Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 19J / a / ii / Solution

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 $\theta^w(t)=\theta(w^{-1}tw)$.


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 Incoming links: Mackey inner product for the finite SL2 principal series