Rational elements in representations of simple algebraic groups, I

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 $(G,g,ϕ)$, where $G$ is a simple algebraic group of type $A_n,B_n$ or $C_n$ over an algebraically closed field of characteristic $p\geq 0$, $g\in G$ is a rational odd order semisimple element and $ϕ$ is an irreducible representation of $G$ such that $ϕ(g)$ has eigenvalue 1.

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