Siegel Modular Varieties and the Eisenstein Cohomology of $\PGL_{2g+1}$

TL;DR

This paper employs a twisted topological trace formula to analyze liftings from symplectic to general linear groups, describing the image in terms of Eisenstein cohomology and providing insights into endoscopic and CAP-representations.

Contribution

It offers a detailed description of the image of the symplectic to general linear lift in terms of Eisenstein cohomology, advancing understanding of automorphic representations.

Findings

Describes the image of the lift from $ ext{Sp}_{2g}$ to $ ext{PGL}_{2g+1}$ in terms of Eisenstein cohomology.

Relates the $L^2$ cohomology of $ ext{Sp}_{2g}$ to intersection cohomology of Shimura varieties.

Provides tools to analyze endoscopic and CAP-representations of symplectic groups.

Abstract

We use the twisted topological trace formula developed in an earlier paper to understand liftings from symplectic to general linear groups. We analyse the lift from $\SP_{2g}$ to $\PGL_{2g+1}$ over the ground field $\Q$ in further detail, and we get a description of the image of this lift for the $L^2$ cohomology of $\SP_{2g}$ (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.