On the image of complex conjugation in certain Galois representations

TL;DR

This paper determines how complex conjugation acts on Galois representations linked to automorphic forms and cohomology classes over totally real fields, providing explicit descriptions of their images.

Contribution

It explicitly computes the image of complex conjugation on Galois representations associated to automorphic forms and cohomology classes for $GL_n$ over totally real fields.

Findings

Explicit description of complex conjugation images in Galois representations.

Applicable to automorphic representations and torsion cohomology classes.

Enhances understanding of Galois actions in automorphic and cohomological contexts.

Abstract

We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $GL_n$ over a totally real field $F$.