Rational elements in representations of simple algebraic groups, I

Abstract

A finite order element $g$ of a group $G$ is called rational if $g$ is conjugate to $g^i$ for every integer $i$ coprime to the order $g$. We determine all triples $(G,g,\varphi )$, where $G$ is a simple algebraic group of type $A_n,B_n$ or $C_n$ over an algebraically closed field of characteristic $p\ge 0$, $g\in G$ is a rational odd order semisimple element and $\varphi$ is an irreducible representation of $G$ such that $\varphi (g)$ has eigenvalue 1.