X\) is Hermitian. Weighted cohomology is the main technical ingredient in the topological trace formula. This is a fixed point formula for the Lefschetz number of a Hecke correspondence operating on any weighted cohomology group [see \textit{M. Goresky} and \textit{R. MacPherson}, Lefschetz numbers of Hecke correspondences, in: The zeta functions of Picard modular surfaces, CRM Workshop, Montréal 1988, 465-478 (1992; Zbl 0828.14029)].
scientific article; zbMATH DE number 554016
scientific article