  Let $G$ be a finite simple group of Lie type, and let $\pi_G$ be the
permutation representation of $G$ associated with the action of $G$ on itself
by conjugation. We prove that every irreducible representation of $G$ is a
constituent of $\pi_G$, unless $G=PSU_n(q)$ and $n$ is coprime to $2(q+1)$,
where precisely one irreducible representation fails. Let St be the Steinberg
representation of $G$. We prove that a complex irreducible representation of
$G$ is a constituent of the tensor square $St\otimes St$, with the same
exceptions as in the previous statement.
