  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
$H^{\bullet}_{\textbf{!}}(Y;\mathcal{E})$ be the image in
$H^{\bullet}(Y;\mathcal{E})$ of the cohomology with compact support. In the
cases where both $H^{\bullet}_{\textbf{!}}(Y;\mathcal{E})$ and
$H^{\bullet}_{\textbf{!}}(Y;\mathcal{E}^*)$ ($\mathcal{E}^*$ being the the
sheaf associated to the representation dual to $E$) are finite dimensional, we
establish a non-degenerate duality between $H^{m}_{\textbf{!}}(Y;\mathcal{E})$
and $H^{n-m}_{\textbf{!}}(Y;\mathcal{E}^{\ast})$. We also show that this
duality is compatible with Hecke operators.
