Endoscopy and the cohomology of GL(n)

TL;DR

This paper investigates the nonvanishing of cuspidal cohomology for GL(n) using endoscopic transfer, constructing new cohomological representations over various number fields and analyzing their implications for the structure of inner cohomology.

Contribution

It introduces methods to construct cuspidal cohomological representations of GL(n) via endoscopic transfer and generalizes Clozel's construction to broader settings.

Findings

Constructed cuspidal cohomological representations of GL(n) over various fields.

Proved nonvanishing of cuspidal cohomology for GL(2n) with parallel weights.

Provided insights into the endoscopic stratification of inner cohomology.

Abstract

In this article we study the nonvanishing of cuspidal cohomology for GL(n). Using endoscopic transfer from various classical groups we construct cuspidal representations of GL(n) of cohomological type while working over a totally real field or a totally imaginary quadratic extension of a totally real field. Generalizing a construction of Laurent Clozel, we also prove nonvanishing of cuspidal cohomology of GL(2n) over any number field but only for coefficient systems coming from parallel weights. Working at an arithmetic level, we also draw some inferences on an endoscopic stratification of inner cohomology of GL(n).