Induction from the diagonal subgroup of upper-triangular SL2
= Induction from the diagonal subgroup of upper-triangular SL2
If $B=U\rtimes T$ is the upper-triangular subgroup of $SL_2(\mathbb F_p)$ and $\theta$ is one-dimensional on $T$, then
$$
\operatorname{Ind}_T^B\theta
$$
is the direct sum of three pairwise nonisomorphic irreducibles. Their restrictions to $U$ have character supports given by the trivial character, the square orbit, and the nonsquare orbit, so their dimensions are $1,(p-1)/2,(p-1)/2$.