Another point in homological algebra: Duality for discontinuous group actions

Abstract

We consider discontinuous operations of a group $G$ on a contractible $n$-dimensional manifold $X$. Let $E$ be a finite dimensional representation of $G$ over a field $k$ of characteristics 0. Let $\mathcal{E}$ be the sheaf on the quotient space $Y=G\setminus X$ associated to $E$. Let $H_{\text{!}}^{\bullet }(Y;\mathcal{E})$ be the image in $H^{\bullet }(Y;\mathcal{E})$ of the cohomology with compact support. In the cases where both $H_{\text{!}}^{\bullet }(Y;\mathcal{E})$ and $H_{\text{!}}^{\bullet }(Y;{\mathcal{E}}^{\ast })$ (${\mathcal{E}}^{\ast }$ being the the sheaf associated to the representation dual to $E$) are finite dimensional, we establish a non-degenerate duality between $H_{\text{!}}^m(Y;\mathcal{E})$ and $H_{\text{!}}^{n-m}(Y;{\mathcal{E}}^{\ast })$. We also show that this duality is compatible with Hecke operators.