We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Let 
$G$ be a connected, reductive algebraic group over a number field 
$F$ and let 
$E$ be an algebraic representation of 
${G}_{\infty } $. In this paper we describe the Eisenstein cohomology 
${ H}_{\mathrm{Eis} }^{q} (G, E)$ of 
$G$ below a certain degree 
${q}_{ \mathsf{res} } $ in terms of Franke’s filtration of the space of automorphic forms. This entails a description of the map 
${H}^{q} ({\mathfrak{m}}_{G} , K, \Pi \otimes E)\rightarrow { H}_{\mathrm{Eis} }^{q} (G, E)$, 
$q\lt {q}_{ \mathsf{res} } $, for all automorphic representations 
$\Pi $ of 
$G( \mathbb{A} )$ appearing in the residual spectrum. Moreover, we show that below an easily computable degree 
${q}_{ \mathsf{max} } $, the space of Eisenstein cohomology 
${ H}_{\mathrm{Eis} }^{q} (G, E)$ is isomorphic to the cohomology of the space of square-integrable, residual automorphic forms. We discuss some more consequences of our result and apply it, in order to derive a result on the residual Eisenstein cohomology of inner forms of 
${\mathrm{GL} }_{n} $ and the split classical groups of type 
${B}_{n} $, 
${C}_{n} $, 
${D}_{n} $.
