Mackey inner product for the finite SL2 principal series

For one-dimensional representations $\theta ,\phi$ of the upper-triangular subgroup $B\le S{L}_2({\mathbb{F}}_p)$,where ${\theta }^w(t)=\theta (w^{-1}tw)$.

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 ${\theta }^2\ne 1$.