Principal series of a finite reductive group
= Principal series of a finite reductive group
For a rational <Borel subgroup> $B=TU$, the principal series is $\operatorname{Ind}_{B^F}^{G^F}\widetilde\theta$, with $\widetilde\theta$ 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.