Residues of Eisenstein series and the automorphic cohomology of reductive groups (Q2841765)

The cohomology of a congruence arithmetic subgroup \(\Gamma\) of a reductive group \(G\) defined over a number field \(F\) can be expressed (up to a twist by an explicit character) as the relative Lie algebra cohomology of the space of automorphic forms with respect to \(\Gamma\). This is a result of \textit{J. Franke} [Ann. Sci. Éc. Norm. Supér. (4) 31, No. 2, 181--279 (1998; Zbl 0938.11026)], building on the regularization theorem of \textit{A. Borel} [Duke Math. J. 50, 605--623 (1983; Zbl 0528.22010)]. The proof is made in the adèlic setting for the direct limit of cohomology over open compact subgroups of \(G(\mathbb{A}_f)\) induced by the obvious covering maps between the corresponding locally symmetric spaces, where \(\mathbb{A}\) is the ring of adèles of \(F\) and \(\mathbb{A}_f\) denotes the finite adèles. Thus, the relative Lie algebra cohomology of the space of all automorphic forms on \(G(\mathbb{A})\) is obtained in the direct limit (cf. [\textit{A. Borel} and \textit{H. Jacquet}, in: Automorphic forms, representations and \(L\)-functions. Proc. Symp. Pure Math. 33, Part 1, 189--202 (1979; Zbl 0414.22020)] for the definition of an automorphic form). This so-called automorphic cohomology \(H^\ast(G,E)\) of the reductive group \(G\), with respect to the coefficient system given by a finite-dimensional algebraic representation \(E\) of \(G\), is the main object of concern of the paper under review.NEWLINENEWLINEThe key idea of the paper is to use the structural description of the space of automorphic forms to gain information about the automorphic cohomology. The same idea was previously applied by the reviewer and the author [Trans. Am. Math. Soc. 365, No. 10, 5199--5235 (2013; Zbl 1298.11049)] in the case of the split symplectic group of rank two defined over a totally real field.NEWLINENEWLINEThe space \(\mathcal{A}\) of automorphic forms on \(G(\mathbb{A})\) has a direct sum decomposition along the cuspidal support. The summands are indexed by pairs consisting of an associate class \(\{P\}\) of parabolic subgroups of \(G\) and an associate class \(\varphi_P\) of cuspidal automorphic representations of the Levi subgroups of the parabolic subgroups in \(\{P\}\). Let \(\mathcal{A}_{\{P\},\varphi_P}\) denote the summand indexed by \(\{P\}\) and \(\varphi_P\). There is a corresponding decomposition in cohomology, and each summand can be studied separately. This paper is interested in Eisenstein cohomology, that is, the contribution of non-cuspidal automorphic forms. In other words, only the summands with \(\{P\}\neq \{G\}\) are considered.NEWLINENEWLINEFrom the work of Franke [loc. cit.], and its refinement by him-self and \textit{J. Schwermer} [Math. Ann. 311, No. 4, 765--790 (1998; Zbl 0924.11042)], the space \(\mathcal{A}_{\{P\},\varphi_P}\) is spanned by the main values of the derivatives of the Eisenstein series (and their residues) constructed from a representative of the associate class \(\varphi_P\). Moreover, there is a finite descending filtration of \(\mathcal{A}_{\{P\},\varphi_P}\) such that the filtration quotients are parabolically induced representations. The filtration is called the Franke filtration.NEWLINENEWLINEAs the lowest filtration step in the Franke filtration of \(\mathcal{A}_{\{P\},\varphi_P}\) always embeds into \(\mathcal{A}\), the embedding gives rise to a map in cohomology from the relative Lie algebra cohomology of the lowest filtration step to \(H^\ast(G,E)\). The main result of this paper provides an explicit constant \(q_{\mathrm{res}}\), depending on \(\{P\}\) and \(\varphi_P\), such that this map in cohomology is injective in degrees lower than \(q_{\mathrm{res}}\). The importance of the contribution of the lowest filtration step to the automorphic cohomology can be seen from the fact that the residual automorphic representations of \(G(\mathbb{A})\) with the cuspidal support in \(\{P\}\) and \(\varphi_P\), if such exist, always form the lowest filtration step in \(\mathcal{A}_{\{P\},\varphi_P}\). Thus, the main result provides important information about the contribution of residual spectrum of \(G(\mathbb{A})\) to automorphic cohomology.NEWLINENEWLINEThe proof uses the fact that the filtration quotients of the Franke filtration of \(\mathcal{A}_{\{P\},\varphi_P}\) can be described explicitly as parabolically induced representations. The cohomology of these quotients can be determined as explained in [\textit{A. Borel} and \textit{N. Wallach}, Continuous cohomology, discrete subgroups, and representations of reductive groups. 2nd ed. Providence, RI: American Mathematical Society (2000; Zbl 0980.22015)]. Using the long exact sequences and the known cohomology of the quotients, starting from the lowest filtration step one can calculate in several steps the cohomology of \(\mathcal{A}_{\{P\},\varphi_P}\) to some extent. In the widest generality, this approach does not give a complete description of cohomology, but provides important new information, such as the main result of this paper.NEWLINENEWLINESince the constant \(q_{\mathrm{res}}\) is rather tedious to compute, the author also provides a weaker upper bound \(q_{\max}\) which is easier to compute. The paper ends with examples in which new insight to the contribution of residual representations to cohomology is gained from the main result of the paper. These include the case of the general linear groups over a number field and its inner forms, as well as the split symplectic and special orthogonal groups over a number field.