Eisenstein cohomology and the construction of mixed motives
type: Item
instance of: scholarly article
title: Eisenstein cohomology and the construction of mixed motives
zbMATH Open document ID: 0795.11024
author: Günter Harder
published in: Lecture Notes in Mathematics
publication date: 1993-12-15T00:00:00Z
review text: Given a connected reductive algebraic group \(G\) defined over \(\mathbb{Q}\), we consider the spaces \(X_{K_ f}:= G(\mathbb{Q}) \mathbb{A}_ G(\mathbb{R})^ 0\setminus G(\mathbb{A})/ K_ \infty K_ f\) where \(K_ \infty\) is a maximal compact subgroup of \(G(\mathbb{R})\), \(A_ G\) the maximal \(\mathbb{Q}\)-split torus in the center of \(G\), and \(K_ f\) an open compact subgroup of \(G(\mathbb{A}_ f)\). Each of the (finitely many) connected components of this space is of the form \(\Gamma\setminus G(\mathbb{R})^ 0/ K_ \infty\) for a suitable arithmetic subgroup \(\Gamma\) of \(G\). Let \((\tau,E)\) be a finite dimensional algebraic representation of \(G(\mathbb{C})\); suppose that \(A_ G\) acts by a central character \(\chi_ E\) on \(E\). One is interested in the cohomology groups \(H^*(G,E)= {\displaystyle \varinjlim_{K_ f}} H^* (X_{K_ f},E)\), defined as the inductive limit over the directed system of cohomology groups \(H^* (X_{K_ f},E)\), \(K_ f\subset G(\mathbb{A}_ f)\). They carry a natural \(G(\mathbb{A}_ f)\)-module structure. One may recover the cohomology of \(X_{K_ f}\) by taking \(K_ f\)- invariants; this is endowed with a Hecke algebra module structure. The cohomology groups \(H^*(G,E)\) have an interpretation in relative Lie algebra cohomology. This leads, by use of a result of Langlands, to a decomposition \[ H^*(G,E)= \bigoplus_{\{P\}\in{\mathcal P}} H^*(G,E)_{\{P\}} \] ranging over the set \({\mathcal P}\) of classes of associate parabolic \(\mathbb{Q}\)-subgroups of \(G\) where \(H_{\{P\}}^*\) corresponds to the space of functions of uniform moderate growth on \(G(\mathbb{Q}) A_ G(\mathbb{R})^ 0 \setminus G(\mathbb{A})\) which are negligible along \(Q\) for every parabolic \(\mathbb{Q}\)- subgroup \(Q\not\in \{P\}\), i.e., the constant term \(f_ Q\) with respect to \(Q\) is orthogonal to the space of cusp forms on the Levi component of \(Q\). The space \(H^*(G,E)_{\{G\}}\) corresponds to the space of cuspidal functions on \(G(\mathbb{Q}) A_ G(\mathb…
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14F99
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 482864
zbMATH Keywords: connected reductive algebraic group
zbMATH Keywords: cohomology groups
zbMATH Keywords: relative Lie algebra cohomology
zbMATH Keywords: Eisenstein series
zbMATH Keywords: Eisenstein cohomology classes
zbMATH Keywords: Euler products
zbMATH Keywords: automorphic \(L\)-functions
zbMATH Keywords: intertwining operators
zbMATH Keywords: mixed motives
zbMATH Keywords: Shimura varieties
zbMATH Keywords: cusp cohomology
zbMATH Keywords: Hecke algebra submodules
zbMATH Keywords: Anderson's mixed motives
zbMATH Keywords: Hodge-de Rham extension class
zbMATH Keywords: Beilinson- Deligne conjectures
zbMATH Keywords: topological trace formula
reviewed by: Joachim Schwermer
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 482864