Steinberg representation of GL2 over a finite field
= Steinberg representation of GL2 over a finite field
{c}
{title2=$\mathrm{St}$}
The action of $GL_2(\mathbb F_q)$ on $\mathbb P^1(\mathbb F_q)$ has permutation representation $\mathbf1\oplus\mathrm{St}$. The Steinberg constituent is irreducible of dimension $q$ and has character values $q,0,1,-1$ on scalar, nontrivial Jordan, split regular semisimple, and nonsplit semisimple elements respectively.