Endoscopy and the cohomology of $GL(n)$

Let $G = {rm Res}_{F/mathbb{Q}}(GL_n)$ where $F$ is a number field‎. ‎Let $S^G_{K_f}$ denote an ad`elic locally symmetric space for some level structure $K_f.$ Let ${mathcal M}_{mu,{mathbb C}}$ be an algebraic irreducible representation of $G({mathbb R})$ and we let $widetilde{mathcal{M}}_{mu,{mathbb C}}$ denote the associated sheaf on $S^G_{K_f}.$ The aim of this paper is to classify the data $(F,n,mu)$ for which cuspidal cohomology of $G$ with $mu$-coefficients‎, ‎denoted $H^{bullet}_{rm cusp}(S^G_{K_f}‎, ‎widetilde{mathcal{M}}_{mu,{mathbb C}})$‎, ‎is nonzero for some $K_f.$ We prove nonvanishing of cuspidal cohomology when $F$ is a totally real field or a totally imaginary quadratic extension of a totally real field‎, ‎and also for a general number field but when $mu$ is a parallel weight‎.