A cohomological injectivity result for the residual automorphic spectrum of \(\mathrm{GL}_n\) (Q2510057)

Let \(F\) be a number field and let \(G=\mathrm{GL}_n / F\). The space of square-integrable automorphic forms of \(G(\mathbb{A})\) decomposes into the space of cuspidal and residual automorphic forms, where the latter are given by square-integrable residues of Eisenstein series. A residual automorphic representation \(\Pi\) of \(G(\mathbb{A})\) is cohomological if the ring of relative Lie algebra cohomology of \(\Pi\) is nonvanishing with respect to some irreducible, finite-dimensional, algebraic representation of \(G\). Let \(q_{{\min}}\) denote the lowest degree in which \(\Pi\) has non-vanishing cohomology. The main results of the paper under review states that, in degree \(q_{\mathrm{min}}\) the cohomology of (cohomological) representation \(\Pi\) always injects into the cohomology of the locally symmetric space attached to \(G\). This result represents extension of the analogous result of \textit{A. Borel} [Prog. Math. 14, 21--55 (1981; Zbl 0483.57026)] for cuspidal representations to all square-integrable representations in this degree and improves the result of \textit{J. Rohlfs} and \textit{B. Speh} [Clay Math. Proc. 13, 501--523 (2011; Zbl 1326.11025)].