On Eisenstein series and the cohomology of arithmetic groups (Q984670)

Automorphic cohomology of a reductive \(\mathbb Q\) group \(G\) is defined as the Lie-algebra cohomology of the space of automorphic forms (twisted by a finite dimensional representation). It is known to decompose as a direct sum over all classes of parabolic subgroups \(P\) according to the cuspidal support. The contributions for \(P\neq G\) can be constructed as residues or principal values of derivatives of Eisenstein series, which justifies the name \textit{Eisenstein cohomology} given to these contributions. The main result of the paper, spelled out for \(\mathbb Q\)-split groups, is that a given evaluation point can only contribute non-trivially to the cohomology, if its coordinates are half-integral, i.e., lie in \(\frac12\mathbb Z\). The proof, which is not given, rests on combinatorial descriptions of the parameter sets for the classical groups, which yield explicit formulae for the Weyl group actions.