Mackey inner product for the finite SL2 principal series
= Mackey inner product for the finite SL2 principal series
For one-dimensional representations $\theta,\varphi$ of the upper-triangular subgroup $B\leq SL_2(\mathbb F_p)$,
$$
\left\langle\operatorname{Ind}_B^G\theta,
\operatorname{Ind}_B^G\varphi\right\rangle_G
=\mathbf1_{\theta=\varphi}
+\mathbf1_{\theta^w|_T=\varphi|_T},
$$
where $\theta^w(t)=\theta(w^{-1}tw)$.