Endoscopy and the cohomology of GL(n)

Let G=ResF/Q(GLn) where F is a number fieldý. ýLet SGKf 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 M˜μ,C denote the associated sheaf on SGKf. The aim of this paper is to classify the data (F,n,μ) for which cuspidal cohomology of G with μ-coefficientsý, ýdenoted H∙cusp(SGKfý,ýM˜μ,C)ý, ýis 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 fieldý, ýand also for a general number field but when μ
is a parallel weightý.