I was wondering if all complex irreducible representations of $PGL(2,q)$ are defined over $\mathbb{R}$? For example this is not true for $PGL(3,2)$. I believe a name for this type of group is 'totally orthogonal'.
I have searched for quite large $q$ in magma and it seems that the complex irreducible character tables all have Frobenius-Schur index 1, which supports the conjecture.
If this is the case, I would really appreciate a reference, or even just a reference to a good treatment of the complex representation theory of $PGL(2,q)$.
Thanks in advance!