Square orbits of additive characters of a finite field

The diagonal subgroup of $S{L}_2({\mathbb{F}}_p)$ conjugates $u(x)$ to $u(a^{2}x)$. It therefore has two orbits, each of size $(p-1)/2$, on the nontrivial characters $\chi (v\cdot )$ of the unipotent subgroup: one indexed by squares and one by nonsquares.

If $B=U⋊T$ is the upper-triangular subgroup of $S{L}_2({\mathbb{F}}_p)$ and $\theta$ is one-dimensional on $T$, thenis 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 $1,(p-1)/2,(p-1)/2$.