A decomposition of spaces of automorphic forms, and the Eisenstein cohomology of arithmetic groups
Let \(G\) be a connected reductive group over \(\mathbb{Q}\) and let \((\nu,E)\) be a finite-dimensional algebraic representation of \(G(\mathbb{C})\). It has been known for long that the space of automorphic forms \(A_E\) attached to \(E\) can be decomposed as \(A_E= \bigoplus_{\mathcal P} A_{E,{\mathcal P}}\), where the sum ranges over all associated classes of \(\mathbb{Q}\)-parabolic subgroups. The class \({\mathcal P}= \{G\}\) corresponds to the cusp forms then. In [Ann. Sci. Ec. Norm. Supér. (4) 31, 181-279 (1998)] the first named author showed, verifying a conjecture of A. Borel, that the \(E\)-cohomology of arithmetic groups can be computed from Lie algebra cohomology with coefficients in \(A_E\). Thus the above decomposition also applies to cohomology of arithmetic groups. The aim of the present paper is to refine this decomposition to \(A_E= \bigoplus_{\mathcal P} \bigoplus_\varphi A_{E,{\mathcal P},\varphi}\), where \(\varphi\) ranges over all cuspidal automorphic representations of a Levi component of some \(P\in{\mathcal P}\). Following a general philosophy there are two definitions for \(A_{E,{\mathcal P},\varphi}\), one by use of the constant term, the other in terms of residues of Eisenstein series. The main results of the paper are the equivalence of the two definitions and the decomposition above. Again this decomposition gives a corresponding one for the cohomology of arithmetic groups which then is used for a closer study of Eisenstein cohomology classes and their behavior under twists.
scientific article; zbMATH DE number 1221872