= Unnormalized parabolic induction
{title2=$\operatorname{Ind}_B^G\chi$}
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>.
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