 Irreducible principal series of finite SL2

Every linear character of the upper-triangular subgroup $B\le S{L}_2({\mathbb{F}}_q)$ with $q\ge 4$ is obtained from a multiplicative character $\theta$ of ${\mathbb{F}}_q^{\times }$. Its induction to $S{L}_2({\mathbb{F}}_q)$ is irreducible exactly when ${\theta }^2\ne 1$.