 Unnormalized parabolic induction

For a smooth linear character $\chi$ of a parabolic subgroup $B$, unnormalized induction consists of smooth functions satisfying $f(bg)=\chi (b)f(g)$, with right-translation action. The quotient $B\backslash G$ is compact for the local general linear groups in question. Normalized induction inserts a square root of the parabolic modulus in the covariance, changing formulas for the inducing linear character.