Another point in homological algebra: Duality for discontinuous group actions

TL;DR

This paper establishes a duality theorem for the cohomology with compact support of sheaves associated to group actions on manifolds, extending homological algebra concepts to discontinuous group actions and showing compatibility with Hecke operators.

Contribution

It introduces a duality framework for the cohomology of sheaves arising from discontinuous group actions on manifolds, generalizing classical duality results.

Findings

Proves a non-degenerate duality between certain cohomology groups.

Shows the duality is compatible with Hecke operators.

Extends homological algebra techniques to 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 $H^{\bullet}_{\textbf{!}}(Y;\mathcal{E})$ 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.