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Cohomology of spaces of automorphic forms (according to J. Franke). (Cohomologie des espaces de formes automorphes (d’après J. Franke).) (French) Zbl 0883.11025
Séminaire Bourbaki. Volume 1995/96. Exposés 805–819. Paris: Société Mathématique de France, Astérisque. 241, 139-156 (1997).
This is a survey on work of J. Franke, who proved the Borel conjecture on automorphic forms, which states that any class in the group cohomology of an arithmetic group can be represented by an automorphic form. More precisely, let \(\Gamma\) be a congruence subgroup of \(G= {\mathcal G} (\mathbb{R})\), where \({\mathcal G}\) is a reductive group over \(\mathbb{Q}\). For the sake of clarity we restrict here to semisimple \(G\). Let \(E\) be a finite dimensional complex representation of \(G\). The objects of interest are the cohomology groups \(H^\bullet (\Gamma,E)\). Using a de Rham resolution one gets \(H^\bullet (\Gamma,E) \cong H^\bullet ({\mathfrak g}, K,C^\infty (\Gamma \backslash G) \otimes E)\), where \({\mathfrak g}\) is the Lie algebra of \(G\) and \(K\) a maximal compact subgroup. The space of automorphic forms \({\mathcal A} (G, \Gamma)\) is a subspace of \(C^\infty (\Gamma \backslash G)\) whose members satisfy growth conditions and are \({\mathfrak z}\)-finite, where \({\mathfrak z}\) is the center of the universal enveloping algebra of \({\mathfrak g}\). The Borel conjecture, now a theorem of Franke, states that the inclusion \({\mathcal A} (G,\Gamma) \hookrightarrow C^\infty (\Gamma \backslash G)\) induces an isomorphism \(H^\bullet (\Gamma,E) \cong H^\bullet( {\mathfrak g}, K,{\mathcal A} (G, \Gamma) \otimes E)\).
A central ingredient of the proof is the theory of Eisenstein series as developed by Langlands. Franke proves that any automorphic form is a linear combination of derivatives of Eisenstein series of cusp forms. Together with a known decomposition of the cohomology of \(\Gamma\) into parts which are attached to cusp forms on Levi components, this gives way for an inductive argument to prove the Borel conjecture.
For the entire collection see [Zbl 0866.00026].

11F75 Cohomology of arithmetic groups
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