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Harmonic analysis in weighted \(L_2\)-spaces. (English) Zbl 0938.11026
This paper contains the proof of the Borel conjecture, 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 \(H^\bullet (\Gamma,E)\cong H^\bullet ({\mathfrak g},K, c^\infty (\Gamma\setminus G)\otimes E)\), 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 \(C^\infty (\Gamma\setminus G)\) 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 \({\mathcal A}(G,\Gamma) \hookrightarrow C^\infty (\Gamma\setminus G)\) induces an isomorphism \(H^\bullet (\Gamma,E)\cong H^\bullet ({\mathfrak g},K,{\mathcal A}(G,\Gamma)\otimes E)\).
The central tool is an acyclicity result as follows: Let \(J\) be an ideal of \({\mathfrak z}\) of finite codimension and let \({\mathfrak {Fin}}_J\) be the functor which associates to a \(({\mathfrak g},K)\)-module its submodule of elements annihilated by a power of \(J\). The author proves that the logarithmically weighted \(L^2\)-space is \({\mathfrak {Fin}}_J\)-acyclic. This assertion is used in an inductive argument to prove the Borel conjecture. It also is proven that any automorphic form is a sum of Laurent coefficients of Eisenstein series. These results are then applied to prove a generalized Manin-Drinfel’d theorem stating that the decomposition of the cohomology of an arithmetic group into contributions of parabolic classes is stable under automorphisms of \(\mathbb{C}\). As a final application explicit expressions for the traces of Hecke operators are given.
See also the Bourbaki Seminar note by J.-L. Waldspurger [Astérisque 241, 139-156 (1997; Zbl 0883.11025)].

MSC:
11F75 Cohomology of arithmetic groups
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
22E41 Continuous cohomology of Lie groups
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