Principal series of a finite reductive group

For a rational Borel subgroup $B=TU$, the principal series is ${\mathrm{Ind}}_{B^F}^{G^F}\tilde{\theta }$, with $\tilde{\theta }$ inflated from a linear character of $T^F$. The Mackey restriction formula and rational Bruhat decomposition give its self-inner-product as the size of the Weyl stabilizer of $\theta$. It is irreducible precisely when that stabilizer is trivial.

For $N=q-1$, write a diagonal-torus character as $\theta (\mathrm{diag}(a,b,(ab{)}^{-1}))=\chi (a{)}^r\chi (b{)}^s$, with $\chi$ a generator of the character group of ${\mathbb{F}}_q^{\times }$. Its Weyl stabilizer is trivial exactly when $r,s,r-s$ are all nonzero modulo $N$ and, if $3\mid N$, the unordered pair $\{r,s\}$ is not $\{N/3,2N/3\}$. The first conditions exclude transpositions; the exceptional pair excludes cyclic permutation of three characters modulo a common twist.