The image of complex conjugation in $l$-adic representations associated to automorphic forms

Abstract

If $F^{+}$ is a totally real field, if $n$ is an odd integer and if $\Pi$ is a regular, algebraic, essentially self-dual, cuspidal automorphic representation of ${GL}_n\left({\mathbb{A}}_{F^{+}}\right)$, then we calculate the image of any complex conjugation under the $l$-adic representations $r_{l,\iota }\left(\Pi \right)$ associated to $\Pi$.