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 PGL2g+1 over the ground field Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Q}}$$\end{document} in further detail, and we get a description of the image of this lift of the L2 cohomology of SP2g (which is related to the intersection cohomology of the Shimura variety attached to GSp2g) 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.