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Residues of Eisenstein series and the automorphic cohomology of reductive groups

Part of: Lie groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  07 May 2013

Harald Grobner
Harald Grobner*
Affiliation:
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Wien, Austria email [email protected]
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Abstract

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} $.

Keywords

automorphic representationcuspidalresidualEisenstein seriesautomorphic cohomology

MSC classification

Secondary: 11F75: Cohomology of arithmetic groups 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F55: Other groups and their modular and automorphic forms (several variables) 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 7 , July 2013 , pp. 1061 - 1090
Copyright
© The Author(s) 2013 

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