Siegel modular varieties and the Eisenstein cohomology of PGL2g+1

We use the twisted topological trace formula developed in Weselmann (Compos. Math. 148:65-120, 2012) to understand liftings from symplectic to general linear groups. We analyse the lift from SP2g to PGL(2g+1) over the ground field in further detail, and we get a description of the image of this lift of the L (2) cohomology of SP2g (which is related to the intersection cohomology of the Shimura variety attached to GSp(2g) ) in terms of the Eisenstein cohomology of the general linear group, whose building constituents are cuspidal representations of Levi groups. This description may be used to understand endoscopic and CAP-representations of the symplectic group.