Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type

Let $G$ be a finite simple group of Lie type, and let $π_G$ 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 $π_G$, 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.

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