Endoscopy and the cohomology of GL(n)

.Abstract. Let G = ResF/Q(GLn) where F is a number field. Let SG Kf denote an ad`elic locally symmetric space for some level structure Kf . Let Mµ,C be an algebraic irreducible representation of G(R) and we let Mfµ,C denote the associated sheaf on SG Kf . The aim of this paper is to classify the data (F, n, µ) for which cuspidal cohomology of G with µ-coefficients, denoted H• cusp(SG Kf ,Mfµ,C), is nonzero for some Kf . 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 µ is a parallel weight