= Irreducibility of the split SL3 principal series
For $N=q-1$, write a diagonal-torus character as $\theta(\operatorname{diag}(a,b,(ab)^{-1}))=\chi(a)^r\chi(b)^s$, 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.
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