Rational Elements in Representations of Simple Algebraic Groups, I

Abstract

An element g of a group G is called rational if g is conjugate to $$g^i$$ for every integer i coprime to |g|. We determine all triples $$(G,g,\phi )$$ , 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 $$\phi $$ is an irreducible representation of G such that $$\phi (g)$$ has eigenvalue 1.