Irreducible principal series of finite SL2
= Irreducible principal series of finite SL2
Every linear character of the upper-triangular subgroup $B\leq SL_2(\mathbb F_q)$ with $q\geq4$ is obtained from a multiplicative character $\theta$ of $\mathbb F_q^\times$. Its induction to $SL_2(\mathbb F_q)$ is irreducible exactly when $\theta^2\ne1$.