A counterexample to a conjugacy conjecture of Steinberg

Abstract: Let $G$ be a semisimple algebraic group over an algebraically closed field of characteristic $p \geq 0$. At the 1966 International Congress of Mathematicians in Moscow, Robert Steinberg conjectured that two elements $a, a' \in G$ are conjugate in $G$ if and only if $f(a)$ and $f(a')$ are conjugate in $\operatorname{GL}(V)$ for every rational irreducible representation $f: G \rightarrow \operatorname{GL}(V)$. Steinberg showed that the conjecture holds if $a$ and $a'$ 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 $u, u' \in G$ such that $f(u)$ and $f(u')$ are conjugate in $\operatorname{GL}(V)$ for every rational irreducible representation $f: G \rightarrow \operatorname{GL}(V)$.