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On the Eisenstein cohomology of arithmetic groups. (English) Zbl 1057.11031

Let \(G\) be a connected semisimple algebraic group over \({\mathbb Q}\) and \(\Gamma\subseteq G({\mathbb R})\) a torsion-free arithmetic subgroup. Denote by \(K\) a maximal compact subgroup of \(G({\mathbb R})\) and by \(X\) the quotient space \(G({\mathbb R}) / K.\) If \((\nu, E)\) is a finite-dimensional representation of \(G({\mathbb R}),\) then the cohomology \(H^\ast (\Gamma, E)\) is that of a local system \({\mathcal E}\) defined on \(\Gamma\backslash X.\) This in turn can be described by certain automorphic data, and one gets the notion of Eisenstein cohomology. This is a part of \(H^\ast(\Gamma\backslash X, {\mathcal E})\) corresponding to cuspidal automorphic forms on Levi components of (proper) parabolic subgroups of \(G\). This decomposition is achieved in [J. Franke and J. Schwermer, Math. Ann. 311, No. 4, 765–790 (1998; Zbl 0924.11042)] as a result of Franke’s proof [Ann. Sci. Éc. Norm. Supér., IV. Sér. 31, No. 2, 181–279 (1998; Zbl 0938.11026)] of a conjecture of Borel.
The authors now aim at refining this decomposition. In particular, they show that if an Eisenstein-class is regular and comes from a cuspidal automorphic representation whose archimedian component is tempered then its cohomological degree is bounded from below by \[ (1/2) [\dim X - \text{rk} (G({\mathbb R})) + \text{rk} (K)]. \] This result gives rise to new vanishing theorems. Moreover, if the \({\mathbb Q}\)-rank of \(G\) is positive and \((\nu,E)\) has regular highest weight, then in degrees larger than \((1/2) [\dim X + \text{ rk} (G({\mathbb R})) - \text{ rk} (K)]\) the cohomology of \(\mathcal E\) is spanned by regular Eisenstein classes.
The results are achieved in a more general setting and the adelic language is being used throughout.

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F75 Cohomology of arithmetic groups
22E40 Discrete subgroups of Lie groups
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