Cohomology of \(S\)-arithmetic subgroups in the number field case
Let \(G\) be a semisimple, absolutely almost simple group over a number field \(k\), and let \(S\) be a finite set of places of \(k\) containing all the archimedean ones. Assume that the \(k\)-rank of \(G\) \(rk_ k(G)>0\); so that \(G(k_ v)\) is noncompact for \(v \in S\). Given an \(S\)-arithmetic subgroup \(\Gamma\) of \(G(k)\) the cohomology of \(H^*(\Gamma,E)\) (with respect to a finite dimensional complex rational representation of \(G(\mathbb{R}))\) is defined. Generalizing to the \(S\)-arithmetic situation the regularization theorem of Borel and a decomposition theorem of Langlands for functions of uniform moderate growth on \(\Gamma \backslash G_ s\) one can derive [cf. \textit{A. Borel}, \textit{J. P. Labesse} and \textit{J. Schwermer}, On the cuspidal cohomology of \(S\)-arithmetic groups of reductive groups over numbers fields (to appear)] that the cohomology space \(H^*(\Gamma,E)\) is canonically a direct sum \[ H^* (\Gamma, E) = \oplus H^* (\Gamma,E)_ P \] where \(P\) runs through the set of classes of associate parabolic \(k\)-subgroups of \(G\). In terms of automorphic forms, the summand indexed by \(G\) can be identified to the differential cohomology \(H_ d\) of \(G_ S\) with coefficients in the cuspidal spectrum \(L^ 2_{cusp} (\Gamma \backslash G_ S)^ \infty_{cusp}\) twisted by \(E\). This space is called the cuspidal cohomology \(H^* (\Gamma,E)_{cusp}\) of \(\Gamma\). The cuspidal spectrum is contained in the discrete \(L^ 2\)-spectrum \(L^ 2 (\Gamma \backslash G_ S)_{dis}\), and one has for the differential cohomology of \(G_ S\) with respect to this space \[ H^*_ d \bigl( G_ S; L^ 2 (\Gamma \backslash G)^ \infty_{dis} \otimes E \bigr) = H^* (\Gamma;E)_{cusp} \oplus H^*_ d \bigl( G(\mathbb{R}), E \bigr). \] In the paper under review it is shown for congruence subgroups of simply connected groups that this cohomology space is equal to the full cohomology \(H^* (\Gamma,E)\), i.e. in terms of automorphic forms, Eisenstein series which depend on a parameter give no contribution to the cohomology. The summand \(H_ d (G (\mathbb{R}),E)\) comes from the constant automorphic representation. This main result relies on the yet unpublished results of J. Franke giving a description of the cohomology of an arithmetic group in terms of automorphic forms.
scientific article; zbMATH DE number 554013
scientific article