A counterexample to a conjugacy conjecture of Steinberg

Let $G$ be a semisimple algebraic group over an algebraically closed field of characteristic $p\ge 0$. At the 1966 International Congress of Mathematicians in Moscow, Robert Steinberg conjectured that two elements $a,a^{\mathrm{\prime }}\in G$ are conjugate in $G$ if and only if $f(a)$ and $f(a^{\mathrm{\prime }})$ are conjugate in $\mathrm{GL}(V)$ for every rational irreducible representation $f: G \rightarrow \operatorname{GL}(V)$. Steinberg showed that the conjecture holds if $a$ and $a^{\mathrm{\prime }}$ are semisimple, and also proved the conjecture when $p=0$. In this paper, we give a counterexample to Steinberg’s conjecture. Specifically, we show that when $p=2$ and $G$ is simple of type $C_5$, there exist two non-conjugate unipotent elements $u,u^{\mathrm{\prime }}\in G$ such that $f(u)$ and $f(u^{\mathrm{\prime }})$ are conjugate in $\mathrm{GL}(V)$ for every rational irreducible representation $f:G\to \mathrm{GL}(V)$.